Minorsky - Introduction to Non-Linear Mechanics - Part IV - Relaxation Oscillations - 1946.pdf
TO NON-LINEAR MECHANICS
N. MINORSKY, Ph.D.
DAVID TAYLOR MODEL BASIN
Captain H.E. Saunders, USN
Comdr. E.R. Tilburne, USN
E.H. Kennard, Ph.D.
Comdr. L.S. Chambers, USN
C .J. Wenzinger
HEAD AERONAUTICAL ENGINEER
Capt. R.A. Hinners, USN
D.F. Windenburg, Ph.D.
ENGINEERING AND DESIGN
Comdr. L.W. Shallenberg, USNR
HEAD MECHANICAL ENGINEER
M .C. Roemer
~-~3~rpap.- .p..l-31^ ll~~L1
The report on the introduction to non-linear mechanics as a whole falls into four major divisions.
Part I, published as David Taylor Model Basin Report
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 Kryloff-Bogoliuboff.
Part III, published as David Taylor Model Basin Report 558 under date of May 1946, deals with the complicated phenomena of non-linear 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
127. INTRODUCTORY REMARKS .......................
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;
131. CONDITIONS OF MANDELSTAM . ...........
132. REMARKS CONCERNING SYSTEMS OF DEGENERATE
DIFFERENTIAL EQUATIONS .......................
. .. .....
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. RC-MULTIVIBRATOR ..........................
136. SYSTEM WITH ONE DEGREE OF FREEDOM DESCRIBABLE BY
TWO DIFFERENTIAL EQUATIONS OF THE FIRST ORDER
MULTIPLY DEGENERATE SYSTEMS . ............
137. MULTIVIBRATOR OF ABRAHAM-BLOCH
138. HEEGNER'S CIRCUIT; ANALYTIC TRAJECTORIES . ...........
139. TRANSITION BETWEEN CONTINUOUS AND DISCONTINUOUS
SOLUTIONS OF DEGENERATE SYSTEMS . ...............
MECHANICAL RELAXATION OSCIILATIONS
140. INTRODUCTORY REMARKS .......................
141. QUALITATIVE ASPECTS OF A MECHANICAL
RELAXATION OSCILLATION .......................
142. MECHANICAL RELAXATION OSCILLATIONS
CAUSED BY NON-LINEAR FRICTION
OSCILLATIONS MAINTAINED BY PERIODIC IMPULSES. . ...
143. INTRODUCTORY REMARKS .......................
144. ELEMENTARY THEORY OF THE CLOCK . ...............
145. PHASE TRAJECTORIES IN THE PRESENCE OF COULOMB FRICTION
146. ELECTRON-TUBE OSCILLATOR WITH QUASI-DISCONTINUOUS
147. PHASE TRAJECTORIES OF AN IMPULSE-EXCITED OSCILLATOR
. . .
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 ....................
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.... . . . . . . . . . ..
.............. . . .. . . .
INTRODUCTION TO NON-LINEAR MECHANICS
The term relaxation oscillations, introduced by Van der Pol (I)t (2) and commonly used at present, generally designates self-excited oscillations exhibiting quasi-discontinuous features. Because of the importance of such oscillations in applications in connection with the so-called "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
s system is constrained by the stops
* 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
of this report.
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
immediately after the extinction of the discharge. The transition from V
is gradual, but the inverse transition from V to V
is quasi-discontinuous. A phenomenon of this kind is represented in the (V,t)-plane by a so-called "saw-tooth" curve.
It was shown in Section 37, Part I, that a similar situation exists for large values of the parameter
in the Van der Pol equation. Figure 37.1C
shows quasi-discontinuous 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 quasi-linear equation
[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
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 non-elastic impacts, the so-called 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.
One can imagine a discontinuous
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
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 quasi-discontinuous 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
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 quasi-discontinuous 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 impulse-excited oscillations those quasi-discontinuous oscillations in which these rapid changes are due to the action of certain external impulsive causes.
On this basis, the quasi-discontinuous oscillations of the container shown in Figure 127.1 are of a relaxation type, whereas the ball rebounding between the walls is an impulse-excited 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 quasi-discontinuous changes of certain variables, whereas in impulse-excited systems, on the contrary, the energy content changes abruptly.
Impulse-excited 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 quasi-discontinuous. 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 d-c 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
t + 0).
Let us assume that we open and close the switch S at some fre-
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
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
= 0 and assume that for t < t
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
+ Ri = E; rC
dV + V = E
[127.2] where E
0 for t
= constant for t
. With these assumptions one finds that
V = E 1
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
Noting that i
can take instead of V the variable i, and state that under the assumed ideal-
izations the functions
and ic undergo discontinuities
r ly. If one takes the plane of the variables (-j-,ic
, the process occurring at
= 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-
dt gent spiral. If, at a later instant t = t,, the switch is opened, another jump will oc-
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
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.
