Minorsky  Introduction to NonLinear Mechanics  Part IV  Relaxation Oscillations  1946.pdf
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REPORT 564
INTRODUCTION
TO NONLINEAR MECHANICS
PART TE
RELAXATION
OSCILLATIONS
BY
N. MINORSKY, Ph.D.
SEPTEMBER 1946
HANN
DAVID TAYLOR MODEL BASIN
Captain H.E. Saunders, USN
DIRECTOR
HYDROMECHANICS
Comdr. E.R. Tilburne, USN
E.H. Kennard, Ph.D.
CHIEF PHYSICIST
AEROMECHANICS
Comdr. L.S. Chambers, USN
C .J. Wenzinger
HEAD AERONAUTICAL ENGINEER
STRUCTURAL MECHANICS
Capt. R.A. Hinners, USN
D.F. Windenburg, Ph.D.
CHIEF PHYSICIST
ENGINEERING AND DESIGN
Comdr. L.W. Shallenberg, USNR
G.A. DeShazer
HEAD MECHANICAL ENGINEER
TECHNICAL INFORMATION
M.L. Dager
SENIOR LIBRARIAN
M .C. Roemer
TECHNICAL EDITOR
0
~~3~rpap. .p..l31^ ll~~L1
0 1111
1ldi l1
FOREWORD
The report on the introduction to nonlinear mechanics as a whole falls into four major divisions.
Part I, published as David Taylor Model Basin Report
534
under date of December 1944, is concerned with the topological methods; its presentation substantially follows the "Theory of Oscillations" by Andronow and Chaikin.
The material is slightly rearranged, the text is condensed, and a number of figures in this report were taken from the book. Chapter V, concerning
Li6nard's analysis, was added since it constitutes an important generalization and establishes a connection between the topological and the analytical methods, which otherwise might appear as somewhat unrelated.
Part II, published as David Taylor Model Basin Report 546 under date of September 1945, gives an outline of the three principal analytical methods, those of Poincarg, Van der Pol, and KryloffBogoliuboff.
Part III, published as David Taylor Model Basin Report 558 under date of May 1946, deals with the complicated phenomena of nonlinear resonance with its numerous ramifications such as internal and external subharmonic resonance, entrainment of frequency, parametric excitation, and the like.
Part IV, published here, reviews the interesting developments of
Mandelstam, Chaikin, and Lochakow in the theory of relaxation oscillations for large values of the parameter p.
Part IV also contains a subject index to all four parts of this treatise.
TABLE OF CONTENTS
PART IV

RELAXATION OSCILLATIONS
127. INTRODUCTORY REMARKS .......................
CHAPTER XX

FUNDAMENTALS OF THE DISCONTINUOUS THEORY
OF RELAXATION OSCILLATIONS ...............
128. SOLUTIONS OF A DIFFERENTIAL EQUATION IN THE
NEIGHBORHOOD OF A POINT OF DEGENERATION . ..........
129. IDEALIZATIONS IN PHYSICAL PROBLEMS . .............
130. CRITICAL POINTS OF DIFFERENTIAL EQUATIONS;
BASIC ASSUMPTION
..........................
131. CONDITIONS OF MANDELSTAM . ...........
132. REMARKS CONCERNING SYSTEMS OF DEGENERATE
DIFFERENTIAL EQUATIONS .......................
. .. .....
CHAPTER XXI

DEGENERATE SYSTEMS WITH ONE DEGREE OF FREEDOM
.
133. PERIODIC SOLUTIONS OF DEGENERATE SYSTEMS
OF THE FIRST ORDER ........................
134. RELAXATION OSCILLATIONS IN A CIRCUIT
CONTAINING A GASEOUS CONDUCTOR
. ................
135. RCMULTIVIBRATOR ..........................
136. SYSTEM WITH ONE DEGREE OF FREEDOM DESCRIBABLE BY
TWO DIFFERENTIAL EQUATIONS OF THE FIRST ORDER
. ......
CHAPTER XXII

MULTIPLY DEGENERATE SYSTEMS . ............
137. MULTIVIBRATOR OF ABRAHAMBLOCH
................
138. HEEGNER'S CIRCUIT; ANALYTIC TRAJECTORIES . ...........
139. TRANSITION BETWEEN CONTINUOUS AND DISCONTINUOUS
SOLUTIONS OF DEGENERATE SYSTEMS . ...............
CHAPTER XXIII

MECHANICAL RELAXATION OSCIILATIONS
..........
140. INTRODUCTORY REMARKS .......................
141. QUALITATIVE ASPECTS OF A MECHANICAL
RELAXATION OSCILLATION .......................
142. MECHANICAL RELAXATION OSCILLATIONS
CAUSED BY NONLINEAR FRICTION
.................
.
.
CHAPTER XXIV

OSCILLATIONS MAINTAINED BY PERIODIC IMPULSES. . ...
143. INTRODUCTORY REMARKS .......................
144. ELEMENTARY THEORY OF THE CLOCK . ...............
145. PHASE TRAJECTORIES IN THE PRESENCE OF COULOMB FRICTION
.
146. ELECTRONTUBE OSCILLATOR WITH QUASIDISCONTINUOUS
GRID CONTROL
.......... .................
147. PHASE TRAJECTORIES OF AN IMPULSEEXCITED OSCILLATOR
. . .
.
.
page
1
20
22
25
30
30
34
35
37
37
16
19
19
38
40
43
43
44
47
50
53
8
8
11
12
15
~~III~III~IIU YI1.

~
MIMII
V
CHAPTER XXV 
EFFECT OF PARASITIC PARAMETERS ON STATIONARY
STATES OF DYNAMICAL SYSTEMS
. ............
148. PARASITIC PARAMETERS
.......................
149. INFLUENCE OF PARASITIC PARAMETERS ON THE STATE
OF EQUILIBRIUM OF A DYNAMICAL SYSTEM.
........
150. EFFECT OF PARASITIC PARAMETERS ON STABILITY
OF AN ELECTRIC ARC
..........................
...
151. EFFECT OF PARASITIC PARAMETERS ON STABILITY
OF RELAXATION OSCILLATIONS ....................
REFERENCES
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
INDEX
.... . . . . . . . . . ..
.............. . . .. . . .
.
58
60
63
66
68
page
56
56
0
IIIIII~~ iii
INTRODUCTION TO NONLINEAR MECHANICS
PART IV
RELAXATION OSCILLATIONS*
127.
INTRODUCTORY REMARKS
The term relaxation oscillations, introduced by Van der Pol (I)t (2) and commonly used at present, generally designates selfexcited oscillations exhibiting quasidiscontinuous features. Because of the importance of such oscillations in applications in connection with the socalled "sweep circuits" in electronics, television, and allied fields, an extensive literature exists on this subject, References (3) through (9).
Ph. LeCorbeiller (10) gives an interesting survey of various devices, both mechanical and electrical, by which these phenomena can be demonstrated; some of these devices have been known for centuries.
The characteristic feature of these phenomena is that a certain physical quantity (such as coordinate, velocity, etc., in mechanical problems, and charge, current, etc., in electrical problems) exists on two levels, remaining on each level alternately for a relatively long time but passing from one level to the other so rapidly that in the idealized representation the passage may be considered as instantaneous. A few examples taken from the paper by LeCorbeiller will illustrate these phenomena.
Figure 127.1 shows a device consisting of a container C, of the form shown, fastened to a support R capable of rotating about an axis A, perpendicular to the plane of the paper, and provided with a weight W sufficient to hold the system against the stop S. The container is slowly filled with water, and at the instant when the moment due to the weight of the water becomes greater than the moment due to the weight W, the system tumbles over against another stop S'. The container then empties, and the weight W brings the system back against the stop S, after which the filling period begins again, and so on. In this system the two levels, previously mentioned, are the angles 0 and 0' at which the s
A
s system is constrained by the stops
Figure 127.1
* The text of Part IV follows the presentation in "Theory of Oscillations," by Andronow and Chaikin,
Moscow, (Russian), 1937. A complete bibliography on the subject of relaxation oscillations appears in this volume.
t Numbers in parentheses indicate references on page
66
of this report.
2
S and S' respectively, and the representation of this system in the (O,t) plane appears as a periodic rectangular ripple with the length of its horizontal stretches determined by the rates of filling and of evacuation of C.
Another familiar example is the charging of a capacitor shunted by a gaseous conductor such as a neon lamp. During the charging period the capacitor's voltage gradually rises. At the point of ionization of the gaseous conductor, the neon lamp flashes and the capacitor is suddenly discharged, whereupon the gaseous conduction ceases abruptly and the charging period begins anew. Here, again, there are two levels, the voltage Vimmediately before the discharge strikes and the voltage V
2
immediately after the extinction of the discharge. The transition from V
2
to V
1
is gradual, but the inverse transition from V to V
2
is quasidiscontinuous. A phenomenon of this kind is represented in the (V,t)plane by a socalled "sawtooth" curve.
It was shown in Section 37, Part I, that a similar situation exists for large values of the parameter
p
in the Van der Pol equation. Figure 37.1C
shows quasidiscontinuous changes in the variable x(t) between the two regions in which it changes but little.
When these periodic phenomena are represented in the phase plane of the variable undergoing rapid changes, they appear as closed curves with regions of very large curvature, such as the curves shown in Figures 37.2c and
37.3c. By idealizing these very rapid changes as discontinuous changes, closed trajectories of this kind become piecewise analytic curves "closed" by the discontinuous stretches.
Available analytical methods are inadequate for a rigorous treatment of these phenomena. In fact, all analytical methods presuppose that the parameter p appearing in the basic quasilinear equation
+ x
=
P=f(x,x)
[127.1] is very small. On the contrary, in some of these oscillations, which are expressible by Van der Pol's differential equation, this parameter is large.
More specifically, in Figures 37.1c, 37.2c, and 37.3c, referred to above, the value of the parameter # is 10.
Attempts have been made to extend the analytical methods to oscillations in which p is large. In Section 36 it was shown that Li6nard succeeded in obtaining certain conclusions regarding the qualitative aspect of the phase trajectories when p was very large. N. Levinson (11) extended the proof of the existence of closed trajectories to cover oscillations in which p is not small. In a recent publication (12) J.A. Shohat has indicated a form of series expansion formally satisfying the Van der Pol equation when p is large.
These various attempts, however, did not result in any complete analytical
;IgWL~ar)i~~_~_
111= theory comparable to the one which has been studied in Part II in connection with oscillations in which p is small.
Moreover, as will appear below, not all known relaxation oscillations seem to belong to the group of equations [127.1] of which the Van der
Pol equation is a particular example. More specifically, it will be shown in
Chapter XXI that relaxation oscillations are frequently observed in systems which are amenable to representation by differential equations of the first order which do not admit any analytic periodic solutions for the simple reason that these equations do not possess singularities, without which no closed analytic trajectories can exist; see Section 25. These difficulties led the school of physicists under the leadership of L. Mandelstam and N. Papalexi to evolve a theory, called by its authors the discontinuous theory of relaxation
oscillations, whose exposition and applications will form the principal topic of Part IV.
The use of the concept of mathematical discontinuities for the purpose of describing a rapidly changing dynamical process, at least during certain instants of its evolution in time, is not new. It is recalled that the classical theory of mechanical impacts uses precisely the discontinuous method by assuming an infinitely small duration of the impact process during which the dynamics of the process is entirely ignored, and the "initial" and "terminal" conditions are correlated on the basis of certain additional informa
tion not contained in the differential equations themselves. This permits obtaining the correct overall effect of the impact without knowledge of its details. For an elastic impact, such additional information is supplied by the theorems of momentum and kinetic energy; to this information is added, for nonelastic impacts, the socalled coeffi
cient of restitution, an empirical factor characterizing the loss of energy during the impact. This coefficient depends on the material of which the colliding bodies
1 are composed.
2u
w
One can imagine a discontinuous
(a)
periodic motion generated by impacts from the following example given by Andronow and Chaikin (13). Let us assume that a
1HINE11116 perfectly elastic ball rolls without friction on a horizontal plane and strikes
'
" rection of its motion, as shown in Figure
127.2a. With the usual discontinuous treatment of mechanical impacts, the phase
4.
Figure 127.2
trajectory of such motion is represented by a "closed curve" ABCDA, a rectangle. On the branches AB and CD the motion is continuous with the constant velocities i = ±vo, respectively; on the branches BC and DA, on the contrary, it is discontinuous.
In the systems of Figures 127.1 and 127.2 we encounter periodic phenomena having certain quasidiscontinuous features. The nature of these discontinuities in the two systems is, however, different. For the ball striking the walls there exists a definite external actuation, the reaction of the constraint, the wall, applied to the dynamical system, the ball; this actuation is properly "timed" by the distance 2a between the walls which determines the
"period" of the motion. On the contrary, no external impact excitation exists during the discontinuities in the motion of the container shown in Figure
127.1. These discontinuities are due rather to a sudden loss of equilibrium between the moment M, of the constant weight and that of the container Mc, occurring at a certain critical value, M
C
= M
W
.
The change of equilibrium position from 0 to 0' is not instantaneous, of course, but in comparison with the long periods of filling and evacuation of the container it may be considered as such in the idealized picture of the phenomenon. We can improve the small; this will render the short time interval during which 0 varies still shorter, which, in turn, makes the relative time intervals of filling and of evacuation still longer.
We find it expedient to define as relaxation oscillations those quasidiscontinuous oscillations in which the rapid changes between certain levels of a physical quantity occur as the result of the loss of a certain internal equilibrium in the system, and as impulseexcited oscillations those quasidiscontinuous oscillations in which these rapid changes are due to the action of certain external impulsive causes.
On this basis, the quasidiscontinuous oscillations of the container shown in Figure 127.1 are of a relaxation type, whereas the ball rebounding between the walls is an impulseexcited phenomenon. The essential difference between the two types of oscillations is that in relaxation oscillations the energy content stored in the system remains constant during the quasidiscontinuous changes of certain variables, whereas in impulseexcited systems, on the contrary, the energy content changes abruptly.
Impulseexcited oscillations do not require any particular additional information for their treatment, as will be seen in Chapter XXIV. For relaxation oscillations proper (see Chapters XXI, XXII, and XXIII) it is necessary to specify the conditions under which the discontinuities are bound to occur in a system; it will be shown that the basic assumption given in


Section 130 provides a criterion sufficiently broad to cover all known types
of relaxation oscillations.
Finally, inasmuch as the representation of a rapidly changing process by a mathematical discontinuity is always an idealization, it becomes necessary to analyze the conditions under which this idealization is justified in practice. In analyzing the behavior of the device shown in Figure 127.1, we have noted that the change from the angle 0 to the angle 9' may be considered as quasidiscontinuous. The smaller the moment of inertia of the system, the more accurate is the approximation. In mechanical systems, such as the examples described above, it is obviously difficult to extend the hypothesis by assuming that the moment of inertia is zero, but in electrical systems neglect of one of the oscillatory parameters is a common practice. In both types of systems, instead of a "full" differential equation of the second order, the abbreviated or degenerate equation of the first order is frequently employed. By using degenerate equations, numerous problems can be treated as discontinuous in the phase plane; this appreciably simplifies their solution.
A simple example will show the application of degenerate equations for this purpose. Let us consider an oscillating circuit shown in Figure 127.3
comprising an inductance L, a capacity C, and resistors R and r as shown. The circuit may be closed on a source of dc voltage by a switch S. It is useful to specify certain idealizations which appear somewhat trivial but which will be found to be of considerable importance in what follows.
We assume first that the left branch of the circuit ALRB has no capacity and that the right branch ACrB has no inductance. In other words, we neglect the effect of small, parasitic, distributed capacities in the inductance L and resistance R; likewise, we neglect the effect of a small inductance accompanying the flow of current in the branch ACrB.
The second assumption will be that the opening and clos ing of S is instantaneous, that is, occurs in an infinitely small time interval (t

0,
t + 0).
Let us assume that we open and close the switch S at some fre
s
L
quency. The process will then be represented by a sequence of very long intervals, when S is either closed or opened, separated by infinitely short intervals of closing and opening. During the long intervals there will generally be a
Figure 127.3
B
"111M~
certain oscillatory process in the circuit (L,C,R+ r) describable by a differential equation of the second order whose phase trajectories are spirals converging toward a focal point, as we know from Section 5. The closing or opening of S will disturb this process by introducing a certain transient.
Without any loss of generality we may consider the first closing of S at
t o
= 0 and assume that for t < t
o
the circuit was "dead."
It is noted that, at the instant of closing, the two circuits ALRB and ACrB are in parallel, and the differential equations are
di dV
Ldi
+ Ri = E; rC
dV + V = E
dt dt
[127.2] where E
=
0 for t
5
t o
and E
= constant for t
2
t o
. With these assumptions one finds that
S
E(
di
=
E eL
R
[127.3]
E
"
T
,
A
V = E 1

e
1
rc
dt
dt rC
It is seen that for t
= t o the solutions of Equations [127.2], i(t) and V(t), are continuous but not analytic in the sense that their first derivatives undergo discontinuous jumps
di=
;
Noting that i
=
C
dV weE
can take instead of V the variable i, and state that under the assumed ideal
dt di
izations the functions
and ic undergo discontinuities
E
and
E ,
respective
dt dL
r ly. If one takes the plane of the variables (j,ic
, the process occurring at
t
= t o is represented by a discontinuous jump of the representative point from the origin to the point A whose coordinates are R and
;
see Figure 127.4.
After the initial discontinuity the subsequent motion of the representative point will follow a continuous trajectory AB which, as the transient dies out, will eventually approach a damped oscillatory motion represented by a conver
di
dt gent spiral. If, at a later instant t = t,, the switch is opened, another jump will oc
A"
cur, but this jump will generally not bring the representative point back to the origin but to some other point
A", and so on. It is thus seen that in the phase plane of these
O1
B
,/
E
ic
A
Figure 127.4 the system will be represented by a sequence of disconnected, spiral arcs "joined" by discontinuous stretches such as OA, BA', ..
We thus obtain a piecewise analytic repre
sentation of such a phenomenon.
_ _^ _____ __~
7
It is important to emphasize once more that such a representation of a quasidiscontinuous phenomenon by discontinuities in the phase plane of certain variables is possible only because we have introduced certain idealizations into the problem.
a. We consider the time interval during which rapid changes occur as an infinitely short interval.
b. The effect of the parasitic parameters is neglected, which enables us to deal with the degenerate equations [127.2] of the first order instead of the full equations of the second order.
di
and ie which are capable of undergoing discontinuities under Assumption b.
The necessity for Assumptions a and b is obvious. As for Assumption c, it is clear that if one selected some other variables, for example,
i and V instead of and ic, discontinuous representation in such a phase
dt,
plane would be lost since these variables are continuous.
In the preceding discussion we have tacitly assumed an impulseexcited phenomenon as previously defined. For a pure relaxation phenomenon we must answer an additional question, namely, how to determine the instants
(in the time representation) or the points (in the phaseplane representation) at which the discontinuity occurs in a system of this kind.
It is impossible to go beyond this point without formulating some kind of a priori assumption, as will be explained in Section 130.
CHAPTER XX
FUNDAMENTALS OF THE DISCONTINUOUS THEORY OF RELAXATION OSCILLATIONS
128. SOLUTIONS OF A DIFFERENTIAL EQUATION IN THE
NEIGHBORHOOD OF A POINT OF DEGENERATION
We now propose to investigate the nature of the solutions of a differential equation of the second order with constant coefficients,for example, ai + bi + kx
=
0
when one of the coefficients approaches zero. In an electrical problem, a
=
L,
b = R, and k = 1/C; in a mechanical one, a = m (mass), b is the coefficient
of "velocity damping," and k is the spring constant.
First, if b approaches zero, one readily sees that the oscillatory damped motion approaches the oscillatory undamped motion. We saw that in the phase plane the solutions of [128.1] with b
*
0 but small are spirals approaching a stable focal point; this remains true as b approaches zero. For
b = 0, the origin is a vortex point and the trajectories are closed. It is
thus seen that there is a definite difference between the qualitative aspect of trajectories when b is very small and that when b is equal to zero. From a practical standpoint, however, there is hardly any difference between the two cases; no discontinuities of any kind exist in the solutions.
Of greater practical interest are the cases when either a

0 or
0. It is apparent that when
a = 0 Equation [128.1] becomes an equation of the first order and its solution is given in terms of one constant of integration, namely,
k
x =
x
0 e
bt
[128.2] where
zo
is that constant. By differentiating [128.2] we obtain
x=
k
 t
oe
k
[128.1]
[128.3]
It is seen that the coordinate z and the velocity i are not independent but are related by Equation [128.3]. In other words, in the phase plane the trajectories of Equation [128.2] are reduced to a single line y
=

z,
the rest of the plane is not involved. This fact can be expressed by stating that the phase space of a differential equation of the first order is uni
dimensional, that is, it is a phase line instead of a phase plane.
The limit case, a = 0, never actually occurs in practice since in any electrical system containing resistance and capacity there is always a small residual or "parasitic" inductance. Likewise, mechanical systems without inertia are only idealizations. For these reasons it is preferable to investigate the effect of a small coefficient a in the solution of Equation
111 III,
[128.1] rather than to drop this coefficient in the differential equation itself.
The solution of [128.1] is
x
=
C
1 e rlt
+ C
2 er2
[128.4] where C
1
and C
2
are the constants of integration and r
1
and r
2
are the roots of the characteristic equation
ar
2
+ br + k
=
0
If the initial conditions t
=
0, x = z
o,
and i = io are given, one obtains
x o
= C
1
+ C
2
;
o =
C,
2
from which one obtains the values of
C,
and C
2
:
[128.51
C,
Z
0 r
2
2

X
r
1 and
C2
1
1
[128.6]
2 where
r
2
1,
=
b b2 k
+
4a2 a

b
2a

b 1
2ak
+4(

2
)
[128.7
[128.7]
In this expression only one term is retained in the expansion of the square root, since a is small. This gives
r
1
k
b
and
r
2
b k
+
a
b

b
a[128.8
[128.8]
If the values [128.6] of the constants and [128.8] of the approximate expressions for the roots r
1
and r
2
are substituted in Equation [128.4], the approximate solution xl(t) of [128.1] is in the form x(t)
=
xo e
Atk ak
A
b a
a
+ b
i o
e
kt

b
e
a
[128.9]
It is to be noted that the solution x
1
(t) is an approximate one because the expansion of the square root has been limited to the first two terms; this is justified by the assumed smallness of a.
On the other hand, for a = 0 the solution of Equation [128.1] of the second order becomes the same as that of the equation of the first order given by [128.2].
To emphasize the fact that the solution [128.2] is the same as that of Equation [128.1] when complete degeneration occurs, that is, when
a = 0, we will write it as
k x(t) = xoe
b
[128.2]
Consider now the function
ak
O(a,t) = zxl(a,t) 
x(a,t)
= 
oe
At
a
e
[128.10]
+ xo
e

e t
128.10
This function represents the difference between the approximate solution zx(t) of [128.1] in the neighborhood of the point of degeneration, where a is very small, and the solution T(t) of a completely degenerated equation [128.1] of the first order: The function 0(t) approaches zero uniformly in the interval
0 < t <
o
when a ,0.
The expression for the derivative of this function is
(a,t)
= i
1
(a,t) 
zo
k
+
)e
a

ak
k
b2 oe
b
[128.11
For very small values of t the function
=
zA
+ j o,
and it is impossible to reduce it by reducing the coefficient a. However, for a sufficiently large t, which is supposed to be fixed, one can always find a value of a small enough so that the value of ¢(t) is smaller than a given positive number e.
We can express this by saying that whereas the function 0(a,t), conwhen a

0, the function ¢(a,t) behaves in a like manner only when the values of t are sufficiently large. For t = 0 the convergence of the function i
1
(t) to the function :(t) when a

0 is not uniform. In other words, to a given a,
0.
however small, one can always assign a value of t = t
I
such that = Xok +
Only in a very special case, when Xo + o
= 0, does thi's nonuniformity of convergence disappear, but this case is of no practical interest.
One can also state that the difference between the approximate solution xl(t) of a quasidegenerate system when a is very small and the corresponding solution z(t) of a completely degenerate system when a = 0 approaches zero in the whole interval 0 < t < o when a