_ _^ _____ __~
It is important to emphasize once more that such a representation of a quasi-discontinuous 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.
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
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 phase-plane 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.
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
when one of the coefficients approaches zero. In an electrical problem, a
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. 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,
is that constant. By differentiating [128.2] we obtain
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
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
[128.1] rather than to drop this coefficient in the differential equation itself.
The solution of [128.1] is
1 e rlt
[128.4] where C
are the constants of integration and r
are the roots of the characteristic equation
+ br + k
If the initial conditions t
0, x = z
and i = io are given, one obtains
from which one obtains the values of
b b2 k
In this expression only one term is retained in the expansion of the square root, since a is small. This gives
If the values [128.6] of the constants and [128.8] of the approximate expressions for the roots r
are substituted in Equation [128.4], the approximate solution xl(t) of [128.1] is in the form x(t)
It is to be noted that the solution x
(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
Consider now the function
O(a,t) = zxl(a,t) -
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 <
when a -,0.
The expression for the derivative of this function is
For very small values of t the function
+ 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
(t) to the function :(t) when a
0 is not uniform. In other words, to a given a,
however small, one can always assign a value of t = t
such that = Xok +
Only in a very special case, when Xo- + o
= 0, does thi's non-uniformity 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 quasi-degenerate 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
(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 quasi-discontinuous 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
The solution of Equation [128.13] is
+ Ce a
where C is a constant of integration. One obtains finally
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
+ b-o 1 -
By forming the functions
S(k,t) = xl(k,t)
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
for which value 0(k,t) approaches the value x
129. IDEALIZATIONS IN PHYSICAL PROBLEMS
In applications, idealizations of quasi-degenerate systems as absolutely degenerate systems are frequently made, as was mentioned in Section
127. Thus, for example, the differential equation of the so-called (R,C)- circuit is usually written in the form
[129.1] where i is the current in the idealized circuit (L
0). The corresponding quasi-degenerate equation, with L very small, is
+ R_ + -i
With the solution of the absolutely degenerate equation [129.1] designated by
i(t) and that of the quasi-degenerate equation [129.2] designated by ij(t),
the functions i(t) and il(t) have the appearance shown in Figure 129.1.
Equation [129.1], with L = 0, the current starts from the point A and decays exponentially thereafter. For the quasi-degenerate 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 quasi-degenerate equation [129.2], with L
0, has a quasi-discontinuity
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
depicted in Figure 127.4 is governed by the full differential equation of the second
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',
... 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
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~
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 quasi-discontinuous 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 non-existence, 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 quasi-linear 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.
Critical points are the points at which the differential equation describing
a phenomenon in a certain domain ceases to describe it.
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
P(x,y) dy Q(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
) for which R(xi,yi) =
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
) = 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
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
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
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 impulse-excited 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
and Vare selected. In the preceding section
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
) 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
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 impulse-excited 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
Ai t to
= 0 [131.1] where (t
- 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 short-circuited so that its energy
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
The usefulness of the conditions of Mandelstam is limited only to relaxation oscillations proper.
In fact, in impulse-excited 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
differential equation of the n order can be reduced to a system of n differential equations of the first order by introducing the variables
dy dt 2
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 two-dimensional, that is, a phase plane, becomes uni-dimensional, 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
it permits obtaining satisfactory qualitative information in a great majority of important practical problems whose solutions are beyond the reach of existing analytical methods.
DEGENERATE SYSTEMS WITH ONE DEGREE OF FREEDOM
133. PERIODIC SOLUTIONS OF DEGENERATE SYSTEMS OF THE FIRST ORDER
A differential equation of the first order
obviously does not possess continuous analytic periodic solutions. Moreover, one can assert that if the function f(x) is single-valued, 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 single-valued. 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
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 i-axis. In this case the trajectory would be formed by stretches AB and CD of analytic arcs "closed" by the discontinuous stretches BC and DA.
The change from one branch of the function f(x) to the other one generally occurs at critical points
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 x-axis. This is not always so, as will be shown later. We will now illustrate this matter by the following two well-known examples.
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.
0 C D
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 non-linear empirical relation il
The differential equations of the circuit obviously are
= E; i
These equations reduce to the following differential equation of the first order
1 [E - V-
This equation is valid only when the discharge exists; during its extinction
0, and we have
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
through the non-linear conductor
across it. We will reserve the term
to mean an
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
= 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 uni-dimensional 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 V-axis. 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 =
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
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
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-
which means that "during" the jump the voltage V across the capacitor remains constant, that is, the jump occurs parallel to the il-axis along the stretch
BC. The process of extinction is thus "explained" on the basis of the discontinuous theory.
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 V-axis 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 limit-cycle 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 semi-intuitive physical argument. In still further examples it would be altogether impossible.
As a second example of a system which can be described by one differential equation of the first order, we shall investigate the so-called RCmultivibrator circuit shown in Figure 135.1. The electron tube V
2 is a non-* linear conductor characterized by the equation I
= q(e). The tube
V, appears here merely as a linear amplifier amplifying the potential difference ri, the feed-back voltage, between the points B and D and applying-the amplified voltage eg = kri to the grid of V
,, 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
-_ , _- 'i
1 11 1 W MION
(R + r)i
+ V = Rq(kri);
i = Cdt
These equations reduce to the equation
(R + r)
The critical point i
is given by the equation
(R + r)] = 0
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
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
= 0 and approaches zero monotonically as
S and assume that RS > R + r, without which, obviously, no self-excitation is possible. The curve V(i) represents the function RO(kri)
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 MN-axis parallel to M'N' and at a distance
(R + r) from the origin 0, the ordinates f(i) of this curve represent the
left-hand 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
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
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
point has reached the point B on the stretch AB, it has to jump dis-
continuously, and the second condition of Mandelstam specifies that the jump must occur parallel to the i-axis, since in such a case the energy stored in the capacitor does not change during the jump. The discontinuous stretch BC so tra-
versed ends at the point C. Here
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 neon-tube oscillator.
The difference between the neon-tube 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.
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 uni-dimensional 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;
e gV vl
Introducing the variables x = kri and y
- and differentiating these equa-
tions, one gets
dx y dt
y krLC kL (x)
The phase trajectories are given by the equation
dy dx x(x) krLCy
The point x
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)
Hence there exist two roots x
for which both and dt become infinite. According to our definition, these roots
z = +x are the
critical points of the system [136.2].
phase plane the values
points of this system. It is to be noted that at these thresholds the tangent to the phase trajectories
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
separate the regions df the phase plane in which the topological structure of trajectories is radically different, we shall simplify the
problem slightly without, however, in-
troducing any qualitative changes. Figure 136.2 represents the characteristic
the electron tube V
We shall exclude the critical
= +zx by drawing parallels
to the y-axis on both
each threshold so as to obtain narrow strips of width
will then be S(x)
> 0 and -L in the
inner and outer intervals, respectively.
This will cause no qualitative changes but will change slightly the form of the
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
appears as a saddle point for the trajectories of the inner interval (-x i
< +zx) and as either a focal point (if r < 2 I) or
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
to - 0
ydt = 0;
to'- O krC to - O
= 0 [136.6]
Applying these conditions to Equations [136.1], one obtains
[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
a curve similar to the curve V(i) of Figure
135.2a. By expressing the condition 0(x
) = 8(x
), one can find
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
-z to +x,. When z
2 is known, the second equation [136.7] permits determining y
for given x
1 and y
The representation of such a quasi-discontinuous oscillation in the phase plane of the variables (z,y) is shown in Figure
136.4. It is noted that the jump from (x,,y,) to (X
) 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
). 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 x-axis 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 limit-cycle 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
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-
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
in Figure 136.5. There are two spiral arcs
BC and DE corresponding to a relatively slow
motion of the electronic beam on the analytic trajectories of the outer interval. The quasi-discontinuous 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
across the inductance and current the current i
in the capacitor undergo quasi-discontinuities, 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 well-known 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.
MULTIPLY DEGENERATE SYSTEMS
137. MULTIVIBRATOR OF ABRAHAM-BLOCH*
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
= E; RI
-' i2 ri
where I' = O(e ) is the non-linear characteristic of the electron tubes V
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)--
+ Rr '(ri
dil di +
RrO'(rij)1Lj + (R + r)-
1. 0 ,
This device is described in Reference (15).
This system reduces to the following equations:
(R + r)
(R + r)
i [137.3] which are of the form
(R + r)-
) (R + r)
2 dil dt
) di dt
One notes the symmetry of Equations [137.3] with respect to the variables i and i
; this symmetry is due, of course, to the symmetry of the circuit.
The phase trajectories in the (i
)-plane are given by the equation
From the explicit values of the functions P and
Equations [137.3], one observes that the origin i] =
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
dQ di m
2(R + r)
Hence no closed analytic trajectories exist here. The nature of the singular point il
0 is determined by the equations of the first approximation.