0 except in a very small neighborhood around the point t = 0; this neighborhood is smaller as a is smaller and in it the difference i
1
(t)  F(t) of the slopes of the two curves xl(t) and X(t) cannot be reduced. This means that the function xl(t) undergoes a quasidiscontinuous jump in this neighborhood.
When the parameter k

0, the problem is treated in a similar manner. First, for a completely degenerate equation with k = 0, Equation [128.1] becomes
ax + bi = 0 [128.12]
Integrating it, one obtains ai
+ bx = M
[128.13] initial conditions, namely, aio + bx
o
=
M
[128.14]
_11MI11111ahl
= M
The solution of Equation [128.13] is
x

M
b
+ Ce a
where C is a constant of integration. One obtains finally
[128.151
_(t)
=
X o
+
1 
ea
[128.16]
If, however, one proceeds with the solution of Equation [128.1] in the neighborhood of its degeneration, where k is very small, the approximate solution is
x
1
(t) =
x
At a
b
+ bo 1 
ea
[128.17]
By forming the functions
S(k,t) = xl(k,t)
x(k,t)
and
b(k,t) = xl(k,t) (k,t)
[128.18] one ascertains by an argument similar to that given in connection with the functions 0(a,t) and (a,t) that for a sufficiently small k the function
(k,t) approaches zero when k

0 uniformly in the interval 0 < t < whereas
0(k,t) approaches zero when k

0 for all values of t except when t
) cc,
for which value 0(k,t) approaches the value x
0
.
129. IDEALIZATIONS IN PHYSICAL PROBLEMS
In applications, idealizations of quasidegenerate systems as absolutely degenerate systems are frequently made, as was mentioned in Section
127. Thus, for example, the differential equation of the socalled (R,C) circuit is usually written in the form
Rdi +
1i
= 0
dt
C
[129.1] where i is the current in the idealized circuit (L
0). The corresponding quasidegenerate equation, with L very small, is
d'i di
1
Ld
+ R_ + i
= 0
Sdt2
dt
C
[129.2]
With the solution of the absolutely degenerate equation [129.1] designated by
i(t) and that of the quasidegenerate equation [129.2] designated by ij(t),
the functions i(t) and il(t) have the appearance shown in Figure 129.1.
For
Equation [129.1], with L = 0, the current starts from the point A and decays exponentially thereafter. For the quasidegenerate equation [129.2], with L very small, the current starts from zero, increases very rapidly, and follows the curve i,(t) which becomes practically identical with the curve i(t) after a very short time; this time is shorter as L is smaller. According to the
absolutely degenerate equation [129.1], with L = 0, the current i(t) undergoes a true mathematical discontinuity OA at t = 0, whereas the current il(t) of the quasidegenerate equation [129.2], with L
0, has a quasidiscontinuity
Although these facts are well known if considered independently of the previous history of the system, as in the above discussion, they appear in a somewhat different form in a quasidiscontinuous stationary relaxation oscillation. Thus, for example, the oscillation
A
depicted in Figure 127.4 is governed by the full differential equation of the second
St
(t)
Figure 129.1
S of the trajectories; these branches are determined in terms of the two constants of integration corresponding to the initial conditions represented by the points A and
A' of the phase plane. During the jumps BA',
B'A",
... the transition is governed by two differential equations of the first order. Since the solutions of each of these equations are determined by one constant of integration, there appears a relation between the variables and ic of the phase plane which does not
dt
exist on the analytic branches AB, A'B',
...
.
As a result of this, the jumps must occur along a certain direction in the phase plane.
Thus the assumption of certain idealizations specified in connection with Figure 127.3 not only permits a discontinuous treatment of the problem but also indicates the direction of the jumps in the phase plane.
It is impossible, however, to proceed beyond this point if one attempts to apply these idealizations to the problem of discontinuous stationary relaxation oscillations. What is lacking in the system of Figure 127.4 is the mechanism by which the phenomenon of "closing" or "opening" the switch is produced spontaneously by the internal reactions of the circuit itself.
In order to be able to formulate this condition and thus to complete the discontinuous theory it will be necessary to introduce an a priori proposition whose validity is justified only by its agreement with the observed facts. This emphasizes once more the physical nature of this theory as distinguished from the contents of Parts I and II in which the argument was purely analytical.
130. CRITICAL POINTS OF DIFFERENTIAL EQUATIONS; BASIC ASSUMPTION
It was shown that, as a result of certain idealizations, discontinuities appear in the mathematical treatment of physical phenomena which exhibit

4 nl~rmuq a~
 li1
i,
13
rapid changes at certain points of their cycles. The use of discontinuities is convenient in some respects but inevitably introduces certain complications.
These facts are too well known from the theory of mechanical impacts to need any emphasis here.
It is obvious that similar difficulties are to be expected if a discontinuous treatment of relaxation oscillations is adopted. For example, in an electronic "sweep circuit" some variables change so rapidly at certain points in the cycle that it is natural to attempt to idealize these changes by mathematical discontinuities. Obviously, any attempt to explain these changes on the basis of some kind of impact is difficult because the energy delivered by the external source, a battery, remains constant, and one cannot very well correlate the apparent continuity of the energy input into the system with the quasidiscontinuous changes of some of its variables. Very frequently a slight change of a parameter causes a disappearance of the phenomenon,and vice versa.
In some particularly simple circuits in which the effect is known to exist, one succeeds in "explaining" it by a more or less elementary physical argument. In more complicated circuits it is impossible to give an account of what actually happens and, still less, to predict theoretically the existence, or nonexistence, of such effects. There exists no analytical theory of these oscillations which would permit a treatment of these phenomena on a uniform basis as was possible for the quasilinear oscillations with which we were concerned in Parts I, II, and III.
In order to be able to find a solution and to correlate the numerous experimental phenomena on a common basis, it becomes necessary to define terms and to introduce some kind of basic assumption, the value of which is to be justified by its agreement with the observed facts.
Definition:
Critical points are the points at which the differential equation describing
a phenomenon in a certain domain ceases to describe it.
Basic Assumption:
Whenever the representative point following a trajectory of the
differential equations describing a phenomenon reaches a critical point, a discontinuity
occurs in some variable of the system.
Since, by virtue of the basic assumption, the occurrence of discontinuities depends on the existence of citical points, it is necessary to specify certain criteria by which their existence can be ascertained. In what follows we will encounter two principal criteria.
1. Let us consider a system of differential equations of the form
dx
P(x,y) dy Q(x,y)
dt = R(x,y)
;
dt R(x,y)
Obviously these differential equations become meaningless and, hence, cease to
describe a physical phenomenon for the points (xi,y
i
) for which R(xi,yi) =
0.
This equation represents a certain locus of critical points, and by virtue of the basic assumption a discontinuity occurs each time the representative point reaches the curve R(xi,y
i
) = 0. It is important to note that, as far as the
trajectory is concerned, the passage through a critical point does not in any way affect its determinateness since R cancels out in the expression dy
dx

P
It is impossible, however, to determine the motion on the trajectory in the neighborhood of a critical point. In this respect the local properties of a critical point are opposite to those of a singular point where the trajectory is indeterminate but the motion is determinate.
2. The existence of critical points or of a locus of such points can sometimes be revealed from the study of trajectories in a certain domain of the phase plane. A typical example in which this can be done is shown in Figure 130.1.
The trajectories arrive at, or
Ldepart from, a certain threshold L from both
M
T'
I
/R
\is,
N
sides, as shown.
If no singular points, that postions of equilibrium, exist in a nar
row domain surrounding L, one can assert that
the line L is a locus of critical points.
It is apparent that the trajectories situated in the regions M and N belong to two different differential equations. Let us assume that the phenomenon is represented by the
Figure 130.1 motion of the representative point R on a trajectory T of the region N. Since the singular points are absent by our assumption, R will reach a point P on L in a finite time. Having reached this point, the representative point finds itself in a kind of analytical impasse from which there is no normal issue, that is, along the integral curves. In fact R cannot pass onto the trajectory T' passing through P nor can it turn back on T since, in both cases, this would be inconsistent with the differential equations prescribing a definite direction on the trajectories of the two regions M and N. Nor can the representative point remain at the point P which is not a pqsition of equilibrium. The differential equations cease to have any meaning at the point P and therefore cease to represent the physical phenomenon. Hence the point P is a critical point, and the line L is a locus of such points. By our basic assumption the discontinuities necessarily occur once the representative point has reached some point on L.
We are going to use the basic assumption extensively in the investigation of relaxation oscillations in relatively complicated circuits in which it is impossible to predict the nature of the phenomenon on a basis of
L"~
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llli.ill
li
IlM
15
elementary intuitive reasoning. It is useful to illustrate the application of the basic assumption to the simple example given previously, Figures 127.3
and 127.4. It is apparent that at the instant when the switch is closed, or opened, the differential equation of the second order ceases to describe the phenomenon since the right and the left portions of the circuit, instead of being in series, become in parallel. Hence this instant corresponds to a critical point, and a discontinuity is to be expected as we have ascertained by the elementary argument. The same argument applies to the ball striking the walls. In these two examples the oscillations are of the impulseexcited type, and the application of the basic assumption does not yield anything of interest.
We shall see that in connection with relaxation oscillations proper the basic assumption will be a useful tool by which the possibility of relaxation oscillations can be ascertained.
131. CONDITIONS OF MANDELSTAM
At the end of Section 127 certain idealizations and a choice of variables were specified so as to be able to introduce a discontinuous treatment of certain problems. It was shown that the necessary condition for such a treatment is the degeneracy of differential equations from the second to the first order if the variables
L
dt
and Vare selected. In the preceding section
dt
we have formulated a sufficient condition for the occurrence of a discontinuity on the basis of a certain basic assumption.
There still remains one question to be settled, namely, the determination of the discontinuities once we have ascertained by the assumption that the discontinuity has to occur. Using the terminology of the phase plane, we can specify this last part of the problem as follows. Let us assume that the representative point R has reached a critical point A(xz,y,). We may question into which other point B(x2,y
2
) the representative point will jump from the point A. In discussing the solutions [127.3] of Equations [127.2] we have already touched this subject and found that in the very special case considered there the jump is from A(0,0) to
B( E
).
L. Mandelstam formulated the conditions of a jump on the basis of certain plausible assumptions regarding the continuity of energy during the infinitely short time interval of the discontinuity. It is to be noted that these conditions of Mandelstam are useful for relaxation oscillations and not for impulseexcited oscillations for reasons which will appear later. The argument of Mandelstam is based on the continuity of the functions i(t), the current through an inductance L, and V(t), the voltage across the capacitor, as was previously mentioned in connection with the expressions [127.3] representing solutions of the degenerate equations of the first order. Since
i(t) and V(t) are continuous, clearly the electromagnetic energy
2 stored in an inductance and the electrostatic energy stored in a capacitor are also continuous functions of time. One obtains the conditions of Mandelstam by writing
to+0
Ai t to
0
=
0;
to+0
AV
t o
0
= 0 [131.1] where (t
o
 0, to + 0) is the infinitely small time interval during which the discontinuity occurs. The important point to be noted in connection with these conditions is that they are applicable to an infinitely small time interval and to circuits with finite dissipative parameters. The first restriction is trivial and is nothing but the expression of a continuity of functions i(t) and V(t). As to the second, it requires an additional remark. One could formulate the following case in which the conditions of Mandelstam apparently do not hold. Let us assume that a charged capacitor is suddenly shortcircuited so that its energy
C
disappears instantly; this seems to contradict the second condition [131.1] of Mandelstam. The fallacy of this reasoning lies in the fact that the only way in which the energy can disappear suddenly is to be totally converted into heat. But in order that this may occur, a finite dissipative parameter must be present. If, however, such a dissipative parameter exists in the circuit, there exists also a finite time constant so that the disappearance of the charge, and, hence, of the energy, cannot be instantaneous, and it is sufficient to define a small time interval consistent with the time constant of the circuit to ensure the validity of the conditions of
Mandelstam.
The usefulness of the conditions of Mandelstam is limited only to relaxation oscillations proper.
In fact, in impulseexcited oscillations the idealization employed is of an entirely different kind, and it is assumed that the energy exchanges between the system and an external source occur instantaneously. Summing up the result of this and of the preceding sections, it can be stated that the basic assumption and the conditions of Mandelstam are useful in studies of relaxation oscillations but are unnecessary for impulseexcited oscillations.
132. REMARKS CONCERNING SYSTEMS OF DEGENERATE DIFFERENTIAL EQUATIONS
In Section 3 it was mentioned that a differential equation of the second order can always be represented by a system of two differential equations of the first order if a new variable y =
d is introduced. Likewise, a th
dt dt
differential equation of the n order can be reduced to a system of n differential equations of the first order by introducing the variables
= Yl,
dy dt 2
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nillgilNMllMI
17
With degenerate differential equations the situation is somewhat different. Thus a completely degenerate equation of the second order is, in fact, a differential equation of the first order. As a result of this, the phase space, instead of being twodimensional, that is, a phase plane, becomes unidimensional, that is, a phase line. Moreover, instead of analytic trajectories, piecewise analytic trajectories become possible.
A system of two differential equations of the second order can generally be reduced to a system of four differential equations of the first order, which means a system of the fourth order.
If, however, each of the original differential equations of the second order degenerates into one equation of the first order, the system of the fourth order reduces to one equation of the second order, and its solutions can be represented by trajectories in a phase plane.
This resultant equation of the second order, however, represents the result of degeneration of the system of the fourth order. We can express this by saying that we have a doubly degenerate system. Since each of the two differential equations of the first order admits discontinuous solutions, the doubly degenerate system of the second order will also possess certain discontinuous stretches in its phase plane so that its trajectories, in general, will be composed of certain analytic arcs joined by these stretches.
Under certain conditions a doubly degenerate system of the second order may degenerate into a single differential equation of the first order; we can call such a case a triply degenerate system. We shall encounter one such system in what follows. In a triply degenerate system one differential equation of the first order represents the result of the degeneration of the system of the fourth order.
Although the use of degenerate equations extends the application of topological methods to a series of important practical cases which could not otherwise be represented in a phase plane, it should be noted that piecewise analytic trajectories of this kind "closed" by discontinuous stretches do not exhibit the features common to the regular analytic trajectories studied in
Part I. We shall encounter, for example, systems in which no singularities exist inside such "closed" trajectories; in some other systems such trajectories include singularities alternating in the course of time between a focal, or nodal, point and a saddle point. Moreover, certain difficulties arise in the formulation of conditions for stability of such degenerate systems, as will be specified in Chapter XXV. These peculiarities are, of course, to be expected. They reflect to some extent the fact that the discontinuous theory of relaxation oscillations is based on certain idealizations; this fact makes it difficult to compare it directly with classical methods making use of analytic functions. The principal usefulness of this theory at present is that
18
it permits obtaining satisfactory qualitative information in a great majority of important practical problems whose solutions are beyond the reach of existing analytical methods.
CHAPTER XXI
DEGENERATE SYSTEMS WITH ONE DEGREE OF FREEDOM
133. PERIODIC SOLUTIONS OF DEGENERATE SYSTEMS OF THE FIRST ORDER
A differential equation of the first order
dx
dt dt
= f(x)
[133.1]
obviously does not possess continuous analytic periodic solutions. Moreover, one can assert that if the function f(x) is singlevalued, no continuous, although not necessarily analytic, periodic solutions are possible. In fact, in order that some sort of periodicity may exist, it is necessary that the system traverse the same line x = x, with two oppositely directed velocities; this, however, is impossible if f(x) is singlevalued. In the example illustrated by Figure 127.2b, we saw that the "closed" periodic trajectory ABCDA is characterized by the fact that the function f(x) = tv
o
is actually doublevalued. It so happens that, in the example referred to, the function f(x) is a constant. It is easy, however, to waive this restriction by assuming that the plane on which the ball rolls, instead of being horizontal, rises toward the right wall. Then the trajectory will be a trapezoid ABCDA, as shown in
Figure 133.1. One could imagine still other cases by assuming that, instead of rolling on a plane, the ball rolls on some kind of cylindrical surface whose generating lines are parallel to the iaxis. In this case the trajectory would be formed by stretches AB and CD of analytic arcs "closed" by the discontinuous stretches BC and DA.
w
f
The change from one branch of the function f(x) to the other one generally occurs at critical points
Figure 133.1
and is discontinuous. Very frequently this is equivalent to saying that the phenomenon is governed by two distinct differential equations during its cycle. During one fraction of the .cycle
the phenomenon is described by one differential equation and during the other fraction by the other equation. The change from one differential equation to the other occurs at the critical points. In the example illustrated by Figure
133.1 the function f(x) happens to have branches which are symmetrical with respect to the xaxis. This is not always so, as will be shown later. We will now illustrate this matter by the following two wellknown examples.
~
20
134. RELAXATION OSCILLATIONS IN A CIRCUIT CONTAINING A GASEOUS CONDUCTOR
In view of the fact that this subject has been explored, we shall omit the familiar details and will endeavor primarily to show the application of the discontinuous theory in order to prepare the groundwork for more complicated cases beyond the reach of elementary theory.

E

F
T
R
C i_
V
i,
I
BL
N EV
Figure 134.1
0 C D
Figure 134.2
V
Figure 134.1 shows a circuit with the usual notations and with the positive directions as indicated; N is a gaseous conductor, such as a neon tube.
Figure 134.2 represents the characteristic* ABD of the gaseous conductor N, which can be represented by a nonlinear empirical relation il
=
0(V).
The differential equations of the circuit obviously are
R(i
+ i
)
dV
+
V
= E; i
=
C dt
[134.1]
These equations reduce to the following differential equation of the first order
dV
dt
1 [E  V
RC
Re(V)]
[134.2]
This equation is valid only when the discharge exists; during its extinction
il
=
q(V)
=
0, and we have
dV dV

dt
1
RC
(E

V)
[134.3]
Since we know that the phenomenon is characterized by alternate
striking and extinction of the discharge,it is apparent that it is represented
* In order to avoid any misunderstanding, we shall use the term characteristic in the engineering sense, that is, to designate a certain experimental curve connecting the values of certain physical quantities, such as current
i
I
through the nonlinear conductor
N
and voltage
V
across it. We will reserve the term
trajectory
to mean an
integral curve
of a differential equation, as we have done previously.
7Pll"" I ~^I~  rCu
alternately by two distinct differential equations, [134.2] and [134.3].
Since these differential equations are of the first order, only one constant of integration is involved in their solutions. But as there are two dynamic variables i
I
= O(V) and V, it is apparent that there must exist a definite
Equation [128.3]. During the time intervals when the discharge exists, this relation is obviously the characteristic ABD.
Since the phase space is unidimensional for this system, it is apparent that during the intervals when the discharge exists the characteristic appears as the phase line and during the intervals of extinction the phase line is the Vaxis. Aside from these phase lines, the plane is not involved.
As is well known from elementary theory, the points of equilibrium are given by the points of intersection of the characteristic and the straight line i =
E V
On the upper branch AB of the characteristic the equilibrium is stable; on the lower branch BD it is unstable, as is easy to ascertain by an elementary procedure. The case when the straight line i = E  V cuts the
R
upper branch of the characteristic is obviously of no interest from the standpoint of oscillations. We shall confine our attention, therefore, to the case when the resistance R has been adjusted to a value at which the straight line and the characteristic intersect at some point S situated on the lower unstable branch BD.
Assume that we start the investigation of the phenomenon at the instant when the discharge has just appeared; this instant is represented by the point A in Figure 134.2. Since
< 0 on the upper branch in this case, the
dt
representative point will move from A to B on the upper branch. The arrows on the characteristic, which, as was just explained, is also the phase line, indicate the positive directions consistent with the differential equation. It follows, therefore, that having reached the point B, the representative point finds itself in a situation which was specified in connection with Figure
130.1, so that the point B is a critical point of the differential equation
[134.2], and by our basic assumption we can assert that a jump must occur at this point. In order to determine the character of the jump we have to apply the condition of Mandelstam. The only form of stored energy is the electro
CV
2
to +0
AV
to
0
= 0
which means that "during" the jump the voltage V across the capacitor remains constant, that is, the jump occurs parallel to the ilaxis along the stretch
BC. The process of extinction is thus "explained" on the basis of the discontinuous theory.
II
Beginning with the point C, Equation [134.3] describes the process of charging the capacitor with no discharge. During that time interval the representative point moves along the Vaxis with a finite velocity until the point D is reached. Here the discharge strikes again, and the representative point is transferred discontinuously to the point A, after which the cycle is repeated. Since at the point D Equation [134.3] ceases to represent the phenomenon, we conclude that D is also a critical point.
It is thus seen that the familiar phenomenon of relaxation oscillations of a circuit containing a gaseous conductor can be treated consistently on the basis of the discontinuous theory and can be represented by a piecewise analytic cycle ABCDA formed by two analytic branches AB and CD "closed" by the discontinuous stretches BC and DA. We note also certain differences between the discontinuous oscillations just studied and the continuous oscillations treated by the classical theory in Part I. The closed trajectory ABCDAis not of a limitcycle type in that the phenomenon jumps, so to say, directly into its cycle without a gradual asymptotic spiraling around it. This cycle resembles in all respects a similar cycle shown in Figures 127.2b and 133.1
describing the idealized behavior of a ball undergoing reflections from constraining walls.
So far, study of such oscillations does not reveal anything new, and the main purpose of this discussion is to illustrate the application of the discontinuous theory of relaxation oscillations to a familiar phenomenon generally treated by elementary methods. As we proceed further, the usefulness of this theory will become more manifest. In the example treated in the following section, for instance, it would be relatively difficult to apply the semiintuitive physical argument. In still further examples it would be altogether impossible.
135. RCMULTIVIBRATOR
As a second example of a system which can be described by one differential equation of the first order, we shall investigate the socalled RCmultivibrator circuit shown in Figure 135.1. The electron tube V
2 is a non* linear conductor characterized by the equation I
a
= q(e). The tube
V, appears here merely as a linear amplifier amplifying the potential difference ri, the feedback voltage, between the points B and D and applyingthe amplified voltage eg = kri to the grid of V
2
,, k being the amplification factor of V,. The circuit is idealized in the customary manner, that is, the effects of the parasitic inductance, the grid current, and the anode reaction are neglected.
The differential equations of the circuit with the positive directions shown are
11400,
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1 11 1 W MION
I 


III I
Figure 135.1
(R + r)i
+ V = Rq(kri);
i = Cdt
These equations reduce to the equation
[135.1]
krR0'(kri) 
(R + r)

i
[135.2]
The critical point i
=
il
is given by the equation
f(i)
=
[krRO'(kri)

(R + r)] = 0
[135.31
where 0'(krij) designates the slope of the characteristic at the point i = ij.
One can also apply the argument given in connection with Figure
130.1.
For this purpose the graphical procedure shown in Figures
135.2a and b will be useful. Figure 135.2a shows the characteristic Ia = l(kri) of V
2 multiplied by a constant factor R. The tube V
2 is supposed to be biased at a point 0 in the middle of the rectilinear part of its characteristic. From the usual form of the characteristic q(kri) it is apparent that its slope d
=
0' is maximum
di
when i
= 0 and approaches zero monotonically as
jil

oo.
Let 0'(0)
=
S and assume that RS > R + r, without which, obviously, no selfexcitation is possible. The curve V(i) represents the function RO(kri)

(R r)i.
The curve shown in Figure 135.2b is the slope curve of the function R(kri) multiplied by a constant factor kr and referred to the axis M'N'. It is apparent that if this curve is referred to the MNaxis parallel to M'N' and at a distance
(R + r) from the origin 0, the ordinates f(i) of this curve represent the
lefthand term of the expression [135.3]. Hence, the points P and Q are critical points, and by transferring these points on the diagram of Figure 135.2a
one obtains the critical points B and D situated on the V(i)curve and symmetrical with respect to the origin under the assumed idealization of the
c v
R
(kri) iA
characteristic. It is apparent also that the inner interval
(il < i < +il) is unstable since
f(i) > 0 in that interval. On the contrary, the outer interval
(i > +i; i < i) is stable since
f(i) < 0 in that interval, as is
Si i
seen at once from Equation [135.2].
The positive directions on the phase line, the V(i) curve, are shown in Figure 135.2a. The argument specified in Section 130 is therefore applicable. Thus, for example, when the representative
S()
point has reached the point B on the stretch AB, it has to jump dis
M
,
R f
R+r
i
I
I
N
continuously, and the second condition of Mandelstam specifies that the jump must occur parallel to the iaxis, since in such a case the energy stored in the capacitor does not change during the jump. The discontinuous stretch BC so tra
M'
I I
Figure 135.2
N'
versed ends at the point C. Here
(b)
the phenomenon is again governed by the differential equation, and the analytic stretch CD is traversed with a finite velocity. At the point D another jump occurs along the stretch DA, so that a closed piecewise analytic cycle ABCDAresults. The phenomenon therefore follows a pattern substantially the same as that which has previously been investigated in connection with the ball striking the walls and also with the neontube oscillator.
The difference between the neontube circuit and the multivibrator circuit is that for the former the phenomenon is described alternately by two separate differential equations [134.2] and [134.3], whereas for the latter the phenomenon is governed by only one differential equation [135.2]. Because of the symmetry of the curve V(i) and that of the critical points, however, the closed cycle is obtained with the origin as the center of symmetry.
I
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.
.
.
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136. SYSTEM WITH ONE DEGREE OF FREEDOM DESCRIBABLE BY
TWO DIFFERENTIAL EQUATIONS OF THE FIRST ORDER
The systems with one degree of freedom so far considered could be expressed by one differential equation of the first order. Chaikin and
Lochakow (14) have investigated an interesting case when a dynamical system with one degree of freedom is describable by two differential equations of the second order possessing critical points. We shall outline briefly the principal features of the relaxation oscillations appearing in this case.
The fact that the system of differential equations is now of the second order will account for the representation of the phenomenon in the phase plane instead of its unidimensional representation by a phase line, as in previous cases. It will be shown that the discontinuous theory permits establishing the principal features of the phenomenon, although a direct intuitive argument, such as is applicable in the simple systems of Sections 133,
134, and 135, is insufficient here.
The circuit investigated by Chaikin and Lochakow is shown in Figure
136.1. It is observed that the only difference between this circuit and that shown in Figure 135.1 is that here the inductance L replaces the resistance R of the circuit of Figure 135.1.
It will be shown now that the behavior of this circuit exhibits features entirely different from those of the circuit of Figure 135.1.
Using the previous notations, we obtain, by Kirchhoff's laws,
I.= (kri) = I + i;
dl
Lt 
ri
.=

dt =
A
I
e gV vl
LC
Figure 136.1
dl
Introducing the variables x = kri and y
=
 and differentiating these equa
tions, one gets
dx y dt
=
(x)'
dy
dt x
1
y krLC kL (x)
[136.2]
where
O(x)
=
'(x)
kr
The phase trajectories are given by the equation
[136.3]
dy dx x(x) krLCy
+ 1
kL
[136.4]
The point x
=
y
=
0
is clearly a singular point. It is apparent from the form of the characteristic of electron tubes that the function O(x) decreases mon
otonically from a positive value 0(0) = S
to a negative value d(x)
=
k
when Jzi
o.
Hence there exist two roots x
=
_xl
,
for which both and dt become infinite. According to our definition, these roots
z = +x are the
critical points of the system [136.2].
In the
phase plane the values
x =
_xl
points of this system. It is to be noted that at these thresholds the tangent to the phase trajectories
(d
d/=
+x
1
kL
[136.5
is determinate. It is impossible, however, to determine the motion of the representative point since the differential equations become meaningless at these points.
In order to show that these thresholds x
=
±x
1
separate the regions df the phase plane in which the topological structure of trajectories is radically different, we shall simplify the
x)/
problem slightly without, however, in
I
AX
Yo
I
troducing any qualitative changes. Figure 136.2 represents the characteristic
of
the electron tube V
2
shown in
136.1.
We shall exclude the critical
_
I
I
x,
I x
I
I
Figure 136.2
thresholds z
= +zx by drawing parallels
to the yaxis on both
sides of
each threshold so as to obtain narrow strips of width
Az.
The function
i(z)
will then be S(x)

kr
> 0 and L in the
kr
inner and outer intervals, respectively.
This will cause no qualitative changes but will change slightly the form of the
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27
trajectories in the neighborhood of the strips; see Section 9, Part I.
Since the equations then become linear, the standard procedure, see
Section 18, Part I, shows that the origin
O
appears as a saddle point for the trajectories of the inner interval (x i
<
x
< +zx) and as either a focal point (if r < 2 I) or
j
outer interval (x < x,
;
x > +x).
We shall assume the existence of a focal point in the outer interval since this is commonly encountered in applications.
The picture of trajectories in this case is shown in Figure
136.3. A continuum of hyperbolic trajectories corresponding to the saddle point fills the inner interval, and a continuum of spirals corresponding to the focal point appears in the outer interval. By an elementary discussion of the sign of in Equation
[136.4] for different quadrants of both intervals, it can be ascertained that the positive directions on the trajectories are oriented as shown by the arrows. It is apparent that these thresholds x
=
±xl are loci of critical points, and discontinuities occur whenever the representative point following a trajectory reaches one of the lines x = ±x,. Since the differential equations become meaningless at the critical points, the determination of discontinuities is made by the conditions [131.1] of Mandelstam, which, in this case, are
AI to+0
=
to  0
to+0
t o
 0
ydt = 0;
AV to+0
1
to

to' O krC to  O
xdt
= 0 [136.6]
Applying these conditions to Equations [136.1], one obtains
(X
) 1
kr
=
X
2
(z
2
)