If we let
= o =
= o =
S and if we assume that RrS > R
Equations [137.3] become
R + r .
; di dt
R + r .
(R + r)
] > 0
The characteristic equation of this system is
2(R + r)
[(R + r) + RrS][(R + r)
= 0 [137.8]
The origin (i, = i
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
2 begin to increase. On the other hand, from the form of the characteristic
= (ri), we know, see Section 135, that
0 when i -- .
] [137.9] which is initially positive, decreases monotonically when il and
increase and is negative when i
are very large and are equal. Hence there are certainly some values of il and i
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.
) = 0 defines a certain curve F
in the plane of the variables i
In order to simplify the notation, let us designate by x and y, the values of ii and i
, respectively, situated on the curve F shown in Figure 137.2.
The equation of this curve is then
(R + r)
A c d b
From the form of the functions ¢'(rx
and 0'(ryl), approximating the slope to the characteristic
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
) on the curve
, a jump must occur, and the point
) 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
(R + r)il;
(R + r)i
If we designate now the terminal coordinates of the jump by (x
), the conditions of Mandelstam obviously are
- --- YYYIIIYIYYIY III i
(R + r)x,
(R + r)x
) (R + r)y,
(R + r)y,
In these notations (xl,y
) represents the critical point on the curve F
from which the representative point is transferred discontinuously to some other point (x
) determined by these equations. Since there is generally a one-to-one correspondence between the points (x
,y) and (x
), we conclude that the locus of the points (x
) is another curve
2 which is also a closed curve symmetrical with respect to both variables il and i
since the circuit is entirely symmetrical.
The phase-plane representation of the behavior of the Abraham-Bloch 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
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 Abraham-Bloch multivibrator approaches a stationary symmetrical state, which is oriented along the bisector of the ii- and i
-axes and which consists of a continuous stretch a# followed by a jump #Q, then by another continuous trajectory 9e followed by another jump
which closes the "cycle." In this case, the cycle is uni-dimensional, 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 steady-state condition was aimed at, a simplification could be made in the differential equations by introducing the conditions of symmetry, that is, i
). With this simplification, one obtains a single differential equation of the first order, namely,
di dt R
(R + r) + Rr&'(ri) i
(R + r)
which reduces to the form
(R + r)]di = i dt
This is identical with Equation [135.2], the equation of an asymmetrical relaxation oscillation. We conclude, therefore, that the transient behavior of the Abraham-Bloch multivibrator is represented by a doubly degenerate system.
As far as the stationary condition described by one differential equation
[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 so-called 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
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.
With the positive directions indicated by arrows, the application of Kirchhoff's laws gives
( R I
= q(krll), as we have previously, and eliminating I, we obtain the following system of the second order:
R + r
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
0. Noting that 0'(0)
S is a maximum and that the function Ib'(krIl)l
c, the characteristic equation of the system
+ R + r
1 X +
It is noted that the singularity here is not a saddle point. Hence, it is either a nodal point, if the roots X, and
2 are real, or a focal point, if they are conjugate complex. In both cases the singularity is unstable if
R + r - RrkS
If the singularity is unstable, self-excitation from rest is possible. By approximating the experimental function
) 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 self-excited oscillations is well known.
Although the Heegner circuit does not produce quasi-discontinuous 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
, 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
be the resistance between the points B and E and
be that between E and D, where
= r and
/r = 8. Proceeding as we did in Section 138, we obtain the following system of equations:
dl _ dt
+ R -
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
and r = 0, we have Heegner's circuit, which produces continuous oscillations.
For sqme intermediate value of 8, the co-factor of (1 - 6) in the denominator of the second equation [139.1] may vanish, which means that-the otherwise continuous oscillation will undergo a dis-
continuous jump parallel to the I2-axis 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
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
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 quasi-discontinuous behavior describable by piecewise analytic trajectories.
These impact-maintained 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 quasi-discontinuous manner by special timing. We shall leave the investigation of impulse-excited systems to a later chapter and will investigate here a simple mechanical system of a pure relaxation type.
Let us consider a non-linear differential equation of the form
[140.1] where F(i) is a certain non-linear function of velocity which we shall identify with friction. If p is small, Equation [140.1] is of a quasi-linear type and can be discussed by the standard methods of Part II.
More specifically, for particular forms of the function f(x) self-excitation 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.
QUALITATIVE ASPECTS OF A MECHANICAL RELAXATION
Since a condition of degeneration, as previously shown, is essential for the appearance of quasi-discontinuous 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
[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.