kr'
yl
2
X

X2
kL
[136.7] where x
2 and y
2 are the coordinates of the representative point immediately after the discontinuity. Since the function
O(x) is an empirical curve, it is preferable to plot first the function 0(x) = q(x)
 as was done in Figure
135.2a. One obtains for
0(x)
a curve similar to the curve V(i) of Figure
x
2
XI
Cd dt
Sgraphical
x_
x
T
x
135.2a. By expressing the condition 0(x
1
) = 8(x
2
), one can find
SX
2 if X is given. We omit this construction and merely mention that for characteristics of electron tubes commonly encountered the jump occurs from +zx to
2
(with z,
1 >
Ix
1
1) c
/
SD\
z to +x,. When z
2 is known, the second equation [136.7] permits determining y
2
for given x
1 and y
,
.
The representation of such a quasidiscontinuous oscillation in the phase plane of the variables (z,y) is shown in Figure
Figure 136.4
136.4. It is noted that the jump from (x,,y,) to (X
2
,y
2
) is the same for all trajectories. Assume that the representative point, following a trajectory T of the outer interval, has reached a critical point A of coordinates (zx,y
1
). The second equation
[136.7] then permits determining the "direction" of the jump AB in the phase plane. At the point B another spiral trajectory begins which encounters at C another critical point resulting in the jump CD, and so on. It can be shown, although we omit the proof, that after one turn of the radius vector the piecewise analytic spiral ABCDE will approach the origin, which means that the point E is below the point A if the latter is sufficiently distant from the origin O. If, however, one applies the same reasoning to a point A' near the xaxis on the critical line x = +xz, one finds, on the contrary, that E' is above A'. Thus the large spirals shrink and the small ones grow with each turn of the radius vector. Hence there exists one and only one piecewise analytic spiral for which the points A and E coincide so that the trajectory becomes closed. Such a closed trajectory may be termed a piecewise analytic
limit cycle, and it is stable. One notes the difference between such a trajectory and the piecewise analytic cycles described in Sections 133, 134, and
135, which are not of' the limitcycle type. The difference is due to the fact that the present phenomenon is expressible by two differential equations of the first order and is represented in the phase plane, whereas in the sections mentioned above it was determined by one differential equation of the first order and was represented by phase lines without involving any other points of the phase plane.
  " IIII
~
oil
29
Another remarkable feature of this analysis is that the unstable hyper bolic trajectories of the inner interval do not appear at all in this representation because that interval is traversed discontinuously. Experiments made by Chaikin and
Lochakow corroborate these theoretical conclusions. In their experiments a cathode
B
E
ray oscillograph was adapted to record the phase trajectories of the system, as was explained in connection with Figure 24.7,
Part I. The record has the appearance shown
C
D
in Figure 136.5. There are two spiral arcs
BC and DE corresponding to a relatively slow
Figure 136.5
motion of the electronic beam on the analytic trajectories of the outer interval. The quasidiscontinuous jumps EB and CD remain unrecorded because of the much higher speed of the beam in the region of the inner interval. It seems logical to assume that when the phenomenon starts from rest, one hyperbolic trajectory of the inner interval is actually traversed, but once the first critical point has been reached, the phenomenon begins to "skip" the inner interval and continues to do so thereafter.
Once the essential features of this phenomenon have been ascertained by this method, it is easy to give a corresponding physical interpretation.
The fact that the jumps occur obliquely in the phase plane of the variables
(x,y) means that at these instants the voltage L
dl
across the inductance and current the current i
=
dt
in the capacitor undergo quasidiscontinuities, although the current I through the inductance and the voltage V across the capacitor remain continuous. In other words, the functions 1(t) and V(t) are continuous but are not analytic at the jumps in the sense that they have discontinuousfirst derivatives. This is in agreement with the elementary theory of an impulsive excitation of the idealized (L,R) and (R,C)circuits. It is apparent, however, that mere knowledge of these wellknown facts would be entirely insufficient for the purpose of establishing a qualitative picture of a complicated phenomenon of this nature if no general method, such as that offered by the discontinuous theory, were available.
CHAPTER XXII
MULTIPLY DEGENERATE SYSTEMS
137. MULTIVIBRATOR OF ABRAHAMBLOCH*
As an example of a system with two degrees of freedom describable by two differential equations of the first order we shall investigate the behavior of the circuit shown in Figure 137.1.
Neglecting the effect of grid current and anode reaction and using the notations and positive directions shown, we obtain the following equations:
S= Ial + il
12 = 2 2
RI
1
+
i
+ ri
1
= E; RI
2
+
' i2 ri
=
E
[137.1]
la, =
q(eg
1
)
=
(ri2)
1a2
=
2
)
=
k(ri
)
where I' = O(e ) is the nonlinear characteristic of the electron tubes V
1 and
V
2
.
Differentiating the second group of equations [137.1] and making use of the other two groups,we obtain the following system of differential equations:
(R + r)
+ 
1
+ Rr '(ri
2
)
=
0
dil di +
1
RrO'(rij)1Lj + (R + r)
+2
i2
=
0
[137.2]
0
1. 0 ,
11IN
I12
A i,
V
iE
\
I_2
Figure 137.1
*
This device is described in Reference (15).
~~~~IIYIIY11
ill
Il,
31
This system reduces to the following equations:
di dt
R
2 r
(R + r)
2

Rr'(ri
2
)
'(ril) '(ri
2
) 
(R + r)
2
i [137.3] which are of the form
di
2
dt R
2 r
(R + r)

Rr '(ril)
2
'(ri
1
) '(ri
2
) (R + r)
2 dil dt
P(i
1
U(i,i
,i
2
)
2
) di dt
2
Q(i
1
,i
2
)
U(i,i)
[1374]
One notes the symmetry of Equations [137.3] with respect to the variables i and i
2
; this symmetry is due, of course, to the symmetry of the circuit.
The phase trajectories in the (i
1
,i
2
)plane are given by the equation
di
2
di
1
Q(i
1
,i
2
P(i
1, i
2
)
)
1375
From the explicit values of the functions P and
Q,
Equations [137.3], one observes that the origin i] =
i
S
=
0 is a singular point, that is, an equilibrium point of the circuit.
We first inquire whether closed analytic trajectories are possible in the system. Applying the negative criterion of Bendixson, see Section 25, one ascertains that
dP
S+
dQ di m
di
2
2(R + r)
C
= constant
[137.6]
Hence no closed analytic trajectories exist here. The nature of the singular point il
=
i2
=
0 is determined by the equations of the first approximation.
If we let
0'(ri)i,
= o =
0'(ri2)2
= o =
S and if we assume that RrS > R
+ r,
Equations [137.3] become
di
I dt
R + r .
M
1
RrS .
M
; di dt
2
RrS .
M
1
R + r .
M
2[1377 where
M
=
C[R
2 r
2
S
2

(R + r)
2
] > 0
The characteristic equation of this system is
2_

2(R + r)
A +
[(R + r) + RrS][(R + r)

RrS]
= 0 [137.8]
The origin (i, = i
2
=
0) is therefore a saddle point. One ascertains easily that, when RrS < R + r, the origin is a stable nodal point, but this case is of no interest from the standpoint of relaxation oscillations. We will assume the existence of a saddle point. Since, initially, RrS > R +
r and the origin
is unstable, the variables i
I
and i
2 begin to increase. On the other hand, from the form of the characteristic
I a
= (ri), we know, see Section 135, that
I'(ri)
+
0 when i  .
The function
U(i,
)
=
C[R
2 r
2
'(ril)'(ri
2
)

(R
+
r)
2
] [137.9] which is initially positive, decreases monotonically when il and
i2
increase and is negative when i
1
and i
2
are very large and are equal. Hence there are certainly some values of il and i
2
for which U= 0. This means that the system [137.3] has critical points so that by virtue of the basic assumption,
Section 130, discontinuities must occur at these points.
The expression
U(i
1
,i
2
) = 0 defines a certain curve F
1
in the plane of the variables i
I
and
i
2
.
In order to simplify the notation, let us designate by x and y, the values of ii and i
2
, respectively, situated on the curve F shown in Figure 137.2.
The equation of this curve is then
R
2 r
2
&'(rx
1
)0'(ry
1
)

(R + r)
2
=
0 [137.10]
B
r2
2
Figure 137.2
A c d b
From the form of the functions ¢'(rx
l
)
and 0'(ryl), approximating the slope to the characteristic
Ia
=
O(eg)
of the electron tube, one ascertains that the curve F is a symmetrical closed curve with the origin at its center and has the form shown in Figure
137.2. As soon as the representative point following a trajectory has reached a point (xl,y
i
) on the curve
F
1
, a jump must occur, and the point
(x
2
,y
2
) into which the representative point jumps is determined by the conditions of Mandelstam which we now propose to apply. Since the only form of stored energy in this case is purely electrostatic, namely, the energy stored in capacitors, the conditions of Mandelstam require that the potential differences V' and V" across the capacitors remain constant during the discontinuity. This gives the conditions
V' =
E

RO(ri
2
)

(R + r)il;
V"
=
E

RO(ri
1
)

(R + r)i
2
[137.11]
If we designate now the terminal coordinates of the jump by (x
2,
Y
2
), the conditions of Mandelstam obviously are
  YYYIIIYIYYIY III i
I
IliJ lll
U miIii
33
R.(ry
1
)

(R + r)x,
= Rq(ry
2
)

(R + r)x
2
R4(rx
1
) (R + r)y,
= Rg(rx
2
)

(R + r)y,
[137.12
In these notations (xl,y
1
) represents the critical point on the curve F
I
from which the representative point is transferred discontinuously to some other point (x
2
,y
2
) determined by these equations. Since there is generally a onetoone correspondence between the points (x
1
,y) and (x
2
,Y
2
), we conclude that the locus of the points (x
2
,y
2
) is another curve
F
2 which is also a closed curve symmetrical with respect to both variables il and i
2
since the circuit is entirely symmetrical.
The phaseplane representation of the behavior of the AbrahamBloch multivibrator can then be described as follows. Assume that the representative point has reached a critical point a on the curve F. From this point it jumps into a corresponding point A on the curve F
2
.
At the point A begins a stretch of a continuous analytic trajectory Ab. The point b is another critical point where another jump bB occurs. At B a second continuous trajectory begins which ends at another critical point c, from which a new jump cC takes place; at C a continuous stretch Cd begins which ends at d, and so on.
By a more detailed analysis of such piecewise analytic trajectories interrupted by discontinuities, it can be shown that the transient behavior of the AbrahamBloch multivibrator approaches a stationary symmetrical state, which is oriented along the bisector of the ii and i
2
axes and which consists of a continuous stretch a# followed by a jump #Q, then by another continuous trajectory 9e followed by another jump
Ea
which closes the "cycle." In this case, the cycle is unidimensional, that is, it is the phase line 9a.
It is apparent that if, instead of assuming the asymmetry of the initial conditions as we did, only a steadystate condition was aimed at, a simplification could be made in the differential equations by introducing the conditions of symmetry, that is, i
I
= i
2
and
0'(ri,)
= 
0'(ri
2
). With this simplification, one obtains a single differential equation of the first order, namely,
di dt R
2 r
(R + r) + Rr&'(ri) i
2
['(ri)
2

(R + r)
2
C
which reduces to the form
[Rro'(ri) 
(R + r)]di = i dt
C
[137.14]
This is identical with Equation [135.2], the equation of an asymmetrical relaxation oscillation. We conclude, therefore, that the transient behavior of the AbrahamBloch multivibrator is represented by a doubly degenerate system.
As far as the stationary condition described by one differential equation
,
, I
34
[137.14) of the first order is concerned, the multivibrator is represented by a triply degenerate system, to use the terminology of Section 132.
138. HEEGNER'S CIRCUIT; ANALYTIC TRAJECTORIES
As a second example of a doubly degenerate system, we shall investigate the socalled Heegner circuit (16) shown in Figure
138.1, which is the same as the circuit of an asymmetrical RC relaxation oscillation shown in Figure 135.1, the only difference being that there is now an additional capacitor
C
2 shunting the resistor R. We propose to show now that the addition of the capacitor C
2 radically modifies the behavior of the circuit. It would be impossible to see this without the criteria of Section 130.
Vi
R
I
I
cI
D
Figure 138.1
With the positive directions indicated by arrows, the application of Kirchhoff's laws gives
Ia
=
I
+
I
+ 2;
1
C1 dt
( R I
rI
) ;
C
2
R dl
I
[138.1
Putting I
a
= q(krll), as we have previously, and eliminating I, we obtain the following system of the second order:
dl
1
dt
_
1
C
1
1
Id
+
rC
2
I
[138.2]
dI dt
2
_
R Rrkq'(krI
1
RrC
1
)
R + r

Rrkk'(krI
1
RrC
2
)
it is observed that this system of differential equations has no critical points, and hence no discontinuities are to be expected. The only singular point is I
I
=
12
=
0. Noting that 0'(0)
=
S is a maximum and that the function Ib'(krIl)l
+
0 when
1 1
1l

c, the characteristic equation of the system
lilY ii
35
[138.2] is
2
+ [
C
+ R + r

RrkS
C
R
1 X +
rC2
1
RrCIC
2

[138.3]
It is noted that the singularity here is not a saddle point. Hence, it is either a nodal point, if the roots X, and
X
2 are real, or a focal point, if they are conjugate complex. In both cases the singularity is unstable if
R + r  RrkS
C2
R

C1
[18.4]
If the singularity is unstable, selfexcitation from rest is possible. By approximating the experimental function
Ia
1
) by a polynomial and by
applying the bifurcation theory, it is possible to establish the existence of a stable limit cycle. We shall omit this calculation since the fact that the Heegner circuit is capable of producing continuous selfexcited oscillations is well known.
Although the Heegner circuit does not produce quasidiscontinuous relaxation oscillations, it is mentioned here as an example of the difficulties which appear if an analysis is attempted on the basis of a more or less intuitive physical argument. In the meantime, this circuit provides an additional example for testing the validity of the basic assumption.
139. TRANSITION BETWEEN CONTINUOUS AND DISCONTINUOUS
SOLUTIONS OF DEGENERATE SYSTEMS
On the basis of the preceding argument, one might ask whether a gradual modification of a given dynamical system, for example, an electric circuit, might cause a transition from continuous performance to quasidiscontinuous performance, or vice versa. It will now be shown that such a transition is indeed possible and that its mathematical formulation can be reduced to the question of the appearance, or disappearance, of critical points as a result of the variation of a certain parameter in the differential equations. In order to show this, let us consider a slightly modified
Heegner circuit such as that shown in Figure 139.1. The modification in question is that the capacitor C
2
, instead of being connected to the point B, is now connected by an adjustable sliding contact to some point E along the resistor r, as shown. Let r
1
be the resistance between the points B and E and
r
2
be that between E and D, where
r i
+ r
2
= r and
r
1
/r = 8. Proceeding as we did in Section 138, we obtain the following system of equations:
dl _ dt
1
)rC
(1 
1
,)rC
2
2
[139.11
[fr
+ R

rR'kr(I,
+
12)]

[r
+ R 
rR'kr(I1 +
I,
2
)]
12
dt
(1

f)[R
+
8r

frRk'kr(I1
+
fl
2
)]
I1UI
eg
I +
I r
I
D
Figure 139.1
It is apparent that for
# = 1, that is, r, r, the circuit reduces to that shown in Figure 135.1 where a discontinuous performance occurs. For
f
=
0
and r = 0, we have Heegner's circuit, which produces continuous oscillations.
For sqme intermediate value of 8, the cofactor of (1  6) in the denominator of the second equation [139.1] may vanish, which means thatthe otherwise continuous oscillation will undergo a dis
I2
continuous jump parallel to the I2axis as shown by the broken lines in Figure
139.2. This generally occurs when the system [139.1] is characterized by a saddle point, and the transition range is at the point where an unstable focal
Figure 139.2
II point degenerates into a saddle point, see Figure 18.1.
We shall not enter into a further study of these complicated and relatively unexplored phenomena. It is sufficient to emphasize once more that their apparent complexity is related to the appearance, or disappearance, of critical points in the differential equations.
r • L " '7 ..... i
"FN
01
'101111
CHAPTER XXIII
MECHANICAL RELAXATION OSCILLATIONS
140. INTRODUCTORY REMARKS
To date, mechanical relaxation oscillations have been studied less than electrical ones. As was mentioned in the introduction to this report, which appears in Part I, two principal reasons account for this. First, the determination of the parameters of a mechanical system is generally more complicated than the determination of the parameters of an electric circuit.
Secondly, mechanical relaxation oscillations always appear as undesirable parasitic phenomena, and the endeavor of engineers is directed toward their elimination rather than their study. These phenomena generally take the form of
"jarring" motions resulting from dry friction, misalignment of machinery, and similar factors. Whether any useful'applications of these effects can be found is difficult to say. Very likely this state of affairs will continue until the laws of friction, which usually account for the appearance of such effects, are better understood and can be controlled with a prescribed degree of reliability. For the time being, the whole subject of mechanical relaxation oscillations is purely academic.
In addition to mechanical relaxation oscillations proper, of which we will indicate examples in Sections 141 and 142, there exists a class of oscillations maintained by periodically timed impacts. These oscillations also exhibit quasidiscontinuous behavior describable by piecewise analytic trajectories.
These impactmaintained oscillations can hardly be considered to be of a relaxation type, although very frequently, and perhaps improperly, they are classified as such. Typical examples of these systems are clocks and quenched spark oscillators, in which the energy of a linear dissipative
'dynamical system is increased periodically in a quasidiscontinuous manner by special timing. We shall leave the investigation of impulseexcited systems to a later chapter and will investigate here a simple mechanical system of a pure relaxation type.
Let us consider a nonlinear differential equation of the form
mi
+
kx
= 
F(i)
=

pf(x)
[140.1] where F(i) is a certain nonlinear function of velocity which we shall identify with friction. If p is small, Equation [140.1] is of a quasilinear type and can be discussed by the standard methods of Part II.
More specifically, for particular forms of the function f(x) selfexcitation may occur, and the oscillation may approach a stable limit cycle, as was studied in Part I in connection with Froude's pendulum; see Section 8.
When p is large, Equation [140.1] cannot be solved by existing analytical methods, and at present the only guide to its solution is the qualitative analysis of Lienard;* see Section 36. We know from his analysis that the periodic trajectory of the differential equation when p is large generally consists of two pairs of branches. On one pair of branches the motion of the representative point is slow and the displacements are large, so the system remains for a relatively large fraction of its relaxation period in that region. On the other pair of branches, however, the motion is very rapid. This phase of the motion is of very short duration so that, in spite of the large values of velocity and acceleration prevailing during these short fractions of the cycle, the coordinate has no time to change appreciably. Li6nard's analysis fails, however, to determine the "corners" connecting these branches in a closed analytic curve; thus a rigorous analytic solution is still lacking.
The mechanical relaxation oscillations caused by friction exhibit the familiar picture previously studied. In fact, there are time intervals when the slow motion becomes a state of rest, followed by an interval of exceedingly rapid motion, followed by another period of rest, and so on. We are thus confronted with a special type of relaxation oscillation appearing as a kind of "jarring" motion.
141.
QUALITATIVE ASPECTS OF A MECHANICAL RELAXATION
OSCILLATION
Since a condition of degeneration, as previously shown, is essential for the appearance of quasidiscontinuous solutions of a differential equation of the second order and since we are trying to approximate the relaxation process by a motion in conformity with the Lienard analysis, we must consider a mechanical system with a very small mass.
Under this assumption, the motion of the system is determined mainly by the balance between the restoring force kx and the friction force pf(f).
When this balance is momentarily lost at a certain instant of the cycle, the acceleration and the velocity of the system may suddenly reach very large values since, by our assumption, the mass is very small.
When m= 0, Equation [140.1] degenerates into the equation
F()
= kx
[141.1] which describes the pair of branches of the Lienard cycle on which.the system remains a relatively long time and the coordinate changes appreciably. The balance between the restoring force and the friction force takes place on these branches. Differentiating this equation, one has
*
Recent research (12) of J.A. Shohat gave hope of bridging this gap, but these efforts have been interrupted by his untimely death.
IIIk,.I.IIII
F'(U)i
= 
ki
[141.2]
If the function F( ) is such that F'(i) < 0 in a certain region, Equation
[141.2] shows that this region is unstable. If at a certain point in this region F'(i) == 0, the acceleration Y may acquire a very large value so that a new balance of forces will appear in which the term mi will play a role in spite of the assumed smallness of the mass m. We thus reach the conclusion that this condition characterizes the second pair of branches of the Lienard cycle which are traversed in a very short time. During this phase of the cycle the acceleration is very large and the velocity varies in a quasidiscontinuous manner by a finite quantity, but the coordinate does not change appreciably because of the short duration of this phase.
The qualitative aspect of the motion in this case is similar in all respects to that which occurs when a ball strikes a wall. However, the underlying dynamical facts are different. For a mechanical impact, the quasidiscontinuities in velocity and acceleration described above are due to an external force, the reaction of constraint when the ball strikes the wall.
In Equations [141.1] and [141.2] these discontinuities appear as a result of the disappearance of the balance of forces expressed by Equation [141.1] and are due to the assumed peculiarities of the function F(i).
The preceding argument can be condensed somewhat by putting i = y in Equation [141.2], which gives
dy
_
dt ky
F'(y)
[141.33
It is apparent that this equation has a critical point whenF(y) = 0 and, hence, ceases to describe the phenomenon at that point. In order to determine the discontinuity by which we idealize the quasidiscontinuity of the physical problem, we must again apply the condition of Mandelstam. In the idealized case, m = 0; hence, the kinetic energy is zero so that the total energy is the potential energy, which is a function of the coordinate x. From the condition of Mandelstam we infer, therefore, that the function x(t) is continuous although it has a discontinuous second derivative, as is to be expected. The situation is thus similar to that which we have studied in
Chapter XXI in connection with electrical relaxation oscillations. It also resembles the results obtained in the theory of mechanical impacts, with the difference, however, that the quasidiscontinuity here arises from the existence of a critical point in the differential equation and is not due to external impulsive excitation as it is for a mechanical impact where the energy also changes abruptly.
142. MECHANICAL RELAXATION OSCILLATIONS CAUSED BY NONLINEAR FRICTION
Since Equation [141.2] has a critical point for F'(i)
=
0, we conclude that relaxation oscillations are possible in a system of this kind whenever the friction force considered as a function of the velocity i possesses extremum values. Moreover, the general condition specified in Section 133, jhat is, that F(i) should be a twovalued function of x, must also be fulfilled.
In practice, these theoretical conditions are frequently encountered. Thus, when a shaft rotates in a bearing, Sommerfeld (17) has indicated that the friction force goes through a minimum for a speed vl given by the equation
1
6
2
P v
15.1
Xr
[142.1] where P is the pressure,
X is the coefficient of viscosity of the lubricant,
r is the radius of the shaft, and
6 is the thickness of the oil film.
Chaikin and Kaidanowski (18) have investigated mechanical relaxation oscillations by the device described in the following paragraphs.
A relatively small mass a forming a
Prony brake engaged frictionally a rotating s shaft K, as shown in Figure 142.1. The mass
a was centralized in a definite position by a rather strong spring S. A definite friction
K
Sthe
Figure 142.1 force was secured by means of another spring not shown, pressing a against the shaft K.
The differential equation of motion of a, neglecting its mass (which results in degeneration of the equation to the first order), is
r(F[(

)r])
=
co
[142.2] where F(v)
=
F[(.

)r] is the friction force, a nonlinear function of the relative peripheral velocity v
= (Q 
)r,
Q is the angular velocity of the shaft K,
4
is the angular velocity of a for small departures 4, and
c
is a constant depending on the spring's strength.
The approximate form of the function F(v) is shown in Figure 142.2. It is noted that the tangent to this characteristic is horizontal at the point
B(v
=
0). When v
=
0, a moves together with K, and 4 = 9. When a is not moving,
# = 0 and v = v
e
=
Dr. To the right of the point
v
=
v o
the mass a and
11,1
the shaft K move in the same direc tion; to the left of v = v o they move in opposite directions. Because of the constraining spring S no con
F
D
tinuous motion of a is possible; there may, however, be a position of equilibrium for which
'
= 0. As long as v = 0, that is,
=
9, the spring
S is gradually stretched, and the friction force F is a static force
B
Figure 142.2
which may have any value whatever provided F : F, where F, is the limit of static friction. When a is moved by the amount o$, for which c€l
=
F
1
r, the limit of static friction is reached. During this phase (v = 0) of the phenomenon, the representative point moves along the axis of F until it reaches the point D, the limit of static friction.
Differentiating Equation [142.2], we have
C
r
2
F'[(9

)r]
= c€
[142.3] where F' designates.dF
It is seen that F' < 0 when and have the same signs, and F' > 0 when they have opposite signs. Moreover, for v = v
o
the quantity
' goes through zero and changes its sign. If the speed
9 of the shaft is relatively small, the point M will be on the part BD of the dynamical characteristic of friction which may be considered here as the phase line of the process, see Section 134. On the part BD of the phase line, F' < 0, so that and € have the same sign; this part of the phase line is therefore unstable, as shown by the arrows. It follows, therefore, that when the representative point has reached the point D during the "static friction period" CD of the process, it cannot pass onto the dynamic characteristic DBA. We conclude that the point D is a critical point.