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 quasi-discontinuous 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
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 quasi-discontinuity 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 quasi-discontinuity 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 NON-LINEAR 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 two-valued 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
[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
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
[142.2] where F(v)
)r] is the friction force, a non-linear function of the relative peripheral velocity v
= (Q -
Q is the angular velocity of the shaft K,
is the angular velocity of a for small departures 4, and
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
0). When v
0, a moves together with K, and 4 = 9. When a is not moving,
# = 0 and v = v
Dr. To the right of the point
the mass a and
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-
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
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
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
[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
' 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.
One can illustrate this also by putting # =
y in Equation [142.3],
dy dt cy r
From this expression it follows that
= c at the point D
since F'[(£ )r] = 0 at this point. At D, the representative point undergoes a discontinuity, the direction of which is parallel to the v-axis 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 non-linear 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 neon-tube circuit if the line
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
' = 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 non-linear differential equation. In both figures the discontinuous stretches DA and BC are determined by the condition of
Mandelstam, and so on. Plotted in
the (0,t)-plane, the curves of the mechanical relaxation oscillations observed by Chaikin and Kaidanowski have a typical "saw-tooth" appearance, see Figure 142.3, characterizing electrical relaxation oscillations of a similar nature.
Figure 142.3 t
This discussion emphasizes the features which all relaxation oscillations have in common.
OSCILLATIONS MAINTAINED BY PERIODIC IMPULSES
143. INTRODUCTORY REMARKS
We have defined "relaxation oscillations" as stationary self-excited oscillations exhibiting quasi-discontinuities 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 limit-cycle 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 so-called
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 quasi-discontinuous 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 quasi-discontinuous 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
+ 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 impulse-excited 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 impulse-excited 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:
An oscillatory dissipative system, for example, an ordinary pendulum, a torsional pendulum with a hair spring, etc.
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 electron-tube 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
' immediately before the second impulse will be v
d. Immediately after the second impulse, the velocity will be v
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,
which gives o
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 x-axis. 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
OA, is a monotonic function u(ro).
Since the spirals S form a continuous family, it is also obvious that the function u(r
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
izv constant and hence are represented by constant jumps a along the x-axis.
follows, therefore, that a steady-state condition will be attained when a radius vector ro
= OA is reached for which u(r
) = 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 limit-cycle 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
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
) 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 self-excitation 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,
In the phase-plane 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=
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
to explain the fact that a clock is a system with a hard self-excitation.
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.
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
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,
kx = + fo;
, these two equations are
+ aw o
when i <
when i >
Introducing the variables x, = x
a for the first equation [145.2] and
x + a for the second, one obtains two equations, namely,
wzX, = 0 when i <
wo2 = 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
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
~ the t-axis. At the point B the velocity i is equal to zero, and the amplitude is Ix
2al with respect to the t-axis. At the point B the change of equations takes place, and the "acting abscissa axis" is now t
displaced a distance -a from the t-axis. With respect to the t
-axis, the motion is again sinusoidal, so that the point C (at which x = 0) is at the same distance from the t
-axis as the point B was originally. With respect to the t-axis, however, the amplitude of the point C is x o =
- 2a. It is seen that with respect to the t-axis 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
y[145.4] when y < 0
a) when y > 0
Integrating these equations, one gets
= 1 when y < 0
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
, 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
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
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
It is thus seen that idealized
Coulomb friction is capable of being represented by piecewise analytic trajecto- ries formed by elliptic arcs of the two families
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 x-axis. At these points the curve has
B a continuous first, but a discontinuous second, derivative as may be expected from the fact that the friction force
o2 changes discontinuously at these points.
Instead of having a point of equilibrium,
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
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"
t, as we did on a number of occasions, Equations [145.3] can be reduced to the form
= 0 when
The trajectories of these equations are two families C
1 and C
2 of concentric circles whose centers are displaced on the x-axis 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
must be equal to or greater than 16f
, where fo is the Coulomb force. This means that a clock is a mechanism with a hard self-excitation. In other words, unless the condition stated above concerning the relation between h
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.