which gives
One can illustrate this also by putting # =
y in Equation [142.3],
dy dt cy r
2
F'[(Q

y)r]
From this expression it follows that
=
= c at the point D
dt
since F'[(£ )r] = 0 at this point. At D, the representative point undergoes a discontinuity, the direction of which is parallel to the vaxis because here the discontinuity occurs at a constant potential energy, as required by the condition of Mandelstam. The jump terminates at the point A where the characteristic is met, and the subsequent motion of the representative point
is again continuous along the branch BC until another critical point is encountered at the point B where F' = 0. Here another jump DA occurs, after which the phase CD of static friction begins anew; during that phase of the motion the representative point again moves continuously along the branch CD of the cycle DABCD.
One notes an analogy between this example of a mechanical relaxation oscillation and the example of the relaxation oscillation of a neon tube, Section 134. In both examples, relaxation oscillations are possible if the phenomenon is confined to an unstable region of the characteristic. In the neontube oscillator this is accomplished by adjusting the resistance so that the straight line E
V cuts the nonlinear characteristic, the phase line of the process, on its unstable branch. With the brake described in this section, a similar effect is obtained by running the shaft K at a relatively slow speed
9 so as to have the point M in Figure 142.2 in the region where F' < 0. There are no relaxation oscillations in a neontube circuit if the line
V inter
R
sects the characteristic on its upper stable branch, see Figure 134.2; likewise, no mechanical relaxation oscillations are observed, see Figure 142.2, if the angular velocity 9 of the shaft is large enough so that the point M' corresponding to v
0
' = Qr is on the stable branch CD of the friction characteristic. The piecewise analytic cycles ABCDA in both Figures 134.2 and 142.2
are indicated by the corresponding letters. Thus, for example, the branch CD corresponds to the period when the capacitor is charged (Figure 134.2) and to the period when the static friction force F increases with no slipping between a and K (Figure 142.1). Branch AB in both examples corresponds to continuous trajectories of a nonlinear differential equation. In both figures the discontinuous stretches DA and BC are determined by the condition of
Mandelstam, and so on. Plotted in
A
the (0,t)plane, the curves of the mechanical relaxation oscillations observed by Chaikin and Kaidanowski have a typical "sawtooth" appearance, see Figure 142.3, characterizing electrical relaxation oscillations of a similar nature.
o
Figure 142.3 t
This discussion emphasizes the features which all relaxation oscillations have in common.
_~~_11
CHAPTER XXIV
OSCILLATIONS MAINTAINED BY PERIODIC IMPULSES
143. INTRODUCTORY REMARKS
We have defined "relaxation oscillations" as stationary selfexcited oscillations exhibiting quasidiscontinuities at some points of their cycle.
One of the fundamental properties.of these oscillations is that their stationary state does not depend on the initial conditions of the system. In that respect they resemble the continuous oscillations of the limitcycle type studied in Chapter IV with the difference, however, that because of the existence of discontinuous stretches their properties are different from those of continuous oscillations.
As a somewhat different type of oscillation appear the socalled
impulseexcited oscillations.
In some respects these oscillations resemble relaxation oscillations as defined above; in some other respects they differ from them.
The feature common to both types of oscillations is that they can be represented in a phase plane by piecewise analytic trajectories. For both types, also, the stationary state does not depend on initial conditions; moreover, the stationary trajectories of both are "closed" by discontinuous stretches.
The essential difference between these two types of oscillations lies in the physical process during the quasidiscontinuous rapid changes, idealized as mathematical discontinuities. In the pure relaxation oscillations with which we have been concerned in preceding chapters of Part IV, the energy stored in the system during the quasidiscontinuous interval does not change appreciably. This results in an idealized picture of these phenomena with an assumption that the energy does not change in the interval (t
o
 0,
t o
+ 0). Moreover, the existence of these discontinuities, as we saw, was formally reduced to that of the critical points in the differential equations describing the system.
For the impulseexcited oscillations which we are going to investigate in this chapter, a difference exists, in that the discontinuous stretches in the representation of the phenomenon by phase trajectories correspond precisely to the impulsive changes of the energy of the system.
In the investigation of relaxation oscillations proper, we had to rely mostly on the theory of electric circuits. We shall begin the study of impulseexcited oscillations by investigating the behavior of a mechanical device known for centuries, the clock.
144. ELEMENTARY THEORY OF THE CLOCK
A clock is a mechanical device consisting of three principal elements:
1.
An oscillatory dissipative system, for example, an ordinary pendulum, a torsional pendulum with a hair spring, etc.
2.
A source of energy, for example, a weight, a main spring, etc., which has to be replenished periodically ("winding" the clock).
3. An escapement connecting periodically the first and second elements.
The purpose of the escapement is to release periodically the energy stored in the second element in the form of an impulse and to apply this impulse to the first element at an appropriate instant of the oscillation.
Since the instant at which the impulse is released by the escapement is uniquely determined by the motion of the system, the system is auton
omous in the sense that all three elements, 1, 2, and 3, of the mechanism are connected so as to form a single unit, just as a thermionic generator forms a single autonomous unit comprising an oscillating circuit, a battery, and an electron tube with its grid control circuit. We shall see later that it is possible to arrange the operation of an electrontube circuit so that it resembles the performance of a clock, but in general the mode of operation of a clock differs in other respects from that of the thermionic oscillators commonly used.
The escapement communicates to the oscillatory system periodically timed impulses which we shall first idealize as instantaneous and of a constant value a. Let v, be the velocity of the system immediately after the first impulse, from which instant we wish to study the motion. If we assume that the system is a linear dissipative one with a decrement d per cycle, the velocity v
2
' immediately before the second impulse will be v
2
'
=
vle

d. Immediately after the second impulse, the velocity will be v
2
=
v
2
'
+
a
=
vled+ a,
where a is the impulsive increment of the velocity, and so on for subsequent impulses. A stationary state will be reached when, beginning with a certain number n of impulses, v,
v,~,
2
"..
v
0
,
which gives o
1 
a
e
 d
[144.1]
The representation of the process in the phase plane is shown in
Figure 144.1, where one spiral trajectory S of the system is indicated. As was shown in Section 5, these spirals form a continuous family so that through every ordinary point of the phase plane passes one and only one such spiral.
Let us designate as u the segment AB of the xaxis. This intercept represents i~J~ii( (i~ri
the change of the initial radius vector
OA after one revolution 27.
From the properties of the logarithmic spiral it is apparent that the intercept u, con sidered as a function of r
o
=
OA, is a monotonic function u(ro).
Since the spirals S form a continuous family, it is also obvious that the function u(r
o)
is a continuous monotonically increasing function of the radius vector r. On the other hand, by our hypothesis, the impulsive changes of the velocity v due to the operation of the escapement are
uA c
B
izv constant and hence are represented by constant jumps a along the xaxis.
It
Figure 144.1
follows, therefore, that a steadystate condition will be attained when a radius vector ro
= OA is reached for which u(r
o
) = a. This stationary condition is represented by a piecewise analytic cycle BCAB shown in Figure 144.1 by a heavy line; this cycle consists of a convergent spiral BCA (since, by our assumption, the system is dissipative) "closed" by a discontinuous jump AB representing the impulsive increase of the velocity v caused by the impact.
It is also clear that such a stationary piecewise analytic cycle BCAB is of
a stable limitcycle type in the sense that, if one starts from the relatively small values of the initial radius vector ro = OA, the value of a is initially greater than the corresponding value of the intercept u(ro) in this region of the phase plane.
Physically this means that the energy communicated by the impulse is greater than the amount of energy which the system is able to dissipate during one cycle
2
n, which will result in the fact that the subsequent radius vectors ro = OA will grow initially. If, on the contrary, the initial velocity of the system is sufficiently great, the dissipation of energy per cycle is greater than the impulsive increments communicated by the escapement so that the spirals will gradually shrink. The stable condition is reached when the impulsive increments a are just equal to the intercepts u(r
o
) in a particular region. In view of the continuity and the monotonic character of the function u(ro), it is clear that there exists one and only one piecewise analytic limit cycle BCAB of the clock and it is stable. This elementary discussion accounts for the fact that the clock's performance is of a limit
cycle type and as such does not depend on initial conditions, see Chapter IV; in other words, the ultimate performance of a clock does not depend on how it has been started. Once it is started the operation of the clock depends
entirely on the parameters of the system and has nothing to do with the initial conditions. The converse is true for a mathematical pendulum.
In one respect the elementary theory discussed above fails to account completely for the observed facts. For, if a clock is wound, this simplified theory indicates that the clock should start by itself even if the initial disturbance is infinitely small. In other words, it predicts a 8oft selfexcitation of the clock. In reality, unless a clock is given a certain minimum disturbance, for example, shaking, it will not start. This threshold is partially due to .the existence of Coulomb friction not taken into consideration here. There is, however, another reason why the elementary theory is not complete, namely, our assumption that the jump a in velocity caused by the impact remains the same whatever the velocity of the system. Andronow and
Chaikin (13) have shown that by a slight refinement of the preceding theory it is possible to explain better the observed facts.
Let us assume that the change of kinetic energy during the impact remains constant, that is,
my
2
2
1
2
mo
= constant
2
In the phaseplane representation this is equivalent to the condition
yz = h2
[144.2] where h is a constant determined by the properties of the escapement and yo and y
1 are the values of the radius vector immediately before and immediately after the impact. It follows, therefore, that the jump a in the phase plane, instead of being constant as was originally assumed, is now given by the equation a=
Y

[144.3]
that is, it decreases with increasing velocity yo according to a hyperbolic law, as shown in Figure 144.2.
This refinement of the theory, although it leads to conclusions concerning the existence of a stable piecewise analytic limit cycle which are similar to those obtained by the original simplified theory, is not yet sufficient
Figure 144.2
to explain the fact that a clock is a system with a hard selfexcitation.
In order to extend the theory still further it is necessary to investigate how Coulomb friction manifests itself in the representation of motion by phase trajectories.
0

111111
145. PHASE TRAJECTORIES IN THE PRESENCE OF COULOMB FRICTION
The customary idealization of Coulomb, or dry, friction consists in assuming that the friction force fo remains constant during the motion and changes its sign with a change of the direction of motion, as shown in Figure
145.1.
The motion of a system with one degree of freedom in the presence of
Coulomb friction cannot be described by one differential equation but requires two equations, one for i < 0 and the other for i > 0, namely,
mx+
kx = + fo;
m
+ kx
=

fo
Putting
=
2
and
Ifol
=
aw
0
2
, these two equations are
+ Wx
S+
w o
2
=
+ aw o
2
=
aw o
when i <
0
when i >
0
[145.1]
[145.2]
Introducing the variables x, = x

a for the first equation [145.2] and
x
2
=
x + a for the second, one obtains two equations, namely,
Y,
+
wzX, = 0 when i <
0
[145.3]
2
+
wo2 = 0
when
' >
0
These equations are obviously identical, with the difference, however, that the center of oscillation is displaced from +a to a, and vice versa, each time i changes its sign. It is to be noted that i itself does not enter into the equations of motion. The "change of equations" occurs at the instant when i
= 0.
In the (x,t)plane such a motion can be represented by the curve shown in Figure 145.2. Assume that initially the system has been deviated to the right (zxe
>
0) and then released with an initial velocity zo,
=
0. The system will then move to the left (i < 0).
As was just mentioned, this is a sinusoidal motion with respect to the axis t
1 displaced a distance +a from
Figure 145.1
Figure 145.2
~ the taxis. At the point B the velocity i is equal to zero, and the amplitude is Ix
0
2
=
IX
0 1

2al with respect to the taxis. At the point B the change of equations takes place, and the "acting abscissa axis" is now t
2
displaced a distance a from the taxis. With respect to the t
2
axis, the motion is again sinusoidal, so that the point C (at which x = 0) is at the same distance from the t
2
axis as the point B was originally. With respect to the taxis, however, the amplitude of the point C is x o =
x02
 2a. It is seen that with respect to the taxis the amplitudes decrease in an arithmetic progression with the constant difference 2a. The motion stops after a certain number of swings.
The time interval between two consecutive maximums (A and C in Figure 145.2) is the same as for a harmonic oscillator, as follows from the preceding discussion.
It is to be noted again that the curve ABC** is a piecewise analytic curve which loses its analyticity at the points A, B, C, ..., at which there is a discontinuity in the second derivative.
The representation of such a motion by phase trajectories is obtained if we observe that each of the equations [145.3] is represented by an elliptic trajectory around the vortex point, see Section 1.
If we letz
=
y, Equations [145.3] become
dt dy
_
_
dx
y[145.4] when y < 0
dx
=dy
o
x+
a) when y > 0
y
Integrating these equations, one gets
(x
2 a)
+ R
y2
= 1 when y < 0
2
+
12
1
when
y >
0
[145.51
where R
i
and R
2
are the constants of integration determined at the end of each preceding interval, that is, when & = 0. Equations [145.5] represent two families of ellipses, F, and F
2
, whose centers are displaced on both sides of the origin by a constant quantity ±a, as shown in Figure 145.3. Let us consider the motion beginning with a point A representing the initial conditions on one of the trajectories
F
2
which is drawn in a heavy line. At the point B, where
y = 0, the change of equations takes place, and the representative point goes onto the trajectory
F, passing through B. The motion on that trajectory will continue up to the point C at which the representative point will go onto a trajectory F
2
which passes through C. On that trajectory the change of equations should occur at the point D. It is apparent, however, that such a trajectory would be confined within the zone limited by the broken lines x = +a and x = a, which characterize the zone of static friction. Hence once a trajectory is reached which does not emerge from that zone, the motion ceases.
~"".ll^o~.~YP~Lss~p. W *ra
___
1,
It is thus seen that idealized
Y
Coulomb friction is capable of being represented by piecewise analytic trajecto ries formed by elliptic arcs of the two families
1
r
2 with centers symmetrically displaced by a constant quantity a on both sides of the origin 0. The junc tion points of the arcs are situated on the xaxis. At these points the curve has
C
I
0
I
B a continuous first, but a discontinuous second, derivative as may be expected from the fact that the friction force
o
+a i
o2 changes discontinuously at these points.
Instead of having a point of equilibrium,
Figure
145.3
systems possessing Coulomb friction have a line of equilibrium 0102, which means that any point on the segment 0102 is a point of equilibrium and that the segmenf 0102 itself is the zone of static
friction.
Making use of these conclusions regarding the form of trajectories in the presence of Coulomb friction, Andronow and Chalkin have further elaboated the theory of the clock. We shall omit the details of their theory and will merely indicate their argument as well as their final conclusions.
By introducing "angular time"
7
=
w
0
t, as we did on a number of occasions, Equations [145.3] can be reduced to the form
d
2
d2
[145.6]
d2 +
x
2
= 0 when
X
> 0
The trajectories of these equations are two families C
1 and C
2 of concentric circles whose centers are displaced on the xaxis on both sides of the origin by a fixed quantity a. By reproducing the preceding argument and by expressing the condition that the ultimate trajectory always emerges from the "dead zone" of static friction, these authors show that, if the trajectory is to emerge ultimately from the dead zone, the impulsive change of kinetic energy h
2
must be equal to or greater than 16f
0
2
, where fo is the Coulomb force. This means that a clock is a mechanism with a hard selfexcitation. In other words, unless the condition stated above concerning the relation between h
2
and 16f
0
2
is satisfied, a clock will not start. If, however, this condition is satisfied, the clock will start and will approach its piecewise analytic limit cycle irrespective of any other aspects of the initial conditions.
~~UL.~
146. ELECTRONTUBE OSCILLATOR WITH QUASIDISCONTINUOUS GRID CONTROL
We will investigate in this section the behavior of an electrontube circuit exhibiting features resembling those of a mechanical system possessing
Coulomb friction and acted on by impulses. The analogy, as we shall see, is purely formal, but it will be helpful in approaching from a somewhat new viewpoint the difficult subject of selfexcited oscillations. The interesting feature of the analysis made by Andronow and Chaikin is that it permits investigating questions concerning the establishment and stability of selfexcited oscillations by a method similar to that used in the preceding section, except that here one may have negative Coulomb friction maintaining the oscillations.
Consider the circuit shown in Figure 146.1 with positive directions and customary notations indicated. The symbol x designates the current in the inductance L. The differential equation of the circuit is
L dt
+ Rx +
C
(x 
I(
) dt = 0
[146.1]
Differentiating and rearranging, we obtain
1 1
Li
+ Ri +
 x
=
C f(e,)
[146.2] where la
= f(eg) is the nonlinear characteristic of the electron tube. We will consider a peculiar performance of this circuit, namely, the one which appears when the coefficient of mutual inductance Mis made very large. As a result of this, the amplitude legl of the gridvoltage variation will also be large; this will cause the electron tube to act more or less as a switch operating between the points A, when I
a
= 0, and B, when I
a
= I,, where I, is the saturation plate current. It is apparent that when the grid potential varies between the limits teg, as shown in Figure 146.2, the interval AzB corresponding to the variations t eg,, which are instrumental in causing the plate current to vary, is small in comparison with the total interval AB, and it is possible to idealize the performance as a discontinuous characteristic such as
a
V, ta XIa
Le
L
x vJ c
9
e
1,
D
Figure 146.1
Figure 146.2
n  ^ ^p~^  *. _~


loll
51 that shown in Figure 145.1. Instead of being described by one nonlinear differential equation [146.2], the phenomenon will be described alternately by two linear differential equations x
+ 2hi + Woz = O;
+ 2hi + W = w [146.31
where 2h =;R
2 = and w021
=
f(+
e). It is obvious that the change from one equation to the other occurs at the point where i changes its sign.
We shall assume that the connections are made so that M > 0, e, > 0 when i < 0, and eg < 0 when i > 0. We shall investigate the second alternative later.
In view of our idealization, the anode current
Ia
will undergo discontinuities in the neighborhood of i = 0. These discontinuities have to pass through the capacitor branch of the oscillating circuit since in the inductive branch no discontinuities of current are possible. However, the discontinuities LL in the voltage across the inductance are possible. We conclude,
dt
therefore, that the quantities which are capable of varying discontinuously are
Ia
in the capacitor branch and Ld across the inductance. The quantity z remains continuous. Another continuous quantity is the voltage V across the capacitor, as was mentioned in Section 131.
It is noted that the change from one differential equation to the other, mentioned above, resembles the situation which we have already encountered in connection with Equations [145.2] describing the behavior of a system possessing Coulomb friction, and may follow a similar argument. We can write the second equation [146.3] as
2
+
2h
+ wox = 0
[146.4] where X
= x 
I,.
In this form the first equation
[146.3] and Equation [146.4] are two linear differential equations possessing focal points on the taxis at a distance
I,
from each other. We can, therefore, repeat the argument used in connection with Figure 145.2. Let us consider the phenomenon from the instant when the amplitude is xz and the electrontube switch is off, which corresponds to the first equation [146.3]. During the first halfcycle the original amplitude zx, see Figure 146.3, will be reduced by damping and will become
x
2
.
It is noted that x
2
< X
1
if referred to the taxis. At the point
B the electrontube current jumps suddenly to the value
I,
and, as we saw, the new abscissa axis is now tj, displaced above the taxis by the quantity I,.
With respect to this tlaxis, the initial amplitude is now x
2
=
X2
+
I, and although during the following halfcycle the amplitude is again reduced with respect to this axis because of the dissipation of energy, the pos4tion of the point C with respect to the taxis may be at a higher level than that of A if
I
A
SI
2 s
/x
i2
T
X3
Figure 146.3
2
l e
hT
2;
3
=
x hT
2hT hT
2 e
2
= xle
2
+ Ie
2;
I,
is large enough. Beginning with the point C, we can resume the same argument which we used at the point
A, and so on.
It is seen that, because of this quasidiscontinuous timing of the electronic switch, the amplitudes in the oscillating circuit may grow, thus outweighing the effect of the dissipation of energy. It is easy to express this condition mathematically. If one designates by x the amplitudes referred to the taxis and by
7 those referred to the tiaxis, one has
=
hT
X l e
2
+ I"
X
3
=
3 +
I
= xl e
2hT
2
hT
+
1,
2
+ I
If the amplitude reaches a stationary value, x,
=
x, = xo, which gives
Xo
hT
I,(1
+e
2
g
1

ehT
)
I,
hr
1 
e 2
[146.51
This shows that a stationary amplitude x
0
is determined solely by the properties of the circuit and is independent of the initial conditions.
The question of the stability of these oscillations can be investigated graphically by comparing their subsequent amplitudes. For instance, take the relation between xz and x
3
, namely,
x
=
xe
hT
+
18(1
+
hT e
2)
[146.6]
In the (x,,
3
)plane, see Figure
146.4, this relation is a straight line with
hT
AhT
a slope tan a = e
T,
and an intercept I,(1 + e
2)
= A. On the other hand, if the oscillation is stationary, xi = x
3
, which represents the bisector of the angle between the x, and x
3
axes. The point M of intersection of these lines determines the stationary amplitude x
0
.
If the initial amplitude x
1
is not stationary, the subsequent amplitude will be x
3
' and the corrected amplitude will be x
3
' on the x
1
axis to which the following amplitude x,' will correspond on the x
3
axis, and its corrected value on the x,axis will be x
5
.
MEI
iIl
53
By the interplay of the subsequent corrections it is seen that any nonstationary amplitude tends to approach the stationary value x
0
.
This argument is also valid when the initial nonstationary ampli tude is larger than the stationary amplitude x o
. Hence the stationary amplitude is stable. It should be noted that if we reverse the connection of the coils, the displacement of the x axis with respect to the xaxis by the quantity
I,
will occur in a direction oppositeI
"
x
x
I
to that shown in Figure 146.3.
This will result in a reduction of
X
1
3
XO x
the subsequent amplitudes similar to that found in connection with
Figure 146.4
Figure 145.2 illustrating the action of Coulomb friction. Viewed from this standpoint, the circuit behaves as if it had a kind of negative Coulomb damp
ing when M > 0 and a positive Coulomb damping when M <
0.
The selfexcitation of the circuit occurs when the circuit is characterized by negative Coulomb
damping.
147. PHASE TRAJECTORIES OF AN IMPULSEEXCITED OSCILLATOR
The property of negative Coulomb damping of the circuit investigated in the preceding section becomes still more striking if we analyze the phase trajectories of the system
[146.3] which we write in the form
i
+ 2hi + w2x = 0
[1471]
xz + 2hil
+
(
0 x2
=
0
where x,
= x  I,. Since each of these equations describes a damped oscillator, each is represented by a spiral trajectory approaching a stable focal point; see Section 5. The focal points of these equations are separated by a shows the phase diagram of the motion corresponding to Equations [147.1].
Point 0 is the focal point of the first equation [147.1], and O that of the second equation. Let us start from a certain arbitrary initial condition, represented, for instance, by a point 1 on the negative abscissa axis. Through
Point 1 passes the spiral trajectory 1m2 having O as its focal point and representing the first equation [147.1]. At the point 2 the "change of equations"
'011~~~
SO. occurs, and the phenomenon is now governed by the second equation
[1147.1] having its focal point at
The second stretch of the spiral trajectory 2n3 ends at the point 3, where the first equation
A_
U
3
1 bo
2 4
Cbegins to describe the phenomenon again; this arc of the spiral tra
S
 jectory 3p4 ends at the point 4, and so on. It can be shown that the piecewise analytic trajectory
2n3p4 .. ultimately approaches
Figure 147.1 a closed curve ABCDA formed by two spiral arcs, ABC with O as its focal point and CDA with 6 as its focal point. One obtains a similar conclusion if, instead of starting with a point 1, one starts with a point 1' exterior to the closed curve
ABCDA and repeats the argument. We shall designate by r the radius vectors with respect to the focal point O when the representative point moves on the upper spiral arcs, and by r the radius vectors with a focal point at
6
when the representative point moves on the lower arcs. For the subsequent radius vectors along the xaxis, we have the following obvious relations:
7rh
7rh
r2
=
rae
w ; r
=
r2e
(
= (r
2
7rh n7h 7rh
+ a)e I
= (r
e
1 +
a)e
..
One obtains easily the following general expressions:
rk
[ a[1
+
e
rh
e
27rh
+
+ e
( k
2) Eh
]
+
rie
(k
l)r
r, = a 1 + e
+
+
(e
+ re

)
r,
=
rk = rk
a
[147.2]
 a
where k is an odd integer and s is an even integer. If k and s increase indefinitely, that is, if the number of turns of the piecewise analytic spiral
1m2n3p4... increases indefinitely, the spiral approaches the closed curve
ABCDA, previously mentioned, in the manner specified in Section 22. One sees this immediately from the expressions given above for rk and
r,
when
k
oo and s
>
oo,
namely, lim
(rk)
a
h
1

e
lir (,)
,o
[147.3]

The conclusions remain exactly the same if one repeats the argument for a piecewise analytic spiral 1'm'2'n'3'p'*** starting at a relatively distant point 1'.
It is thus apparent that the closed curve ABCDA formed by the two arcs of the logarithmic spirals is a piecewise analytic limit cycle which the oscillation approaches as t 
=. Moreover, this limit cycle is stable. It is useful to emphasize once more that this limit cycle characterizing stationary periodic motion is determined exclusively by the constant parameters a = I,
h, and w of the oscillatory system and is entirely independent of initial conditions. This is an essential property of selfexcited oscillations, as was pointed out in Section 22.
It is also worth mentioning that we have obtained these results of the classical theory by replacing the actual nonlinear equation [146.2] by a system [147.1] of two alternating linear differential equations. Such a procedure becomes possible only after we have idealized the relatively complicated nonlinear characteristic, shown in Figure 146.2, as a discontinuous stepfunction like that shown in Figure 145.1. In connection with this it may be useful to recall the statement made in Section 3 that the really important difference between linear and nonlinear systems is their behavior in the
large and that local properties are of relatively minor importance. This argument is a typical example of a situation of this kind where a certain idealization of the local properties of trajectories does not change their properties in the large and merely simplifies a problem which would otherwise be extremely complex.
,
WAIN
CHAPTER XXV
EFFECT OF PARASITIC PARAMETERS ON STATIONARY STATES OF DYNAMICAL SYSTEMS
148. PARASITIC PARAMETERS
The study of relaxation oscillations in the preceding sections was
made under the assumption of certain simplifying idealizations which resulted in degenerate equations instead of complete ones containing both oscillatory parameters (L and C in electrical problems and m and k in mechanical ones).
Although the advantages of such simplifications, as we saw, are numerous and the results obtained in this manner are generally found to be in agreement with experimental facts, the introduction of these simplifying assumptions is not without theoretical difficulties and, in some cases, which we will analyze here, may lead to certain complications. In spite of the fact that such cases appear generally as rare exceptions, it is important to analyze this matter in greater detail now that we are acquainted with the general method, at least in its present scope.
One of the principal difficulties, noted on several occasions, is the inconsistency between the number of integration constants appearing in a degenerate problem and that in the corresponding nondegenerate one. Thus, for example, if the nondegenerate problem involves a differential equation of the second order, two constants of integration are necessary to determine the solution; these constants appear as certain definite physical "initial conditions."
In the corresponding degenerate problem, where the differential equation is of the first order, one constant of integration is sufficient to determine a solution. If, however, there still exist two physical factors to which some initial values can be assigned, these factors cannot have entirely aroitrary values but must readjust themselves eventually so that a definite relation exists between them, as was explained in Section 128. The discontinuous theory of relaxation oscillations ignores this rather delicate passage from the solutions of one form to those of the other form, just as the classical theory of mechanical impacts ignores the unknown dynamics of the collision process. Both theories follow a somewhat similar argument, namely, the rapidly changing motion is idealized during a very short time interval as a mathematical discontinuity occurring in the infinitely short time interval (t  0,
t + 0), and the loss of information resulting from intentionally overlooking
t~
dynamics of the process is supplemented by additional physical information which permits solution of the problem in the large although certain local details are inevitably lost in such a procedure. In the theory of mechanical impacts this additional physical information is provided by the theorems of
57
' momentum and kinetic energy as well as by defining a certain empirical coeffi
cient of restitution characterizing the instantaneous dissipation of energy during the impact; likewise, in the discontinuous theory of relaxation oscillations this additional information appears in the form of the conditions of
Mandelstam.
The use of the concepts "infinitely short time interval," "infinitely large acceleration," and so on, instead of the more correct terms "very short time interval," "very large acceleration," and so on, while convenient for the description of a phenomenon in the large, is sometimes capable of introducing serious errors in a theoretical argument and of leading to conclusions at variance with experimental facts.
Thus, for instance, in dealing with an idealized (L,R)circuit, we describe its behavior by the equation
L di dt
+ Ri = E(t)
[148.1]
This equation describes the phenomenon with adequate accuracy in a range sufficient for practical purposes (on the scale, say, of milliseconds). If, however, the behavior of the same circuit is studied in a different range (say, on the scale of microseconds), this equation may not give a correct answer. In fact, any resistor or inductance coil inevitably has a small parasitic capac
ity CP, and if we take this capacity into account a more correct equation will be
RLC d
+
L + Ri
1
= E(t)
[148.2]
It was shown in Section 128 that under certain conditions the solutions i(t) and il(t) of Equations [148.1] and [148.2] may have entirely different features locally, although in the large these solutions are practically indistinguishable. For that reason an electrical engineer would prefer to use the simpler equation [148.1] rather than the more complicated one [148.2]. These facts are too well known to need further emphasis here. However, they are frequently very troublesome. Thus, for instance, a student attempting to investigate the behavior of a simple circuit by means of a cathoderay oscillograph, instead of observing the pattern of curves in accordance with his differential equation, usually observes what is commonly called "hash," that is, a far more complicated pattern of numerous harmonics with cross modulation, and so on.
The reason for this is that too many parasitic parameters have been neglected in forming the differential equation, and consequently the true differential equation is far more complicated than the assumed one.
In most practical cases such discrepancies are not very serious.
However, in some special instances, analyzed in what follows, difficulties may arise which lead to paradoxes already noted by certain authors (19) (20).
A further remark may be useful. Consider the differential equation of an (L,R,C)circuit,
1
La + R4 +
q = 0
[148.3] and assume that L is very small. If L is neglected, one has a differential equation of the first order. The expression "L is very small" is, however, rather indefinite in the sense that it is not important to know that L is small but rather that the consideration of the problem is restricted to the range in which the term L" is negligible compared with the other two terms.
Still more confusing is the effect of the degeneration when C

O0,
leads to a meaningless result. If, however, we make C
+o,
at the limit we obtain
L
di dt
+ Ri
=
0
[148.4] which is an absolutely degenerate equation of an idealized (L,R)circuit.
Here the degeneration occurs not because a parameter is small but because it is large. This confusion disappears if, instead of the parameter C, the capac
1 ity, we use the elastance k
=
C as a parameter. It is convenient, for this reason, to consider parameters as parasitic if they are small or if the corresponding terms in the differential equation are small. The first condition is merely a definition, whereas the second condition specifies the range in which a small parameter can be neglected.
149.
INFLUENCE OF PARASITIC
PARAMETERS ON THE STATE
OF EQUILIBRIUM OF A DYNAMICAL SYSTEM
Consider a system with several degrees of freedom expressible by the differential equation
(n) aox
(n
(n i)
+
alx n
+ ** + an_z + an_l +
azn
= 0
[149.1] re
dnx
d
, and so on. The corresponding characteristic equation is ao n
+
ax
+
+
a,
2 n1X
+ a
n
=
0
[149.2]
The equilibrium is stable if the real parts of the roots of [149.2] are negative, which can be ascertained by the wellknown criteria of RouthHurwitz:
As was previously stated, we never know the exact form of the differential equations in a physical problem because of a number of parasitic parameters either entirely unknown or known only approximately.
Let us assume that Equation [149.1] is a degenerate one derived from some other equation describing more correctly the behavior of the system by taking into account a parasitic parameter a. There are two ways in which this parasitic parameter may appear, depending on whether it is of the "inductance" type or of the "elastance" type.
I I,
". 4 
1 
tj  I
^~ll1ii1
o4illi1
59
If it is of the inductance type, the nondegenerate equation will be
ac
(n)
+ aox + az
(ta1)

+ + an,_
+ ax = 0
[149.3)
If it is of the elastance type, it will be
aox
+ ax
+    + an,_
Differentiating this equation, we get
+ ax
+
afz
dt = 0
[149.4]
aox(n
+ ax(n) +
+ an_
+
anx
+
ax
=
0
[149.5]
For both types of parasitic parameter the order of the original differential equation has been raised by one unit.
The corresponding characteristic equations are a,,n
+
ao
an1
+
*
+ a,_
+ a = 0
[149.6] ao n+1
+ al
+
a
2
+
+anA + a =0
[149.7]
They now have (n + 1) roots, whereas the original characteristic equation
[149.2] had only
n
roots. Since a is small, by our assumption, the old
n
roots change but little owing to the appearance of the new root, and we may neglect this change. The matter hinges, therefore, on the new root
X n
+1.
We can write Equation [149.2], in terms of its roots, as
ao(X

X,)(

X
2
)
. . .
(X

Xn)
=
0
[149.8]
The new characteristic equation [149.6] of the "parasitic inductance" type can accordingly be written as
a(A

A)(A

A
2
)
(

Xn)(XA 
n+)

a[
A+
+ a
1
n1
+
+ anI
+ a n
](X 
X+
1
)
[149.9]
Expanding the rignthand side of this equation and identifying it with Equation [149.6], we get
Sa°
[149.10]
Applying a similar argument to Equation [149.7], we obtain
Xn+1
=
a
a
n
[149.11]
The sign of the new root
n
+1
indicates its effect on the stability of equilibrium. The general integral of the new system is
x
=
Ciet
+ C
2 et
+
"
+ C ne
t
Cn+e n+
[149.12]
,1101~~~~~~
~~
If the conditions of stability for the original degenerate equation [149.1] are fulfilled, the stability of the new equation [149.3] or [149.5] will depend solely on the new root
,
1
.
If this root is stable, that is, if its real part is negative, the equilibrium will still be stable. If, however, the root A,, is unstable, the nondegenerate system will be unstable although the corresponding degenerate system is stable. For a complete discussion, one must apply the RouthHurwitz criteria, as usual.
In practical problems the existence of parasitic parameters is generally unknown; still less known is the magnitude of such parameters. For these reasons in a complicated physical problem one is never certain about the actual conditions of stability. This is why one occasionally observes somewhat puzzling departures from theoretical predictions based on a simplified study of a system expressed in terms of differential equations with'known parameters.
Conclusions obtained from the simple discussion given above may appear to some extent paradoxical. Thus, for instance, if the parasitic parameter is of the "inductance" type, see Equation [149.10], the new root X,,+ is larger as the parasitic parameter a is smaller. If, however, it is of the
"elastance" type, see Equation [149.11], the magnitude of the root XA,+ is proportional to a.
The rather erratic "floating" of a system around its theoretical position of equilibrium can sometimes be traced to the effects of parasitic parameters, as we shall see from a few typical examples in the following sections.
150. EFFECT OF PARASITIC PARAMETERS ON STABILITY OF AN ELECTRIC ARC
The question of the stability of an electric are was the subject of considerable controversy in the past (19) (20). In Section 21 this matter was investigated in detail by means of Liapounoff's equations of the first approximation on the basis of a differential equation with finite parameters L, C, and p. The behavior of the arc is far more complex if some of these parameters are small, a condition which results in a degenerate form of the differential equation describing the process.
Using the notation of Section 21, we recall that the differential equations are
di
L V 
#(i); dt
C
dV
E

V

Ri
dt
R
[dV50.1
where i is the are current, V is the potential difference across the capacitor, and 0(i)
=
Va is the nonlinear characteristic of the arc, see Figure
150.1. As was explained in Section 21, there are either three equilibrium
~
N
E
61
R
V
C
L V
Vo
A
A
B
ERi=V
Va
(i)
Figure 150.1
3
points (Points 1, 2, and 3) or one such o point (Point 3), as shown in Figure
Figure 150.2
150.2, depending on the orientation of the straight line E  Ri with respect to the characteristic 0(i).
It was shown that for small departures, v of voltage and
j
of arc current, from the corresponding equilibrium values, the equations of the first approximation are
dv dt v
RC j
C dj dt v
L p
L[150.2] where p = '(io) is the slope of the tangent to the characteristic 0b(io) at the point i = i
o
.
The characteristic equation of the system is
2+(
RC
+
L
+
LC
1 +
R
= 0
[150.31]
When there are three points of equilibrium, it is observed that Point 1 corresponds to p = 0'(io) > 0. This point is always stable because the real parts of the roots of Equation [150.3] are negative. It is, therefore, either a stable nodal, or a stable focal, point according to whether the roots are real or conjugate complex. At Point 2 the equilibrium is unstable because p < 0 and, moreover, Jpl > R. This point, therefore, is a saddle point. Point 3 is a point of stable equilibrium because, although at this point
p
<
0, IpJ
< R.
Having recalled these conclusions of Section 21, we propose to investigate now what happens to these various conditions of equilibrium when the system undergoes a degeneration, that is, when one of the two oscillatory
parameters L and C approaches zero. We may distinguish two cases of degeneration, namely, Cdegeneration when C

0 and Ldegeneration when L
+
0.
In order to investigate the Cdegeneration, it is convenient to write Equation [150.3] as
Ck + (
R
+
L
X +
LR
(R + p)
=
0 [150.4]
For the investigation of the Ldegeneration we will write it as
LX
2
+(
+ p)X +
(R + p)
=
0
[150.51
In this manner we shall be able to use the conclusions of Section 128. Putting
C
=
0 in Equation [150.4], we obtain
=
[150.6]
Putting L
= 0 in Equation [150.5], we have
1
S
CR
(R + p)
CRp
[150.7]
Let us consider the equilibrium at Point 2 of Figure 150.2. It is observed that in the Cdegeneration this point is still unstable. In the L degeneration, it is stable. A considerable controversy existed on this subject in the technical literature before this question was completely understood.
The following explanation was suggested by S. Chaikin (20) (21).
Let us consider the characteristic equation
a
2
+ bX + c = 0
[150.8] with a > 0 and c < 0, which characterizes a saddle point; see Section 18. We can define saddle points as positive when b > 0 and as negative when b < 0.
In the case of adegeneration, a positive saddle point gives tise to a positive root and, hence, to unstable equilibrium. If, however, the saddle point is negative, the degenerate system has stable equilibrium at this point. Thus the adegeneration reverses the stability at Point 2 of Figure 150.2. Since in both cases Point 2 corresponds to a saddle point of the system, we conclude that the only way to obtain stability under this condition is to have the representative point follow a stable separatrix.*
If, however, it follows the stable separatrix, there exists a fixed relation between x and y = i in the phase plane. But this is precisely what happens in a degenerate system described by a differential equation of the first order, see Equation [128.3].
The condition of degeneration thus imposes this singular trajectory, the only one which is stable in the neighborhood of a saddle point. In this case, therefore, a threshold exists similar to that which exists for the asymptotic motion of a pendulum approaching a point of unstable equilibrium, see Section
4, Case 3; such a case is never obtained in practice since there exist no absolutely degenerate systems, that is, systems in which a = 0.
*
Only in a purely theoretical case when the motion of the representative point takes place along the stable separatrix or asymptote of a saddle point, may the latter be considered as a stable singularity.
If the actual motion of the representative point occurs in the neighborhood of this theoretical motion, it may appear that the saddle point is a stable singularity, at least for a limited time.
In view of this, one could characterize the saddle point as an almost unstable singularity.

U ii
63
For practical purposes, if precautions are made to reduce the parasitic parameter to a very low value, the representative point will follow a trajectory very closely to the stable separatrix so that one has the impression that the phenomenon would ultimately "settle" at Point 2 as if it were a point of stable equilibrium. In reality, sooner or later the representative point will depart from this point as a result of accidental disturbances.
By a similar argument one finds that Point 3 of stable equilibrium for a nondegenerate system may become unstable in the case of Ldegeneration, as follows from Equation [150.1] in which p <
0 and
Ipl
< R.
Point 1 is stable both for the nondegenerate and the degenerate systems. It is thus seen that the assumption of absolute degeneration may lead to conclusions at variance with experimental facts as far as the question of stability is concerned.
151.
EFFECT OF PARASITIC PARAMETERS ON STABILITY OF RELAXATION OSCILLATIONS
As another example of the same sort, we shall investigate the behavior of the circuit shown in Figure 135.1, taking into account the presence of the small parasitic inductances L and L' shown in Figure 151.1. Kirchhoff's equations for this circuit are
dl
I + i = Ia;
L + RI 
ri  Ld
di
1
t
Cidt
[151]
=0
151.1
Moreover,
Ia
=
0(kri) as before. Elimination of
Ia
and I between these equations results in the equations
di dt
Ro(kri) 
L + L'

(R + r)i V
Lkr'(kri) ; dV dt i
[151.2]
The conditions of equilibrium are obviously
i = 0 and RO(O)

V
=
0. On the
Figure 151.1
r 1
L'
other hand, for small departures from equilibrium, we can write
(kri)
=
0(0) + krio'(0) + . . .
Limiting the expansion to the first term and noting that 0'(0) = S,
Liapounoff equations of the first approximation are the
di dt
6.
_ 
y
V
Y dV i
dt C
[151.3] where
6
= r + R(1 
Skr)
and
= L' + L(1 Skr)
[151.4]
The characteristic equation of the system [151.3] is
2
2
6
1
+ 6X +
1 = 0
Y
yC
and its roots are
X1,2

6+ 62
2C
4
[151.51
[151.6]
Figure 151.2
For y < 0 the roots are real and of opposite sign; hence, the singularity is a saddle point. For y > 0, the singularity is either a nodal point or a focal point, depending on whether the roots are real or conjugate complex. If they are conjugate complex, with 6 > 0, the singularity is stable; with 6 < 0, it is unstable. The distribution of singularities is shown in Figure 151.2, which is the same as Figure 18.1 with the exception that now we make a distinction between "positive" and "negative" saddle points as was explained
For an absolute degeneration, in Section 150.
L
= L' =
=
0. Multiplying the first equation [151.3] by y, one has di
+
V=
0, that is, i = .
do
The second equation gives
dV dt
V
6C
[151.71
tem. The equilibrium is stable for 6 > 0 and unstable for 6 < 0. The points of stable equilibrium are indicated by solid points on the positive axis of ordinates and those of unstable equilibrium by circles on the negative axis a

SI
m
65
of ordinates. It is seen that the region of stable equilibrium is localized in the region of "negative" saddle points and that of unstable equilibrium in the region of "positive" ones. It was shown, however, in the preceding section that such stability is of theoretical interest only, because it is impossible in practice to prescribe the initial conditions accurately enough to make the representative point follow the stable separatrix, Just as it is impossible to impart to a pendulum a velocity Just sufficient to produce An asymptotic motion. Moreover, no physical system is completely degenerate, but is only quasidegenerate.
The situation is still further complicated by the fact that stability conditions are influenced not only by the existence of parasitic parameters but also by their relative importance. In order to show this, assume, for instance, that L * 0, L'
=
0, 1  Skr < 0, but R is sufficiently small to render 6 = r + R(1  Skr) positive. In such a case the equilibrium is unstable for any value of L, however small, because the real part of the roots is positive, that is, 6 > 0 and 7 < 0. In a degenerate system, where L
=
L'
=
0, the equilibrium is stable, however, as follows from Equation [151.7]. Since any physical system is only quasidegenerate, we conclude that the equilibrium is unstable in this case.
If we consider now the second case, when L' * 0, the situation is different. Let us assume, as previously, that 1  Skr < 0. It is obvious that, by giving L' a suitable value, the quantity y = L' + L(1  Skr) can be made positive and since 6 = r + R(1  Skr) is also positive, as previously assumed, it follows from Equation [151.6] that the equilibrium is stable. Hence, the equilibrium which was unstable for L * 0, L' = 0 becomes stable for L = 0,
L' * 0. Thus the change from stability to instability depends here on the ratio L/L' of the parasitic parameters.
Since in practice one is. seldom sure of the existence of parasitic parameters, much less of their relative values, the predetermination of stability conditions for a circuit of this kind becomes altogether meaningless.
It is likely that floating, erratic excitation and similar phenomena still little explored can be explained on this basis.
To sum up the results of this analysis, it can be stated that, in general, the effect of parasitic parameters is negligible in practice unless the system happens to be in the neighborhood of a branch point of equilibrium.
If the system is in such a neighborhood, even a small "cause" may exert an appreciable "effect" on the system, and its behavior then becomes entirely unpredictable.
ilYI__ _ _____~.._ ___~ ___~~_
REFERENCES
(1)
"On Relaxation Oscillations," by B. Van der Pol, Philosophical
Magazine, 7th Series, Vol. 2, November 1926.
(2) "Uber Relaxationschwingungen" (On Relaxation Oscillations), by B. Van der Pol, Zeitschrift fUr Hochfrequenztechnik, Vol. 28, 1926, and
Vol. 29, 1927.
(3) "Uber Kippschwingungen insbesondere bei Elektronenrbhren" (On
Sweep Oscillations in Electron Tubes), by E. Friedlhnder, Archiv fUr Elektrotechnik, Vol. 17, No. 1, September 1926, and Vol. 18, No. 2, October 1926.
(4) "Reissdiagramme von Senderrbhren" (Graphical Diagram for
Transmitting Tubes), by H. Rukop, Zeitschrift fUr Technische Physik, Vol. 5,
1924.
(5) "Eine neue Schaltung zur Erzeugung von Schwingungen mit linearem Spannungsverlauf" (New Wiring Connection for Generating Oscillations with Linear Voltage Curvature), by G. FrUhauf, Archiv fUr Elektrotechnik,
Vol. 21, No. 5, February 1929.
(6) "Zur Theorie des rUckgekoppelten RUhrensenders" (On the Theory of Regenerative Coupled Vacuum Tube), by F. Kirschstein, Zeitschrift fUr
Hochfrequenztechnik, Vol. 33, No. 6, June 1929.
(7) "Uber Schwingungszeugung mittels eines Elektronrdhrensystems be' welchem die Kapazitht von untergeordneter Bedeutung ist" (On the Excitation of Oscillations by Means of a VacuumTube System in Which Capacity Is of
Subordinate Importance), by K. Heegner and I. Watanabe, Zeitschrift fUr Hochfrequenztechnik, Vol. 34, No. 2, August 1929.
(8) "Uber symmetrische Kippschwingungen und ihre Synchronisierung"
(On
Symmetrical Relaxation Oscillations and Their Synchronization), by H.E.
Hollmann, Elektrische Nachrichtentechnik, Vol. 8, No. 10, October 1931.
(9) "On the Theory of the Frthauf Circuit," by L.P. Cholodenko,
Journal of Technical Physics, USSR, (Russian), Vol. 5, 1935.
(10) "Les systemes autoentretenues et les oscillations de relaxation" (SelfExcited Systems and Relaxation Oscillations), by Ph. LeCorbeiller,
Hermann, Paris, 1931.
(11) "On the Existence of Periodic Solutions of Second Order Differential Equations," by N. Levinson, Journal of Mathematics and Physics,
Vol. 22, 1943.
1 ,
t'
""."
11111
1ImmIilll
67
(12) "On Van der Pol'sand Related NonLinear Differential Equations," by J.A. Shohat, Journal of Applied Physics, Vol. 15, 1944.
(13) "Theory of Oscillations," by A. Andronow and S. Chaikin, Mos
cow, (Russian), 1937.
(14) "Oscillations discontinues" (Discontinuous Oscillations), by
S. Chaikin and L. Lochakow, Journal of Technical Physics, USSR, (Russian),
Vol. 5, 1935.
(15) "Mesure en valeur absolue des p6riodes des oscillations 6lectriques de haute frequence" (Absolute Value of Periods of HighFrequency
Electric Oscillations), by H. Abraham and E. Bloch, Annales de Physique,
Vol. 12, No. 9, 1919.
(16) "Uber Schwingungszeugung mittels eines Elektronr5hrensystem welche Selbstinduction nicht erhalten" (On the Excitation of Oscillations by Means of an ElectronTube System without SelfInduction), by K. Heegner,
Zeitschrift fUr Hochfrequenztechnik, Vol. 29, 1927.
(17) "Zur hydrodynamischen Theorie der Schmiermittelreibung" (On the Hydrodynamic Theory of LubricatingOil Friction), by A. Sommerfeld,
Zeitschrift fUr Mathematik und Physik, Vol. 50, 1904.
(18) "Mechanical Relaxation Oscillations," by S. Chaikin and
N. Kaidanowsky, Journal of Technical Physics, USSR, (Russian), Vol. 3, 1933.
(19) "Warum kehren sich die fUr den Lichtbogen gUltigen Stabilittsbedingungen bei Elektronenr'dhren um?" (Why Do the Conditions of Stability
Valid for an Electric Arc Light Reverse in Electron Tubes?), by H. BarKhausen,
Physikalische Zeitschrift, Vol. 27, No. 2, February 1926.
(20) "Uber Stabilittsbedingungen und ihre Abhgngigkeit von Steuerorganen" (On Stability Conditions as Functions of Control Elements), by
E. Friedlnder, Physikalische Zeitschrift, Vol. 27, 1926.
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5,
1935.
''11111
68
INDEX
This treatise has been published in four parts. In this index the roman numerals indicate the parts, the arabic numerals indicate pages in the respective parts.
AbrahamBloch, multivibrator of, IV 3034
Absolute stability, zone of, I 61
Analogies between electrical and mechanical systems,
III 710
Analytical definition of stability, I 4142
Analytical impasse, IV 14