146. ELECTRON-TUBE OSCILLATOR WITH QUASI-DISCONTINUOUS GRID CONTROL
We will investigate in this section the behavior of an electron-tube 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 self-excited oscillations. The interesting feature of the analysis made by Andronow and Chaikin is that it permits investigating questions concerning the establishment and stability of self-excited 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
+ Rx +
) dt = 0
Differentiating and rearranging, we obtain
+ Ri +
[146.2] where la
= f(eg) is the non-linear 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 grid-voltage 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
= 0, and B, when I
= 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
V, ta XIa
x vJ c
n- - -^- ^p~-^- -- *. _~
51 that shown in Figure 145.1. Instead of being described by one non-linear 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
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
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,
therefore, that the quantities which are capable of varying discontinuously are
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
+ wox = 0
[146.4] where X
= x -
In this form the first equation
[146.3] and Equation [146.4] are two linear differential equations possessing focal points on the t-axis at a distance
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 electron-tube switch is off, which corresponds to the first equation [146.3]. During the first half-cycle the original amplitude zx, see Figure 146.3, will be reduced by damping and will become
It is noted that x
if referred to the t-axis. At the point
B the electron-tube current jumps suddenly to the value
and, as we saw, the new abscissa axis is now tj, displaced above the t-axis by the quantity I,.
With respect to this tl-axis, the initial amplitude is now x
I, and although during the following half-cycle 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 t-axis may be at a higher level than that of A if
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 quasi-discontinuous 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 t-axis and by
7 those referred to the ti-axis, one has
X l e
= xl e
If the amplitude reaches a stationary value, x,
x, = xo, which gives
This shows that a stationary amplitude x
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
In the (x,,
)-plane, see Figure
146.4, this relation is a straight line with
a slope tan a = e
and an intercept I,(1 + e
= A. On the other hand, if the oscillation is stationary, xi = x
, which represents the bisector of the angle between the x,- and x
-axes. The point M of intersection of these lines determines the stationary amplitude x
If the initial amplitude x
is not stationary, the subsequent amplitude will be x
' and the corrected amplitude will be x
' on the x
-axis to which the following amplitude x,' will correspond on the x
-axis, and its corrected value on the x,-axis will be x
By the interplay of the subsequent corrections it is seen that any non-stationary amplitude tends to approach the stationary value x
This argument is also valid when the initial non-stationary 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 x-axis by the quantity
will occur in a direction oppositeI
to that shown in Figure 146.3.
This will result in a reduction of
the subsequent amplitudes similar to that found in connection with
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 <
The self-excitation of the circuit occurs when the circuit is characterized by negative Coulomb
147. PHASE TRAJECTORIES OF AN IMPULSE-EXCITED 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
+ 2hi + w2x = 0
xz + 2hil
= 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"
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
Cbegins to describe the phenomenon again; this arc of the spiral tra-
-- 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
when the representative point moves on the lower arcs. For the subsequent radius vectors along the x-axis, we have the following obvious relations:
w ; r
7rh n7h 7rh
+ a)e- I
One obtains easily the following general expressions:
r, = a 1 + e
rk = rk
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
oo and s
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 self-excited 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 non-linear 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 non-linear characteristic, shown in Figure 146.2, as a discontinuous step-function 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 non-linear 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.
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 non-degenerate one. Thus, for example, if the non-degenerate 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
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
' 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
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)
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
L- + Ri
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 cathode-ray 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,
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
leads to a meaningless result. If, however, we make C
at the limit we obtain
[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.
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
+ ** + an-_z + an_l +
, and so on. The corresponding characteristic equation is ao n
The equilibrium is stable if the real parts of the roots of [149.2] are negative, which can be ascertained by the well-known criteria of Routh-Hurwitz:
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.
". 4 -
tj - I
If it is of the inductance type, the non-degenerate equation will be
+ aox + az
+ + an,_
+ ax = 0
If it is of the elastance type, it will be
+ - - - + an,_-
Differentiating this equation, we get
dt = 0
+ ax(n) +
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
+ a = 0
[149.6] ao n+1
+anA + a =0
They now have (n + 1) roots, whereas the original characteristic equation
[149.2] had only
roots. Since a is small, by our assumption, the old
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
We can write Equation [149.2], in terms of its roots, as
. . .
The new characteristic equation [149.6] of the "parasitic inductance" type can accordingly be written as
+ a n
Expanding the rignt-hand side of this equation and identifying it with Equation [149.6], we get
Applying a similar argument to Equation [149.7], we obtain
The sign of the new root
indicates its effect on the stability of equilibrium. The general integral of the new system is
+ C ne
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
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 non-degenerate system will be unstable although the corresponding degenerate system is stable. For a complete discussion, one must apply the Routh-Hurwitz 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
L V -
where i is the are current, V is the potential difference across the capacitor, and 0(i)
Va is the non-linear characteristic of the arc, see Figure
150.1. As was explained in Section 21, there are either three equilibrium
points (Points 1, 2, and 3) or one such o point (Point 3), as shown in Figure
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
of arc current, from the corresponding equilibrium values, the equations of the first approximation are
dv dt v
C dj dt v
L[150.2] where p = '(io) is the slope of the tangent to the characteristic 0b(io) at the point i = i
The characteristic equation of the system is
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
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, C-degeneration when C
0 and L-degeneration when L
In order to investigate the C-degeneration, it is convenient to write Equation [150.3] as
Ck + (
(R + p)
For the investigation of the L-degeneration we will write it as
+ p)X +
(R + p)
In this manner we shall be able to use the conclusions of Section 128. Putting
0 in Equation [150.4], we obtain
= 0 in Equation [150.5], we have
(R + p)
Let us consider the equilibrium at Point 2 of Figure 150.2. It is observed that in the C-degeneration 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
+ 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 a-degeneration, 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 a-degeneration 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.