Analytical method, I 11, 111; II 1111; III 75; IV 2 of
Kryloff and Bogoliuboff,
I 4 of Poincare, I 4, 87, 99, 104105; II 432 of topology of phase trajectories,
,I
2629 of Van der Pol, I 4
Analytical periodic solutions, IV 3
Analytical trajectories, IV 3, 31, 3334
Andronow, A., I 24, 63, 75; II
1, 3, 14;
IV
1, 3,
46, 4950
Andronow and Witt presentation of, III 96 topological method of, III 94
Aperiodic damped motion, I 106 equation of, I 14 phase trajectories of,
I
1819
Aperiodic damping,
III 14
Aperiodic motion, I 3, 36, 48
Appleton, III 93
Approximation equations of first, I 3, 4952, 5556,
65, 81,
87
94, 97, 99,
101, 103, 114, 120, 127; II 3334,
39, 42, 145, 5254, 59, 83, 90, 95, 102, 110;
III 11, 23, 27 29 31, 33, 338, 4045, 5357,
63, 6769, 74;
IV 31, 6o61, 4 linear, I 49 first, I 27; II 55, 95 III
17, 30, 32, 3436, .49
improved first, II 760 90, 98; III
4647 application of, II 8065 equations of,
III 40 linear,
I
1, 22, 100 method, II 1 method of first, II 6366, 88; III 26 method of Li'nard, I 111 of first order, III 41 of higher orders, II 3, 7598; III 40 of zero order, III 41 definition of, II 56 order of, II 12 quantitative method of, I 23; II 12 advantage of, I 3 second, II 76 example of, II 94 theory of first, II 3, 37, 4975, 99, 107; III 10 applied to nonlinear conservative system,
11
5556
third,
II
76
Asymptotic motion, I 3, 28, 3031, 36 definition of,
I 16 of pendulum,
I
16; IV 62
Asynchronous action conditions for, III 50 on selfexcited system, III 52
Asynchronous excitation, II 100; III 47, 50
Asynchronous motion, I 125
Asynchronous quenching, II 100; III 47, 5051
Asynchronous selfexcitation, III 51
Autonomous equation, II 21
Autonomous system, IV 44 definition of, I 10 nonlinear nonconservative, I 62 quasilinear, III 41 with one degree of freedom, I 105
Autoparametric excitation, III 107108, 126129 condition for, III 126
Autoparametric nonlinear coupling, III 127
Autoparametric selfexcitation, III 127, 129
Autoperiodic excitation, III 4950, 57
Autoperiodic frequency, III 4951, 77, 94,
101
Autoperiodic oscillation, III 53, 97, 100, 102 amplitude of, III 102 conditions for stability of, III 58 definition of, III 47 existence of, III 4752 selfexcitation of, III 52 stability of, III 58 stability of stationary, III 56 synchronized with heteroperiodic oscillation, III 55 synchronous, III 56
Autoperiodic selfexcitation, III 50, 52
Autoperiodic solution, III 58
Autoperiodic state, III 59 of nonlinear systems, III 47 selfexcitation of, III 60
Autoperiodic variables, III 50
Barkhausen, III 12
Barrier, II 43, 69
Beats, III 55, 9394, 103,
106 application of first theorem of, I 7982 criteria of, I 75, 7779, 105; III 97; IV 31 first theorem of, I 75, 777d; III 97 negative criterion of, I 75, 7778; 11I 97 second theorem of, I 75, 7879; III 100101
Bibliography;
see
References
Bifurcation of limit cycles, I
87
Bifurcation point, I 31,
58
(see also Critical point)
Bifurcation theory, IV 35 for quasilinear systems,
II
2627 of Poincar6, I 87104
Bifurcation values of parameter, I 87
BiotSavart law, I 34
Birkhoff, G.D., I 4, 65
Bogoliuboff, I 23, 67; II
1, 3,
47, 76,
100 (see also
Kryloff and Bogoliuboff) method of equivalent linearization of, II 3, 75, 99 notation of, III 6, 15
100, 110; III 1, 10,
83, 94 theory of first approximation of, II
49, 5254, 75;
III 10,
41, 75 treatise of, III 3, 40, 55
Boltzmann, theory of,
I 5
Bowshewerow, I 73
Branch point of equilibrium,
I
3, II
31;
IV 65 of limit cycles, I 88
Brillouin, M., III 107
Canonical equation, I 45
Canonical form, III 81 of linear equations, I 4244 of nonlinear system, I 52
Cartan, E. and H., I 105; II 47
CartanLi6nard, conditions of, II 48
Cauchy's theorem of existence for solution of differential equation,
I 2, 9 of existence of linear differential equation, I 2
0
 INNIII
r 111
.IO oi
Cauchy's theorem  continued of phase trajectories, I 11, 14 of uniqueness, I 24
Chaikin, E., III 103
Chaikin, S.
I 34,
75;
II 1, 3; IV 1, 3, 25, 29, 40,
42,
46,
4950, 62
Change of equations, IV 4748, 53
Characteristic equation,
I
14, 18, 42,
4445, 47,
5758, 102, 114, 120; II 4, 39, 45;
III 63,
65,
80,
110; IV 9, 31, 6162, for transient state of quasilinear sytem,
III 16 of linear circuit, III 21 of nonlinear system, III 20
Characteristic exponents, I 8385; II
2021;
III
81
Characteristic parameter,
III
75
Classification of singularities, I 42,
47
Clock elementary theory of, IV
4446 limit cycles in, I 6768 selfexcitation of, IV 46,
49 theory of, IV 49
Coalescence of equilibrium points, I 33 of frequencies, III 31, 93, 103 of limit cycles, I 7273; II 32 of singular points,
I
127; III 101102 of singularities, I 29, 34, 3637, 39
Coefficient of restitution, IV 3, 57
Combination frequencies, III 2526, 3031, 38, 42, 48,
53
Combination harmonics, III 2526, 28, 38, 46
Combination oscillations, III 47
Combination tones, definition of, III 24
Complex admittance, III
Complex amplitudes, III method of,
III
6
34, 78,
1011,
14
Complex impedance, III 5, 8
Conditional stability, range of, I
61
Conservative system, I 7, 23, 25, 38, 40, 42, 106
128;
II
27, 51, 55, 90;
III
19, 114 127126 cylindrical phase trajectories of, I 118119 equilibrium of, I 40 motion of, I 42 oscillations of, II 80 periodic motion in, I 6263 points of equilibrium in, I 34
Constant linear parameter, III
69
Constant parameter,
II
67;
III
4, 122; IV 55 linear dissipative system with, III 10
Continuum of circles, III
78 of closed curves, I 62, 119 of closed trajectories, I 1213 62; II 67 of phase trajectories,
I
62, 12; IV 27
Coulomb damping, II 61 negative, IV 53 positive, IV 53
Coulomb friction, IV 4647, 4951,
53 existence of, IV 46 negative, IV 50 phase trajectories in presence of, IV
4749
Coupled electronic oscillators, stability of, II 24
Coupled frequencies, III 17, 30
Coupling autoparametric nonlinear, III 127 between degrees of freedom, III 5, 22 critical value of, III 92 inductive,
III
12 lack of, III 6 nonlinear,
III
127
Coupling factor,
III 5
Criteria for existence of critical point, IV 1314 of focal point, I 13 of limit cycle, I 7577 of nodal point,
I 13 of saddle point, I 13 of Bendixson,
I 75, 7779, 105;
III 97; IV 31 application of, I 7982 of Kaufmann for stability, I 61 of Liapounoff for stability, I 4849; III
6364 application of, I
4955 of Poincard, I 105; II 31 of Poincar6Liapounoff for stability, III 6364 of RouthHurwitz for stability, II
22; IV 58, 60 of stability, I 2223, 33; III 85 applied to nonlinear system, I 3, 55 of equilibrium, I 6061 of singularities, I 47
Critical damping, III
123
Critical points, I
31,
102; IV 15, 19, 2126,
2829, 3235, 3943 criteria for existence of, IV 1314 locus of,
IV
14, 2627 of differential equations, IV 1215 theory of, I
130
Critical threshold, I 3, 11; III
46, 125; IV 26
(see also
Divide)
Critical value of amplication factor, I 102 of amplitude, II 69 of coupling, III 92 of damping, 1 123 of equivalent parameter, II
100;
III
30 of index of modulation,
III
64 of negative resistance, III 15 of parameter, I 3135, 39, 87, 97, 104; II 2627,
6970 of resistance,
III
18, 21 of selfexcitation, III
49 of transconductance, III
54 of variable, II
63
Curve of contacts of Poincar6, I 75
Cylindrical phase space, I 116130
Cylindrical phase trajectories of conservative system, I
118119 of nonconservative system, I
119121
Damped oscillator, differential equation of, I 16
Damped oscillatory motion, equation of, I 13
Damping aperiodic, III 14
Coulomb, II 61 negative,
IV
53 positive, IV
53 critical, III123 critical value of, I
123 linear, II 5962 negative, I 105; II 4445, 63, 65 definition of, I 21 examples of, I
2122 nonlinear, I 5 nonlinear dissipative, II 5862,
106107 nonlinear variable, I 105; II
4445, 6365 positive, I 18, 105; II
44, 63 quadratic,
II
6062, 106 variable, I 105; II
47, 8283 deforest, Lee,
I
5
Degenerate differential equation, IV 1617 systems of, IV 1618
Degenerate equation, IV 10,
56, 58, 60 absolutely, IV 1112, 58 of first order, I 112; IV
5, 7, 15, 17
Degenerate limit cycle,
I
70
Degenerate phase trajectory, I
12
Degenerate system, II 14; IV 60, 62, 65 absolutely, IV
11, 62 completely, IV
10 conditions for stability of, IV 17
I  I
Degenerate system 
continued
of first order, periodic solution of, IV 19 transition between continuous and discontinuous solutions of, IV 3536 with one degree of freedom, IV 1929
Degeneration absolute, IV 64 complete, IV 910 condition of, IV 38 of equations, I 45; IV 15,
40 of system, IV 61 solutions of differential equations in neighborhood of, IV 811
Demultiplication of frequency, III 2425
Differential equation, I 16, 1011, 14, 22, 24,
3132,
35, 37,
4243, 46,
49, 51 62, 64, 6667,
72,
80,
2
11
4, 58,
~1,
107; III 75 linear, I 1
57,
90, 9596,
critical points of, IV 1215 degeneration of, I 5; IV 1517
101, i06, 112, 117;
Cauchy's theorem for existence of, I 2 with constant coefficients, I 1 linearized, I 67, 87 nonlinear; see Nonlinear differential equation of damped oscillator,
I
16 of dynamical system, I 20 of electromechanical system, I 117118 of first order,
I
17;
IV
89, 12, 28, 33, 58 degenerate, IV 5, 7, 15, 17 describing system with one degree of freedom,
IV 2529 discontinuous solution of, IV 17 of harmonic oscillator, I 7 of mechanical system, III 9 of motion, I 11 of synchronous motor, I 126 of nondissipative circuit, III 115 of nth order, I 9 of pendulum, II 56 of second order, 1 10, 17, 24; IV 68, 12, 15,
56 full, IV 5 quasidiscontinuous solution of, IV 38 of synchronous motor, I 126 of Van der Pol, III 9496 quasilinear system of,
II 14 singular points of, I 17; III 96 singularities of, I 3, 11 solutions of
Cauchyos theorem of existence for, I 2, 9 in neighborhood of point of degeneration, IV 811 represented by phase trajectories, III 2 systems of degenerate,
IV 1618 theory of,
I
2,
4 with periodic coefficients, III 2, 56, 74, 108112,
124125, 127
Discontinuities, IV 1315, 3234, 41, 43, 48, 51 occurrence of, IV 13, 1516, 27
Discontinuous jump,
II 32; IV 6, 12, 15, 2122, 24,
28, 3233, 36, 4142, 4546
Discontinuous periodic motion, IV 3
Discontinuous stationary relaxation oscillations,
IV 12
Discontinuous solutions of degenerate system, transition between continuous and, IV 3536 of differential equations of first order, IV 17
Discontinuous stretches, IV 2, 4243
Discontinuous theory of relaxation oscillations, IV 3,
818, 2022, 25, 29, 5657
Dissipative parameter,
III 21; IV 16
Dissipative system, II 67, 73; IV 45 heteroparametric excitation of, III 121123
Divides,
I
3, 11, examples of,
I
72 of common topography,
I
32
Dreyfuss,
I 126
Dry friction, I
22; IV 47
Durand, Dr. W.F., II 61
Dynamical equilibrium,
III 34
Dynamical system, I
1011,
32,
51, differential equation of, I 20 equilibrium of, IV 5860 stationary states of,
I 40; IV 5665
Electrical oscillations in circuit containing iron core, II 58 in circuit containing saturated core, II 108
Electrical system,
III
710
Electrodynamical system, I 87
Electromechanical system differential equation of, I 117118 selfexcitation of, I 99,
102104 selfexcited oscillations in, I 99, 102104
Entrainment of frequency,
I 4; III 1, 93106 acoustic, III 94, 103104 artificially produced, III 104 band of, III 104 definition of, III 93 mechanical, III 94 other forms of,
III
104106 pure,
III
100 state of, III 102 steady state of, III 102 transient state of, III 98, 102 zone of,
III
93, 100, 103104,
106
Equations absolutely degenerate, IV 1112, 58 autonomous, II 21 canonical, I 45 change of, IV 4748,
53 characteristic; see Characteristic equation degenerate, IV 10, 56, 58, 60 of first order, I 112; IV 5,
7,
15, 17 degeneration of, I 45; IV 15,
40 differential; see Differential equation
Lagrangian III
126127 linear, I II 23, ,0; approximate, I 100 canonical form of, I 4244 equivalent,
II
101102 harmonic solution of, II 33
Mathieu, III 124
MathieuHill, III 108 of first approximation, I
49 system of, I
18, 23, 80, 83 with constant coefficients, I 1 linearized, I 67, 87, 100 equivalent, II 110 nondegenerate, IV
59 nonlinear, I 1, 118;
II 5, 4748, 108 of oscillation, II 58 nonlinear differential; see Nonlinear differential equation of aperiodic damped motion, I 14 of damped oscillatory motion,
I 13 of first approximation, I 3, 4952, 5556, 65, 81,
87,
94, 97, 99, 101oi, 103, 114, 120,
127;
II
3334,
39, 42,
45, 5254, 59, 83, 90, 95,
102, 110; III 11,
23,
27,
29, 31, 33, 3735,
40i45, 5357, 63, 6769, 74; IV 31, 6061, 4 linear,
I 49 of first order,
I 9; IV 3, 1617, 20, 22 of Froude for rolling of ship, II 61, 107 of Hill,
III
113114 definition of, III 109 of HillMeissner, III 114115, 117 topology of, III 114117 trajectories of, III 114, 117 of improved first approximation,
III 40 of isoclines,
I
121 of Li6nard, I 107, 109110 of limit cycle, II 46 of Mathieu, III 109110 112114, 124, 127128 definition of, III 108 linear,
III
124 parameter of, III 127 stable and unstable regions of, III 111112 of MathieuHill, III 108109, 113114, 124125 linear, III 108 stability of,
III
113 subharmonic resonance on basis of, III 125, 129 of nth order, IV 16
_
_II/
,
I
ial
10
*
Equations

continued
of phase trajectory, I 66, 88, 128 of Poincar6, II 53 of
Rayleigh, II 4445, 4748
65 of second order, IV 1617 36 of separatrix, I 3536, 36 of unstable motion, I 13 of Van der Pol, I 79,
85, 97, 103, 105 108
111,
113; II 27,
III 4546,
41, 47,
100;
53,
IV 23
6364, 66,
83,
65;
quasidegenerate, IV 1112 quasilinear; see Quasilinear equation variational, I 83; II 19; III 3536, 6364, 70, 80 with both linear and quadratic damping, II 61
Equilibrium branch point of, I
3, 42,
48,
55, 60, 82; II
31;
IV 65 conditions for stable, II 27 conditions of, I 11, 29, 60, 80, 94; II 71; IV 63 dynamical, III 34 halfstable, I 51
Kaufmann's criterion for stability of, I 61 line of, IV 49 loss of, IV 4 motion in neighborhood of, I 40 neighborhood of, III 87 of circuit containing nonlinear conductor, I
5561 of conservative system, I 40 of dynamical system, IV 5860 of electromotive forces in circuit, II 58 of point, I
32 point of, I 11,
99, 125;
II 3637;
III 87, 106; IV
21,
49,
6061, 6364 coalescence of, I 33 disappearance of, I 34 in conservative system, I
34 motion in vicinity of, I 29 stability of, III 106 stability of motion in vicinity of, I 24, 26 position of, I 18, 22, 32, 38,
IV 60 change of, IV 4
49; II
51;
III
54; stability of, I 9 stability of, I
3335, 4061, 71, 85, 87
93
127;
II 3536, 68; III 106, 110 IV
21, according to Liapounoff, I 4b61 criteria for, I 6061 theorems for, I 29 state of,
I
22 types of,
I
58
Equilibrium phase angle, III 106
Equiphase interval, III
118119
Equitime interval,
III
119
Equivalent Balance of Energy, Principle of, II
99, 102104 .
Equivalent linear equation, II 101102
Equivalent linear impedance, III 33
Equivalent linear system, II 102; III 33, 38
47,
Equivalent linearization for multiperiodic systems, III 2630 method of, II 3, 99102; III 21, 31, 40, 49, 66 applied to steady state of quasilinear system,
III 1014 applied to transient state of quasilinear system, III 1419 examples of application of, II 105111 in quasilinear systems with several frequencies,
III
3740 principle of, II 105; III 10, 27, 29, 61
Equivalent linearized equation, II 110
Equivalent linearized system,
III
66 selfexcitation of, III 30
Equivalentparameter, II
102105, 107;
III
3335,
62, 66, 6869 critical value of, II 100; III 30 definition of, II 100101; III determination of, II
99102
38 nonlinear, III 29 of multiperiodic system,
III
27
49,
Euler's identity, III 3
Exchange of stability,
I
34, 38
Excitation autoparametric, III 107108, 126129 condition for, III 126 autoperiodic, III 4950, 57 external,
III
41, absence of, III
8996 system with, III
41 external periodic, III
46, 65, 69, 72, 87,
94, 128 of quasilinear systems, III
4152 system with, III
75 heteroparametric, III 107108, 112, 115, 125126 conditions of, III 108, 123 dependence on frequency and phase of parameter variation, III 117121 of dissipative system, III 121123 principal features of,
III
119121 heteroperiodic, III 50, 107108 absence of, III 51 nonresonant external, of quasilinear system,
111
4647 parametric, I 4; III 1, 6065, 107129 definition of, III 60 of critically damped or overdamped circuit,
III
123 periodic nonresonant, III 4144
Existence of autoperiodic oscillation, III 4752 of closed trajectories, I
21, of critical point, criteria for, IV 1314 of focal point, criteria for, I 13 of limit cycles, I 99; II 4748, 64, 6669, 9095;
III
100 condition for, II 35, 39, 42, 45, 65, 73, 93 criteria for, I 7577 proof of, I 79 of nodal point, criteria for, I 13 of periodic solution, I
105115, 121;
II
1011, 86,
90 geometrical analysis of, I 105115 in proof of, I 105 of principal solution, III 85 conditions for, III 87 of saddle point, criterion for, I 13 of selfexcited oscillations, II 47
Expansions of Poincar6, II 7
External excitation, III 41,
8788, 92 absence of,
III
8996 nonresonant, III
4647 system with, III 41
External frequency, III
59, 93,
100, 103, 105106
External periodic excitation, III 1, 41, 46, 65, 69,
72,
87, 94, 128 of quasilinear systems, III 4152 system with, III
75
Externally applied frequency, III 88, 92
FaradayMaxwell theory, I 124
Fleming, I 5
Floquet, theorem of,
III
109
Focal point, I 16, 18, 22, 6970, 7577, 79, 101103,
114, 120, 123, 125, 127;
III 99,
101; IV 6, 8,
17, 27, 3536, 51, 5354, 61,
64 criterion for existence of, I 13 definition of, I
13 neighborhood of, I 1 stability of, I 4748, 5760, 889, 9192, 97, 99;
11 3132, 4143, 68 stable, I 13; II 68 unstable, I 13; III 63
Fractionalorder resonance,
III
53, 55, 5761, 125
Franck, A.,
I 6
Frequency acoustic entrainment of, III 94, 103104 autoperiodic, III 4951, 77, 94, 101 coalescence of, III 31, 93, 103 combination, III 2526, 3031, 38, 42, 48, 53 correction, II 40, 59, 8385, 106 coupled, III
17, 30
, 1 iiju
, ,In
I, ail
Frequency

continued
demultiplication, III 2425 demultiplication network, III 104 entrainment of, see Entrainment of frequency external,
III
59, 93, 100 103, 105100 externally applied, III 8, 92 harmonics of, III
48 heteroperiodic, III 4953, 100101 of pendulum,
III
94 of quasilinear oscillation, II 83 of ripple,
III 115, 117, 119120 of selfexcited nonlinear system, III 17 of selfexcited oscillations, III 13 of subharmonic oscillation, III 88 of thermionic generator, II 2425 synchronization of,
III
104,
106
Friction
Coulomb, IV 4647, 4951, 53 existence of, IV 46 negative, IV 50 presence of, IV 4749 dry,
I
22; IV 47 nonlinear, IV 40 static,
IV 4849 zone of, IV 49
Froude's equation for rolling of ship,
II
61,
Froude's pendulum, I
2122, 67; II 4446; IV 37
Gaponow, III 100101
Gauss's theorem, I 77
Generating solution,
I
86
II
710, 1213,
15, 181
24, 2627, 38 474,
50,
56, 65,
89, 102,
101
III 45,
78 amplitude of,
II 12 definition of, II 7 of Poincar6, III 102
;
Geometrical analysis of existence of periodic solutions, I
105115
Geometrical definition of stability, I 41
Geometrical theory of limit cycles, I 6386
Gorelik, III
126127
Graphical method of topology of trajectories, I 25
Gylden, II 3 method of, II 75
26
Halfstable equilibrium, I 51
Halfstable limit cycle, I 70
Halftrajectory, I
6970, 7879
Hard selfexcitation,
I 74, 99; II 32, 41, 44,
69;
III 72, 83,
91; IV 46, condition for, II 3132 definition of, I 71
49 example of, I
87 subharmonic resonance of order onehalf for,
III 8891
Harmonic Balance of Energy, Principle of, II 99,
104105;
III
38, 49, 53, 62
Harmonic motion,
I 106
Harmonic oscillator, I 86; III 128; IV 48 differential equation of, I 7 motion of, I
86 nondissipative, I 23
Harmonic solution of linear equation, II 33
Harmonics, III 2829, 35, 42, 5354, 62, 66 combination, III 2526, 28, 38, 46 of current,
III
32 of frequencies, III 48
Heegner's circuit, IV 3436
Helmholtz,
III
24
Hertz, I 5
Heterodyning, III 100
(see
also Beats)
Heteroparametric excitation, III 107108, 112, 115,
125126 conditions of, III 108, 123
Heteroparametric excitation  continued dependence on frequency and phase of parameter variation, III 117121 of dissipative system, III 121123 principal features of, III 119121
Heteroparametric generator, III
108
Heteroparametric machine of Mandelstam and Papalexi,
III 124
Heteroparametric selfexcitation, III 116
Heteroparametric oscillations, conditions for selfexcitation of, III 120
Heteroperiodic excitation, III 50, 107108 absence of, III 51
Heteroperiodic frequency, III 4953, 100101
Heteroperiodic oscillations, III 53, 87, 9092, 97,
102 definition of, III 47 existence of, III 4752 possibility of, III 58 stationary, III 100 synchronized with autoperiodic oscillation, III 55
Heteroperiodic solution, III 8586, 96
Heteroperiodic states, III 96 condition of, III 57 of nonlinear systems, III 57 stability of, III 57
Heteroperiodic variables, III 50
Hill equation, III 113114 definition of, III 109
HillMeissner equation, III 114,115, 117 topology of, III 114117 trajectories of, III 114, 117
"Hunting," I 101102
Hurwitz; see RouthHurwitz
Huygens, III 93
Hysteresis, III 61 of oscillation, II 2728 of resonance, III 72
Hysteresis cycle, II 28
III 4647 application of, II 8085 equations of, III 40
Impulseexcited oscillations, IV 4, 7, 1516, 43
Impulseexcited oscillator, phase trajectories of,
IV
5355
Indices of Poincar6, I 7577 theory of, I 7578
Index of modulation, III 123 critical value of, III 64 of singularities, I 7677, 117; III 97 of stepwise modulation, III 115 of trajectory, I 76
Instability, regions of, I 3
"Introduction to NonLinear Mechanics," I 3; II 1;
III 55
Island of trajectories, I 3032, 62, 119, 128129
Isochronism, condition of, I 42
Isochronous motion, I 42; II 12
Isochronous oscillations, II 5556, 68, 105
Isochronous system, II 37;
III
17
Isoolines application of method of, I 108 definition of, I 20 equation of, I 121 method of, I 2021, 105, 113
Jacobian, II 6
I IIIIIiilii
Jump, III 65, 7273, 91 discontinuous, II 32; IV
3233, 36, 4142, 4546 quasidiscontinuous, III 72; IV 10, 29
Kaden, III 104
Kaidanowski, IV 40, 42
Kaufmann's criterion for stability of equilibrium,
I 61
Kennard, Prof. E.H., I 6
Kryloff and Bogoliuboff, I 23, 67; II 1, 3, 47,
7576, 100 analytical method of, I 4 argument of, III 60 condition of selfexcitation, III 64 method of equivalent linearization of, II notation of, III 6, 15
3, quasilinear method of, III 83 quasilinear theory of, I 4; II 100, 110; III 1, 10,
83, 94 theory of first approximation of, II 49, 5254, 75;
III 10, 41, 75 treatise of, III 3, 40, 55
Lagrange method of variation of constants of, II 3, 49, 99 theorem of, 1 29
Lagrangian equation,
III
126127
Langmuir, I
5
LeCorbeiller, Ph., I 114; IV
1
Lefschetz, Prof. S., I 4, 6; II 49
"Les m6thodes nouvelles de la m6canique c6leste," I 2
Levenson, M., I 6
Levinson, N., I 70, 105, 108109; IV 2
Liapounoff, I 4; II 39 criteria of stability of, I
4849;
III 6364 application of, I 4955 equation of first approximation of, I 114; II 42,
60, 64 method of, I 49, 56, 105 stability in theorem of, I 3, 29,
49,
5155, 61
78 advantage of, I 55 proof of, I 49, 5155 treatise of, II 27
Lienard, I 105; II 47;
IV 2 analysis of, I 4 approximation method of, I 111 conditions of Cartan, II 48 cycle of, IV 3839 equation of, I 107, 109110 method of, I 106108 plane of, I 107115 limit cycles in, I 113115 qualitative analysis of, IV 38 theory of, I 107
Limit cycle, I 9092 II 35, 38,
100; 11I 54, 78, 9697 amplitude of, II 32 analytical examples of,
I
6266 bifurcation of,
I
87 branch points of,
I
88 coalescence of,
I
7273;
II
32
43, 48,
65, 9495, condition for stationary oscillation on, II 66 definition of, I 23, 62 degenerate, I 70 disintegration of, I 87 equation of, II
46 existence of, I 99; II 4748, 64, 6669, 9095;
III 100 condition for, II 35, 39, 42, 45, 65, 73, 93 criteria for, proof of, I 79
1 7577 geometrical theory of, I 6386 halfstable, I 70 in case of polynomial characteristic, II 7274 in clock, 1 6768
Van der Pol plane, I 113115 inwardly stable, I 70
Limit cycle
motion on, II 37 nature of, I 99 of first kind, I 125,
130 definition of, I
116117 of Poincar6, I 6286 of second kind, I 123,
125, 129130 definition of, I
116117 of thermionic generator, II 2425 outwardly stable, I 70 physical examples of, I
6668 piecewise analytic,
IV 28, 4546, 49, 55 possibility of, II 63 properties of, I 75, 105 representation of, II 37 stability of, I 62, 6466,
6873, 75, 85,
9394,
9899, 102; II
27, 3132, 3537, 39, 41,
66, 6972;
III 97, 101 stable, IV 35, 37, 55 systems with several, II 6669
4345, theorem of, I 7275; II 71'72 theorem of stability of, I 70 topology of trajectories in presence of, I 6875
Lindstedt, II 3 method of, I 5; II 76 method of Gylden and, II 75
Linear approximation,
I
1, 22, 100
Linear circuit characteristic equation of, III 21 theory of, III 47, 11
Linear damping, II 5962
Linear differential equation, I 1
Cauchy's theorem of existence of, I 2 with constant coefficients, I
1
Linear dissipative circuit with constant parameters,
III 10
Linear equation, I 80; II
23, 5, 45, 52; IV 27, 51 approximate, I 100 canonical form of, I 4244 equivalent, II 101102 harmonic solution of, II 33
Mathieu, III 124
MathieuHill, III 108 of first approximation, I 49 system of, I 83 with constant coefficients, I 1
Linear parameter, III 17
Linear resonance, III 1, 69
Linear system, I 23, 28, 67;
II
4; 27, 36 phase trajectories of, I 723
RouthHurwitz theorem for, I 3 with several degrees of freedom, III 30
Linearized differential equations, I 67, 87
Linear dissipative system with constant parameters,
III 10
Linearized equation, I 67, 87, 100 equivalent, II 110
Linearized motion, I 27
Ludeke, III 73
Mandelstam, I
2, 4; III
113 conditions of, IV 1516, 21, 24, 27, 32, 39, 4142,
57
Mandelstam and Papalexi, III 103, 107108, 124, 129 conditions of stability of, III
82 discontinuous theory of relaxation oscillations of,
IV3 heteroparametric machine of, III 124 method of, III 41, 75 school of, I 4; III 1, 75, 88 theory of, III 1
Marconi, I
5
Mathieu equation,
III
109110, 112114, 124, 127128 definition of, III 08 linear, III 124 parameters of, III 127 stable and unstable regions of, III 111112
1II0I
I WiNNAW
ul um
I m
Iahieu function, III 109, 113
1lathieu1ill equation,
III
108109, 113114, 124125 linear,
III 108 stability of, III 113 subharmonic resonance on basis of,
III 125, 129
Maxwell, theory of,
I
5
Mechanical system,
I
99,
124; III 710; IV 37 differential equation of, III 9 parameter of, IV 37 selfexcited oscillations of, I 99102 stability of,
I
102
Meissner, III 114 (see also HillMeissner equation)
Melde,
III 128 experiment of,
III
107
Migulin,
III 129
Modulation critical value of index of, III 64 index of, III 123 index of stepwise, III 115
Mller, III 93
Motion aperiodic, I 3, 36, 48 aperiodic damped, I 106 equation of, I 14 phase trajectories of,
I
1819 asymptotic, I 3, 28,
3031,
36 definition of, I 16 of pendulum, I 16;
IV 62 asynchronous, I
125 damped oscillatory, equation of, I 13 differential equation of, I 11 discontinuous periodic, IV 3 harmonic, I 106 in neighborhood of equilibrium,
I
40 in neighborhood of focal point,
I
13 in vicinity of equilibrium point, I 29 in vicinity of saddle point, I 26 isochronous,
I
42;
II
12 laws of, I
23 linearized, I 27
Method of Small, I 1, 66 nonisochronous, II 38 nonstationary, I 40 of conservative system, I 42 of elastically constrained currentcarrying conductor, I 3437 of harmonic oscillator, I 86 of pendulum, I
1415, 2930,
100,
118, 123 of representative point,
I
10 of rotating pendulum, I
3739 of synchronous motor, differential equation of,
I 126 of system with one degree of freedom, I 82 on limit cycle, II 37 oscillatory dan1ped,
I
48, 106 phase trajectories of,
I 1618 periodic, I 3, 26, 31, 85,
105 discontinuous, IV 3 in conservative system, I 6263 stability of,
I
8286 periodic stationary, I 63 quasiisochronous, frequency of, II 83 stable, II 21 stability of, I 24, 4142, 85; II 21; III 111112,
125 in neighborhood of singular point, I 4041 in vicinity of critical value of parameter,
I 3334 in vicinity of equilibrium point,
I 24, 26 of harmonic oscillator, I 86 stationary, I 23, 4041; II 14, 24;
III
54 stability of,
II
27;
III
110 stationary periodic, I
61; IV 55 unidimensional real, I 8 unstable, I
2829, 37 equation of,
I 13 phase trajectories of,