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 non-degenerate system may become unstable in the case of L-degeneration, as follows from Equation [150.1] in which p <
Point 1 is stable both for the non-degenerate 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.
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
I + i = Ia;
L- + RI -
ri - Ld
0(kri) as before. Elimination of
and I between these equations results in the equations
L + L'
(R + r)i- V
Lkr'(kri) ; dV dt i
The conditions of equilibrium are obviously
i = 0 and RO(O)
0. On the
other hand, for small departures from equilibrium, we can write
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
Y dV i
= r + R(1 -
= L' + L(1 Skr)
The characteristic equation of the system [151.3] is
+ -6X +
1 = 0
and its roots are
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' =
0. Multiplying the first equation [151.3] by y, one has di
0, that is, i = -.
The second equation gives
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
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 quasi-degenerate.
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
0, the equilibrium is stable, however, as follows from Equation [151.7]. Since any physical system is only quasi-degenerate, 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__ _ _____~.._ ___~ ___~~_
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(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.
(21) "Effect of Small Parameters on the Stationary State of a
Dynamical System," by S. Chaikin, Journal of Technical Physics, USSR,
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.
Abraham-Bloch, multivibrator of, IV 30-34
Absolute stability, zone of, I 61
Analogies between electrical and mechanical systems,
Analytical definition of stability, I 41-42
Analytical impasse, IV 14
Analytical method, I 11, 111; II 1-111; III 75; IV 2 of
Kryloff and Bogoliuboff,
I 4 of Poincare, I 4, 87, 99, 104-105; II 4-32 of topology of phase trajectories,
26-29 of Van der Pol, I 4
Analytical periodic solutions, IV 3
Analytical trajectories, IV 3, 31, 33-34
Andronow, A., I 2-4, 63, 75; II
1, 3, 14;
Andronow and Witt presentation of, III 96 topological method of, III 94
Aperiodic damped motion, I 106 equation of, I 14 phase trajectories of,
Aperiodic motion, I 3, 36, 48
Appleton, III 93
Approximation equations of first, I 3, 49-52, 55-56,
94, 97, 99,
101, 103, 114, 120, 127; II 33-34,
39, 42, 145, 52-54, 59, 83, 90, 95, 102, 110;
III 11, 23, 27 29 31, 33, 3-38, 40-45, 53-57,
63, 67-69, 74;
IV 31, 6o-61, 4 linear, I 49 first, I 27; II 55, 95 III
17, 30, 32, 34-36, .49
improved first, II 76-0 90, 98; III
46-47 application of, II 80-65 equations of,
III 40 linear,
1, 22, 100 method, II 1 method of first, II 63-66, 88; III 26 method of Li'nard, I 111 of first order, III 41 of higher orders, II 3, 75-98; III 40 of zero order, III 41 definition of, II 56 order of, II 12 quantitative method of, I 2-3; II 1-2 advantage of, I 3 second, II 76 example of, II 94 theory of first, II 3, 37, 49-75, 99, 107; III 10 applied to non-linear conservative system,
Asymptotic motion, I 3, 28, 30-31, 36 definition of,
I 16 of pendulum,
16; IV 62
Asynchronous action conditions for, III 50 on self-excited system, III 52
Asynchronous excitation, II 100; III 47, 50
Asynchronous motion, I 125
Asynchronous quenching, II 100; III 47, 50-51
Asynchronous self-excitation, III 51
Autonomous equation, II 21
Autonomous system, IV 44 definition of, I 10 non-linear non-conservative, I 62 quasi-linear, III 41 with one degree of freedom, I 105
Autoparametric excitation, III 107-108, 126-129 condition for, III 126
Autoparametric non-linear coupling, III 127
Autoparametric self-excitation, III 127, 129
Autoperiodic excitation, III 49-50, 57
Autoperiodic frequency, III 49-51, 77, 94,