I
1416
M~ltiperiodic system equivalent linearization for, III 2630 equivalent parameters of,
III 27 resonance in, III 23
Multiply degenerate system, IV 3036
Multivibrator of AbrahamBloch, IV 3034
RC,
IV
2224
Negative criterion of Bendixson, I
75, 7778; III
97
Negative damping, I 105; II
4445, 63, 65 definition of,
I
21 examples of,
I
2122
Nodal point,
I 1819, 22, 4546, 59, 69, 7677, 8182,
101103, 114 127;
III
97, 99,
101; IV 17, 27, definition of, I 13 criterion for existence of, I 13 stability of, I
13, 45, 4748, 58, 6061;
II 68;
III 63 theorem for occurrence of,
I 45
Nonconservative system,
I
17, 23, 106 cylindrical phase trajectories of,
I 119121 existence of periodic solutions in, II 9095 with nonlinear variable damping, II 6365
Nondegenerate equation, IV 59
Nondegenerate system,
IV
6364
Nondissipative circuit, differential equation of,
III 115
Nondissipative harmonic oscillator, I 23
Nondissipative system,
III
20
Nonisochronous motion, II 38
Nonisochronous oscillation,
II 5758
Nonisochronous system, II 106; III 23
Nonlinear circuit, selfexcitation of, I 1
Nonlinear conductors,
I
1;
II
108109
Nonlinear conservative system, I 23, 128 behavior of, I 3233 definition of, I 24 examples of,
II
5658 general properties of, I 24 phase trajectories of, I 2439 theory of first approximation applied to,
II
5556 with cubic term,
II 8889
Nonlinear coupling, III 127 autoparametric, III 127
Nonlinear damping,
I 5
Nonlinear differential equation, I 1, 24, 63, 66, 75,
85; II
1, 3, 7, 67, 82, 100;
III
24; IV 37, 51 conditions for periodicity of solutions of, II 419 exact solutions of,
II
1 of dissipative type, II
45 of oscillating circuit, II 28 of second order, II 24 of selfexcited oscillations, I 66 of torsional oscillation of shaft, II 57
Nonlinear dissipative damping, II 5862, 106107
Nonlinear dissipative system, II 48
Nonlinear equation, I
1, 118;
II
5, 4748, 108 of oscillation, II 58 stability of, I 50, 52
Nonlinear equivalent parameter, III 29
Nonlinear mechanical vibrations, I 1
Nonlinear nonconservative system, I 23, 40, 63, 105 autonomous, I 62 higher approximations for, II 8995
Nonlinear oscillation, I 100; II 3, 99 selfexcited, II 63
Nonlinear oscillatory system, III 24
Nonlinear parameter,
II
109;
III
17, 61,
Nonlinear resonance, I 4; III 1129 external,
III 5374 stability of III
65,
72 internal, III
36 undamped,
III
73
Nonlinear restoring force,
II
105109
mml0
i11RI
Nonlinear selfexcited oscillations, II 63
Nonlinear selfexcited system, III 103
Nonlinear system, I 23, 42, 55; III 36, 39 autoperiodic state of, III 47 canonical form of, I 52 characteristic equation for, III 20 criteria of stability for, I 3, 55 frequency of selfexcited, III 17 heteroperiodic and autoperiodic states of,
III 4752, 57 periodic solution of, II 4, 6 stability of, I 49, 52 with one degree of freedom, III 30 with several degrees of freedom, III 3031
Nonlinear variable damping, 1 105; II 4445, 6365 nonconservative systems with, II 6365
Nonresonance, III 23, 42
Nonresonant oscillations, III 27
Nonresonant selfexcitation of quasilinear system,
II 1921
Nonresonant system, III 29, 33, 55
Nonstationary motion, I 40
Ordinary point, I 1214, 20 definition of, I 11
Ordinary value of parameter, I 32
Oscillating system, experiments with, III 60
Oscillation hysteresis, II 2728
Oscillations autoperiodic, III 53, 97, 100, 102 amplitude of, III 102 conditions for stability of, III 58 definition of, III 47 existence of, III 4752 selfexcitation of, III 52 stability of, III 58 stability of stationary, III 56 synchronized with heteroperiodic oscillation,
111
55
synchronous, III 56 combination, III 47 electrical circuit containing iron core, II 58 in circuit containing saturated core, II 108 heteroparametric, conditions for selfexcitation of,
III 120 heteroperiodic, III 53, 87, 9092, 97, 102 definition of, III 47 existence of, III 4752 possibility of, III
53 stationary, III
100 synchronized with autoperiodic oscillation,
III
55 impulseexcited, IV 4, 7, 1516, 43 in stationary state, III 11 isochronous, II 5556, 68,
105 maintained by periodic impulses, IV 4355 nonisochronous, II 5758 nonlinear, I 100; II 3, 99 selfexcited, II 63 nonlinear equation of, II 58 nonresonant, III 27 of conservative system, II 80 of limitcycle type, IV 43 of pendulum, I 1 of synchronous motor, 1 124130 quasidiscontinuous, IV 28,
37 quasiharmonic, II 49 quasilinear, I 128 II 24; IV 13 frequency of, II
3
relaxation; see Relaxation oscillations selfexcitation of,
I 5859, 66 selfexcited; see Selfexcited oscillations stability of, III
3637, 84, 124; IV 52 stationary,
6667,
71; III 15,
21, on limit cycle, I166 subharmonic, III
67, 9092 frequency of, III 88 selfexcitation of,
III
107 synchronized, stable condition of, III
36
Oscillator coupled electronic, stability of, II 24 damped, differential equation of, I 16
Oscillator  continued electrontube, III 48 harmonic, I 86; II 128; IV 48 differential equations of, I
7 motion of, I 86 nondissipative, I 23 impulseexcited, phase trajectories of, IV 5355
Oscillatory damped motion, I 48 106 phase trajectories of, I 1616
Oscillatory parameter, III 60; IV 5, 56, 61
Oscillatory system, I 68; III 23 nonlinear, III 24 resonance of, III 23
Papalexi, I 2, 4 (see
also
Mandelstam and Papalexi)
Parameter, III 36, 8687, 98, 103, 111; IV 10, 13 bifurcation values of, I 87 characteristic, III 75 constant, II 67; III 4, 122; IV 55 linear dissipative system with, III 10 constant linear, III
69 critical value of, I
3135, 39, 87, 97, 104;
II 2627, 31,
6970 dissipative, III
21; IV 16 equivalent, II
102105, 107; III
3335, 49, 62, 66,
6869 critical value of, II 100; III 30 definition of, II 100101; III 38 determination of, II 99102 nonlinear, III
29 of multiperiodic system, III 27 finite, IV 60 fixed, III 81 large values of, I 4, 111113; IV 2 linear, III 17 method of small, II 132 nonlinear, II 109; III 17, 61, of circuit, III 14; IV 37 of Mathieu equation, III 127 of mechanical system, IV 37 of system, III 23; IV
46 ordinary value of, I
32 oscillatory, III 60;
IV 5, 56, 61 parasitic, IV 7 effect on stationary states of dynamical systems,
IV 5665
existence of, IV 60 influence on equilibrium of dynamical systems,
IV 5860 periodic variation of, III 60, 74, 108 small values of, I 85, 128; II 38, 50, 53; III 38,
40, 48, 79; IV 2, 58 variation of, I
9192; III
36, 62, 107, 117, 121,
124125, 128129; IV
35
Parametric excitation, I
4;
III 1, 6065, 107129 definition of, III 60 of critically damped or overdamped circuit,
III 123
Parametric selfexcitation,
III
113
Parasitic capacity, IV 57
Parasitic parameter, IV
7 effect on stationary states of dynaical systems,
Iv 5665
existence of, IV 60 influence on equilibrium of dynamical systems,
Iv
5860
Pendulum,
III 112; IV 65 as example of nonlinear conservative system,
11 5657 asymptotic motion of, I 16; IV 62 differential equation of,
II
56 elastic, III 126 energy of, I 26 frequency of, III 94
Froude's, I 2122, 67; II 4446; IV
37 mathematical, IV 46 mechanical, III 94 motion of, I 2930, 100, 118, 123 in neighborhood of unstable equilibrium, I 1415 of clock, IV
44 oscillations of,
I 1
Periodic coefficients, III 81, 114 differential equations with, III 2, 56, 74, 108112,
124125, 127
t
i i3iI lIIIE II
Periodic impulses, oscillation maintained by, IV
4355
Periodic motion, I 3, 26, 31,
39, 4142, 6263, 7778,
85, 105 discontinuous, IV 3 in conservative system, I 6263 stability of,
I 8286
Periodic nonresonant excitation, III 4144
Periodic solution, I 66, 79, 82, 84, 86, 122123;
II 12, 7, 12, 1819, 2526,
95;
III 125 analytical, IV 3 condition for stability of, II 8082 existence of,
I
105115, 121;
II 1011, 86,
90
condition for,
II
18
geometrical analysis of, I 105115 proof of, I 105 of degenerate system of first order, IV 19 of nonlinear problem, II
6 of nonlinear system,
II 4, 6 of quasilinear equation,
II 87; III 7880 stability of,
II 1924
Periodicity,
I
116;
II
2, 13 condition of,
I 122, 129;
II
419
Perturbation, II 50; III 70, 80, 125 in phase angle, III 63 of amplitude,
III 63
Perturbation method, II 15, 19
Perturbation term,
II
51
Perturbation variables, III 64
43
Phase angle, I 7; II 4, 25, 38, 82; III 6265, 120 critical, III 120 equilibrium, III 106 of ripple,
III 118119 perturbation in, III 63
Phase line,
IV
8, 17, 21,
Phase plane,
I 78,
1011, 14,
1617, 2122, 2430,
32, 36, 43, 4547
, 263, 6869, 7172
7475, 77,
80o
82,
11 67, 1 ,
353
,105,
116117, 11?;
4041, 64; III
7 , 96,
100101, 114, 116, 118, 122; IV 2, 58, 12,
1415, 17, 2526, 2829, 33, 4346, 53,
62 definition of, I 78
Phase space,
I
2, cylindrical, I 116130 unidimensional, IV
8, 21
Phase trajectories, I 23, 723, 4041, 43,
4546, 62,
90, 94, 97, 102,
105107, 109110, 113114, 11711 , 125,
128;
II 33, 3536, 64; III 78, 96, 101, 112, 116118,
122; IV 2, 4, 6, 1214, 17, 1920, 22, 2627,
4244, 46, 4849, 55, 6263 analytical, IV 3, 31, analytical method of topology of,
I 2629 behavior of, in neighborhood of singularities,
I 1214
Cauchy's theorem of,
I
11, continuum of,
I
62, 128; IV 27 continuum of closed,
I
1213, 62;
II
67 cylindrical of conservative system,
I
118119 of nonconservative system,
I
119121 definition of, I 8 degenerate,
I
12 equations of, I 66, 88, 128 existence of closed, I 21, 108113, 129 graphical method of topology of, I 2526 index of,
I
76
I
6875
islands of, I 3032, 62, 119, 128129 of aperiodic damped motion, I 1819 of HillMeissner equation, III 114, 117 of impulseexcited oscillator, IV 5355 of limitcycle type, IV 22 of linear system, I 723 of nonlinear conservative system, I 2439
.of oscillatory damped motion,
I 1618
Phase trajectories  continued of second kind, I 121123, 129 of unstable motion,
I
1416 of Van der Pol equation, III 100 proper, I 12 sink for, I 69,
79 solution of differential equations represented by,
III 2 source of, I
69, 78 topology of,
II 37;
III
100; IV 26 analytical method of, I 2629 graphical method of, I 2526 in neighborhood of singular points,
I
2529 in phase plane,
I
2932 in presence of singularities and limit cycles,
I
6875
Phase velocity,
I 9, 1617, 24 definition of, I 9
Piecewise analytic curves,
III
117;
IV
2,
48
Piecewise analytic cycle, IV 22, 24, 42, 45
Piecewise analytic limit cycle, IV 28, 4546, 49, 55
Piecewise analytic representation of phenomenon,
IV'6
Piecewise analytic spiral,
III
122; IV 28, 5455
Piecewise analytic trajectory, III 117; IV 17, 33, 37,
43, 49, 54
Poincar6, H.,
I
4, 34, 38;
II
1, 15;
III
1, 54, 79,
96, 107108 analytical method of, I 4 87, 99, 104105; II 432 bifurcation theory of, I 7104 classification of singularities according to, I 42,
47 condition of, II 24 criteria of, I 105; II 31 criteria of stability of Liapounoff and, III 6364 curve of contacts of, I
75 equations of, II 53 expansions of,
II
7 functions of, II 13, 35 generating solution of, III 102 indices of, I 7577 limit cycles of, I 6286 method of, I 5; II 132, 38, 50, 96; III
41, applied to systems with several degrees of freedom,
II
14 notation of, II 20 quantitative method of approximation,
I 2 research of, I 2 rule for ascertaining stability of motion in vicinity of critical value of parameter, I 3334 theorems of indices, I 7677 theory of,
I
32, 63,
9394, 117;
II
23,
7,
28, 35,
41,
44,
66, 7592, 113, 125126;
III 41,
44, topological methods of, I 2; II 69 variational equations of, III 80
Poisson, II
49 method of, II 50 application of, II 52
Positive damping,
1
18, 105;
II
44, 63
Proper trajectory, I 12
Quadratic damping,
II 6062, 106
Qualitative analysis of
Lienard, IV 38
Qualitative methods, II 1
Quantitative method of approximations, I 23;
II 12 advantage of, I
3
Quasidegenerate equation, IV 1112
Quasidegenerate system, IV 1011, 65
Quasidiscontinuity, IV 39, 43
Quasidiscontinuous jump, III 72; IV 10, 29
Quasidiscontinuous oscillation, IV 28, 37
Quasidiscontinuous relaxation oscillation, IV 35
Quasidiscontinuous stationary relaxation oscillation,
IV 12
Quasidiscontinuous solution of differential equations of second order, IV 38
Quasidiscontinuous timing of electronic switch, IV 52
Quasiharmonic oscillation, II 49
EMEM1111=11MUNME1
I
019
Quasiharmonic theory,
II
59
Quasiisochronous motion, frequency of, II 83
Quasiisochronous system, III
21
Quasilinear equation, II 33, 49, 52, 55 58, 66, 80,
85,
89, 96, 99102, 105; III
65, 66;
IV 2, 37 definition of, II 2 of system with external excitation III 41
87; III 7680 with forcing term, periodic solutions of, III 7880
Quasilinear method of Kryloff and Bogoliuboff,
III
83
Quasilinear oscillations,
1
128;
II
24; IV 13 frequency of, II 83
Quasilinear system, II 2627, 95, 99, 103; III 27,
30, 53 autonomous, III 41 bifurcation theory for, II
2627 condition of resonance of, III 23 external periodic excitation of, III
4152
KryloffBogoliuboff theory of, III 41 method of equivalent linearization applied to steady state'of, III 1014 applied to transient state of, III 1419 nonresonant external excitation of, III 4647 nonresonant selfexcitation of, III 1921 of differential equations, II 14 resonance in, III 23 resonant selfexcitation of, III 2123 selfexcitation of,
III
41 with several degrees of freedom, III
323, 26 with several frequendies, III 3740
Quasilinear theory of Kryloff and Bogoliuboff,
I 4;
II
100, 110;
III 1, 10, 83, 94
Quasilinearity, condition of, II 29
Rayleigh, Lord,
III
93, 103, 107 equation of, II
4445, 4748, 65 experiments with oscillating systems, III 60
RCmultivibrator, IV 2224
RC oscillations in thermionic circuits,
I
111
References, I 131133; II 112113; III 130132;
Iv 6667
Reich, H.J., III 25
Relaxation oscillations, I 68, 128, 130; IV 165 definition of, IV 1, 4 discontinuous stationary, IV 12 discontinuous theory of, IV 3, 818, 2022, 25, 29,
56
7 examples of,
I
97, 115 mechanical, IV 3742 quasidiscontinuous, IV 35 quasidiscontinuous stationary, IV 12 stability of, IV 6365 theory of, I 4
62, 69, 73, 94, 112, 115, 125; II 38, 40;
III 116117, 119;
IV 6, 2122, 24, 2628,
3233, 38, 4142, 48, 54, 6263, 65 motion of, I 10
Resonance external, III 5374 definition of, III 1 fractionalorder, III 53, 55, 5761, 125 in in quasilinear system, III 23 internal, III 33, 36, 53 definition of, III
1, 31 of order one, III 3637 linear, III 1, 69 nonlinear, I 4; III 1129 external, III 5374 internal,
III
36 undamped, III
73 of order n, III 75 of oscillatory system, III 23 of quasilinear system, condition of, III 23 subharmonic;
see
Subharmonic resonance
Resonance hysteresis, III 72
Resonant selfexcitation of quasilinear system,
III
2123
Resonant system, III 33, 5357
Restitution, coefficient of, IV
3,
57
Richardson, Dean R.G.D., I 56
Ripple,
III
111112, 121123 capacity, III 116 frequency of,
III
115, 117, 119120 phase angle of,
III 118119 rectangular, III
114115, 117; IV 2 timing of, III 114
RouthHurwitz criteria of, II 22; IV 58, 60 theorem for linear system, I 3
Saddle point, 1 23,
2831, 3436,
3839, 41, 4648,
5559, 6162, 7677, 81
2, 127;
III 97, 99, 101; IV 17, 27, 31, 3536, 62,
64 criterion for existence of, I 13 definition of, I 12 example of,
I 16 motion in negative, IV 6465 positive, IV 6465 theorem for occurrence of,
I
46
Saturation voltage, II 29; III 95
Savart; see BiotSavart law
Secular terms,
II
7,
13, 25, 51,
87 appearance of, II
13 condition for absence of, II 8889, 9293 condition for elimination of, II 97 definition of, II 2 effect of, II 49 elimination of, II
3, 9294,
9798 in solutions by series expansion,
II
4952 presence of, II 52
Sekerska, III
128
Selfexcitation, I 5859, 73; II
25, III 3031, 60,
112, 126 asynchronous, III 51 autoparametric, III 127, 129 autoperiodic, III 50, 52 condition of, I
80, 99;
II
39, 7071; III
16, 1819,
21,
6364, 68,
108, 123 critical value of, III 49 disappearance of, I 74 existence of,
III
44, 120121 hard, I 74, 99; II
32, 41, 44, 69;
III
72, 83, 91;
IV 46, 49 condition for, II 3132 definition of, I 71 example of, I 87 subharmonic resonance of order onehalf for,
111 8891 heteroparametric, III
116 lack of, III
37, 43, 46, 90 nonresonant, of quasilinear system,
III
1921 occurrence of, IV 37 of autoperiodic oscillation, III 52 of autoperiodic state, III 60 of circuit, IV53 of clock, IV 46, 49 of electromechanical system I
99, 102104 of electronic circuits, I 66 of electrontube circuit, III 107 of equivalent linearized system, III 30 of heteroparametric oscillation, III 120 of nonlinear circuit, I 1 of oscillation, I 5859, 66 of quasilinear systems,
III
41 of shunt generator, II 71 of simple circuit, III 1419 of subharmonic oscillations, III 107 of system, II 69; III 86, 90 of thermionic circuits, I 87; II 38 of thermionic generators, I
9599;
II
2732; III
45 parametric, III 113 point of, III 92 possibility of, III 59,
122; IV 23, 35 prevention of, II 4344; III 51 resonant, of quasilinear system,
III
2123 condition for, II 3031; III 5152 definition of, I 71 example of, I 92, 102 subharmonic resonance of order onehalf for,
III 8387 zone of, III 49
Selfexcited oscillations, 1 6668, 97, 103; II 41,
4445, 48; II11
28; IV 1, 35, 55 amplitude of, 1 104 existence of, II 47 frequency of, III 13 nonlinear, II 63 nonlinear differential equation of, I 66 of electromechanical system, I 99, 102104 of mechanical system, I 99102 of nonlinear circuit, I 1 stationary, III 1314; IV 43
Selfexcited state, III
12
Selfexcited system, III
46,
92 asynchronous action on, III
52 nonlinear, III 103 frequency of, III 17
Selfexcited thermionic generator, III 45
Separatrix, I
3,
29, 3132, 3536, 3839, 62, 119,
123, 128129; III 101; IV 62 bifurcation of limit cycles from, 1 87 equation of, I 3536, 38 of second kind, I 123 stable,
I
41; IV 63, 65
Series generator, I 102104 parallel operation of, I 7982
Shohat, J.A., I 6; 38
Shottky,
I 5
Shunt generator, selfexcitation of, II 71
6970,
72,
97, 116117, 119120, 123, 127;
II 39,
III
54, 9698; IV 14, 26, 31, 34 classification of, I 42, 47 coalescence of, I 127;
III
101102 coalescence with limit cycles, I 73 definition of, I 11 nature of, III 97 number of, I 31 occurrence of, III 63 of differential e uations, I 17; III 96 stability of, I 38,
91;
III
97, 102
I 2529, 6875
Singularity, I
2, 41, 61,
III
63, 87;
IV 3,
7879,
17, 64
121; almost unstable, I
41;
IV
62 classification of, I 42, 47 coalescence of, I 29,
, 3637,
39
II 42,
4546; distribution of, III 98102; IV 64 index of, I
7677, nature of, III 98102 of differential equation, I
3, 11 simple, I
29 stability of, I
38,
117; II 97
41,
69, 87,
9192, 103; II 32,
39, 41; II 100101; IV 35,
62, 64 criteria for, I 47 transition of, I 58, 88, 91 zone of, I 82
Sink, I 87;
III 100 for trajectories, I 69, 79
Smith, O.K., I 70, 105,
108109
Sommerfeld, IV 40
Source, I 87; III 100 of trajectories, I 69, 78
Stability,
I 3, 125, 127 absolute, zone of, I 61 analytical definition of, I 4142 conditions of, 1 55, 99; II 14; 2224; III 6972,
82,
8587, 89,
97,
99, 113;
IV 60 conditional,
I
61 criteria of, I 2223, 33; III 85 applied to nonlinear system, I 3, 55 of equilibrium, I
6061 of singularities, I 47 defined for motion in neighborhood of singularities,
140 definition of, I 4041 exchange of, I 34, 38 geometrical definition of, I 41 in sense of Liapounoff, I
4042, 4861; II 21;
111 78
Kaufmann's criteria of, I 61
Stability

continued
Liapounoff's criteria of, I 4849; III 6364 application of, I 4955 of autoperiodic oscillation, III 58 conditions for, III 58 of coupled electronic oscillators, II 24 of degenerate system, conditions for, IV 17 of electric arc, IV 6063 of equilibrium, I 3335, 4061, 71, 85, 87, 93, 127;
II
3536, 68;
III
106, 110 IV 21, 5865 according to Liapounoff, I
461
Kaufmann's criteria for, I
61 theorems for, I 29 of focal point, I 4748,
5760,
8889,
9192, 97,
99; 1 3132,
4143, 69 of heteroperiodic state, III 57 of limit cycle, I 62, 6466, 6873, 75, 85, 9394,
9899, 102; II 27, 3132, 3537, 39, 41, 4345,
66,
6972; III 97, 101 theorem of, I 70 of MathieuHill equation, III 113 of mechanical system, I 102 of motion, I 24, 4142, 85; II 21; III 111'12, 125 in neighborhood of singular point, I 4041 in vicinity of critical value of parameter,
I 3334 in vicinity of equilibrium points, I 24, 26 of harmonic oscillator, I 86 of nodal oint, I 13, 45, 4748, 58, 6061; II 68;
11 63 of nonlinear equation, I 50, 52 of nonlinear external resonance, III 65, 72 of nonlinear system, I
49,
52 of oscillations, III 3637, 84, 124; IV 52 of periodic motion, I 8286 of periodic solution, II 1924 conditions for, III 8082 of point of equilibrium, III
106 of position of equilibrium, I
9 of relaxation oscillations, IV 6365 of singular point, I 38, 91; III 97, 102 of singularity, I 38, 41, 69, 87,
9192, 103; II 32,
39, 41;
III 100101;
IV 35,
2,
64
of stationary autoperiodic oscillation, III 56 of stationary motion, II 27; III 110 of stationary state, III 44, 7071
Poincar6Liapounoff criteria of, III 6364 regions of, I 3
RouthHurwitz criteria for, II 22; IV 58, 60 threshold of, 1 127
Static friction, IV 4849 zone of, IV 49
Stationary motion, I 23, 4041;
II
14, 24;
III 54 stability of,
II
27;
III
110
Stationary periodic motion, I 61; IV 55
Stationary selfexcited oscillation, III 1314; IV 43
Stationary oscillation, I 6667, 71; III 15, 21, 35,
49, 55, 96 on limit cycle,
II 66
Stationary solution, I 64; II 14; III
65
Stationary state, III 70; IV 4344 effect of parasitic parameters on, IV 5665 of dynamical system, I 40; IV 5665 of motion, I 23, 67, 69 oscillation in, III 11 stability of, III 44,
7071
Stationary value, IV 5253
"Struggle for Life," I 68
Strutt, III 111114
Subharmonic oscillation,
III
67, 9092 frequency of, III 88 selfexcitation of, III 107
Subharmonic resonance, III 75 external, I 4; III 125 for underexcited system, III
87 internal,
I
4;
III
3035 of
nth
order, III 129 of order onehalf for hard selfexcitation,
III 8891 for soft selfexcitation,
II 8387 for underexcited system, III 8788 of order onethird, III 9192 on basis of MathieuHill equation, III 125, 129
1111161
Subharmonic resonance  continued on basis of theory of Poincar6, III 7592 phenomenon of, III 129
Subharmonic solutions, III 60
Subharmonics, III 2425
Superregenerative circuit, III 50
"Sur les courbes d6finies par une 6quation diff6rentielle," I 2
Synchronization, III 3536 of autoperiodic with heteroperiodic oscillation,
111
55
of frequencies, III 104, 106 zone of, III 36
Synchronized oscillations, stable condition of, III 36
Synchronous motor differential equation of, I 126 oscillations of, I 124130
System absolutely degenerate, IV 11, 62 autonomous, IV 44 definition of, I 10 nonlinear nonconservative, I 62 quasilinear, III 41 with one degree of freedom, 1 105 completely degenerate, IV 10 conservative, I 7, 23, 25, 38, 40, 42, 106 128;
II 27, 51, 55, 90; III 19, 114, 127126 cylindrical phase trajectories of, I 118119 equilibrium of, I 40 motion of, I 42 oscillations of, II 80
I 6263 points of equilibrium in, I 34 degenerate, II 14; IV 60, 62, 65 conditions for stability of, IV 17 of first order, periodic solutions of, IV 19 transition between continuous and discontinuous solutions of, IV 3536 with one degree of freedom, IV 1929 degeneration of, IV 61 dissipative, II 67, 73; IV 45 heteroparametric excitation of, III 121123 doubly degenerate, IV 17, 3334 dynamical, I 1011, 32, 51, 69, 82, 8788, 107 differential equation of, I 20 equilibrium of, IV 5860 stationary states of,
I 40; IV 5665 electrical,
III
710 electrodynamical,
I
87 electromechanical differential equation of, I 117118 selfexcitation of, I 99, 102104 selfexcited oscillations in, I 99 102104 equivalent linear, II 102;
III
33, 36 equivalent linearized, III 66 selfexcitation of, III 30 isochronous, II 37; III 17 linear, I 23, 28, 67; II 4; III 27,
36 phase trajectories of, I 723
RouthHurwitz theorem for, I 3 with several degrees of freedom, III 30 linear dissipative with constant parameters, III 10 mechanical, I 99, 124; III 710; IV 37 differential equation of, III 9 parameter of, IV 37 selfexcited Oscillations of, I 99102 stability of, I
102 multiperiodic equivalent linearization for,
III
2630 equivalent parameters of,
III
27 resonance in,
23 multiply degenerate, IV 3036 nonconservative, I
17, 23, 106 closed trajectories of second kind in,
I
121 cylindrical phase trajectories of,
I 119121 existence of periodic solutions in,
II 9095 with nonlinear variable damping, II 6365 nondegenerate, IV 6364 nondissipative, III 20 nonisochronous, II 106; III 23 nonlinear; see Nonlinear system nonlinear conservative; see Nonlinear conservative system nonlinear dissipative,
II
48
System 
continued
nonlinear nonconservative,
I
23, 40, 63, 105 autonomous,
I 62 higher approximations for,
II
8995 nonresonant,
III
29, 33, 35 of degenerate differential equations, IV 1618 of equations of first order, I 9 of first order, I 97 of linear equations, I 18, 23, 80, 83 oscillating, experiments with, III 60 oscillatory, I 68; III 23 nonlinear, III 24 resonance of, III 23 parameter of, III 23; IV 46 quasidegenerate, IV 1011, 65 quasiisochronous, III 21 quasilinear; see Quasilinear system resonant, III 33, 5357 selfexcitation of, II 69;
III
86, 90 selfexcited, III 46,
92 asynchronous action on, III
52 nonlinear, III
103 triply degenerate, IV 17, 34 underexcited, III 87 subharmonic resonance for, III 8788 with external excitation,
III 41 with external periodic excitation, III 75 with internal resonance, III 33 with more than one degree of freedom,
I
75
with nonlinear variable damping,
II
6365 with one degree of freedom,
I
10, 82; IV 47 conditions of periodicity for, II
414 differential equation describing, IV 2529 motion of, I 82 with several degrees of freedom, II 14
Poincar6 method applied to, II 14 with several limit cycles, II 6669 with two degrees of freedom, II 1419; IV 30 with variable damping, I 105
Theodorchik, K., III 103
"Theoretical Mechanics,"
I
1516
"Theory of Oscillations,"
I 34, 75; II
1, 3; IV 1
Thermionic circuits, I 101
RC oscillations in, I 111 selfexcitation of, I 87; II 38
Thermionic emission, I 5
Thermionic generator, I 5, 73;
II 3, 109111; IV 44 amplitude of oscillation in, I 67 frequency of, II 2425 limit cycle of, II 2425 selfexcitation of, I 9599; II 2732; III 45 condition for hard,
II 3132 condition for soft,
II
3031 selfexcited, III
45
Threshold, I
48; III
122;
IV
14, 27 critical, I 3, 11;
III
46, 125; IV 26 of stability, I 127
Threshold condition, E 82
Topological methods, I 7130 of Andronow and Witt, III 94 of Poincar6, II 69 of qualitative integration,
I
24 advantage of, I 2 limitations of, I 3
Topological representation, II 14
Topological structure of trajectories, IV 26
Topological study of trajectories of Van der Pol's equation, III 100
Topology of HillMeissner equation, III 114117 of phase trajectories, II 31; III 100; IV 26 analytical method of, I 2629 graphical method of, I 2526 in in phase plane, I 2932 in presence of singularities and limit cycles,
I 6875 of Van der Pol plane,
II 3538, 41
Trajectories; see Phase trajectories
Transition of singularities,
I 58
Underexcited system, III 87 subharmonic resonance for, III 8788
Unidimensional phase space, IV 8, 21
Unidimensional real motion, I 8
Uniqueness, Cauchy's theorem of, I 24
Van der Mark, I 68
Van der Pol, I
1, 4, 2021, II
47;
III
9394, 97, 104,
111113; IV 1 abbreviated equation of, II 54; III 44 analytical method of, I 4 differential equation of, III 9496 equation of, I 79, 85, 97, 103,
105, 108,
111, 113;
47, 53, 6364, 66, 83, 85; III 4546,
100; IV 23 phase trajectories of, III 100 method of, II 3348 plane,
I
113115; II
33, 3538,
41 limit cycles in, I 113115 solution, II 63; III 100102 theory of, I
1;
II
44, 66; III
44
Van der Pol  continued theory of performance of heart, I 68 variables of, II 35, 3738
Variable damping, I
105; II
47,
8283
Variation of constants, method of, II 3, 49, 99
Variational equations, I 83; II 19; III 3536, 6364,
70 of Poincar6, III 80
Velocity field,
I 11
Vincent, III 93
Vlasov, I
124125, 130 theory of, I 125
Volterra, V., examples of limit cycles,
I 68
Vortex point, I 23 2631
3336, 3839, 41, 6263,
76, 119; IV definition of, I 12
48
Weaver, Dr. W.W., I 6
Witt, I 4; II 14; III 94, 126127 (see also Andronow and Witt)
Wronskian, III 109
0
111101
ll
0
39080
02754 0621
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