Autoperiodic oscillation, III 53, 97, 100, 102 amplitude of, III 102 conditions for stability of, III 58 definition of, III 47 existence of, III 47-52 self-excitation of, III 52 stability of, III 58 stability of stationary, III 56 synchronized with heteroperiodic oscillation, III 55 synchronous, III 56
Autoperiodic self-excitation, III 50, 52
Autoperiodic solution, III 58
Autoperiodic state, III 59 of non-linear systems, III 47 self-excitation of, III 60
Autoperiodic variables, III 50
Barkhausen, III 12
Barrier, II 43, 69
Beats, III 55, 93-94, 103,
106 application of first theorem of, I 79-82 criteria of, I 75, 77-79, 105; III 97; IV 31 first theorem of, I 75, 77-7d; III 97 negative criterion of, I 75, 77-78; 11I 97 second theorem of, I 75, 78-79; III 100-101
Bifurcation of limit cycles, I
Bifurcation point, I 31,
(see also Critical point)
Bifurcation theory, IV 35 for quasi-linear systems,
26-27 of Poincar6, I 87-104
Bifurcation values of parameter, I 87
Biot-Savart law, I 34
Birkhoff, G.D., I 4, 65
Bogoliuboff, I 2-3, 67; II
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, 52-54, 75;
41, 75 treatise of, III 3, 40, 55
Boltzmann, theory of,
Bowshewerow, I 73
Branch point of equilibrium,
IV 65 of limit cycles, I 88
Brillouin, M., III 107
Canonical equation, I 45
Canonical form, III 81 of linear equations, I 42-44 of non-linear system, I 52
Cartan, E. and H., I 105; II 47
Cartan-Li6nard, 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
Cauchy's theorem - continued of phase trajectories, I 11, 14 of uniqueness, I 24
Chaikin, E., III 103
II 1, 3; IV 1, 3, 25, 29, 40,
Change of equations, IV 47-48, 53
14, 18, 42,
57-58, 102, 114, 120; II 4, 39, 45;
110; IV 9, 31, 61-62, for transient state of quasi-linear sytem,
III 16 of linear circuit, III 21 of non-linear system, III 20
Characteristic exponents, I 83-85; II
Classification of singularities, I 42,
Clock elementary theory of, IV
44-46 limit cycles in, I 67-68 self-excitation of, IV 46,
49 theory of, IV 49
Coalescence of equilibrium points, I 33 of frequencies, III 31, 93, 103 of limit cycles, I 72-73; II 32 of singular points,
127; III 101-102 of singularities, I 29, 34, 36-37, 39
Coefficient of restitution, IV 3, 57
Combination frequencies, III 25-26, 30-31, 38, 42, 48,
Combination harmonics, III 25-26, 28, 38, 46
Combination oscillations, III 47
Combination tones, definition of, III 24
Complex admittance, III
Complex amplitudes, III method of,
Complex impedance, III 5, 8
Conditional stability, range of, I
Conservative system, I 7, 23, 25, 38, 40, 42, 106
27, 51, 55, 90;
19, 114 127-126 cylindrical phase trajectories of, I 118-119 equilibrium of, I 40 motion of, I 42 oscillations of, II 80 periodic motion in, I 62-63 points of equilibrium in, I 34
Constant linear parameter, 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 12-13 62; II 6-7 of phase trajectories,
62, 12; IV 27
Coulomb damping, II 61 negative, IV 53 positive, IV 53
Coulomb friction, IV 46-47, 49-51,
53 existence of, IV 46 negative, IV 50 phase trajectories in presence of, IV
Coupled electronic oscillators, stability of, II 24
Coupled frequencies, III 17, 30
Coupling autoparametric non-linear, III 127 between degrees of freedom, III 5, 22 critical value of, III 92 inductive,
12 lack of, III 6 non-linear,
Criteria for existence of critical point, IV 13-14 of focal point, I 13 of limit cycle, I 75-77 of nodal point,
I 13 of saddle point, I 13 of Bendixson,
I 75, 77-79, 105;
III 97; IV 31 application of, I 79-82 of Kaufmann for stability, I 61 of Liapounoff for stability, I 48-49; III
63-64 application of, I
49-55 of Poincard, I 105; II 31 of Poincar6-Liapounoff for stability, III 63-64 of Routh-Hurwitz for stability, II
22; IV 58, 60 of stability, I 22-23, 33; III 85 applied to non-linear system, I 3, 55 of equilibrium, I 60-61 of singularities, I 47
Critical damping, III
Critical points, I
102; IV 15, 19, 21-26,
28-29, 32-35, 39-43 criteria for existence of, IV 13-14 locus of,
14, 26-27 of differential equations, IV 12-15 theory of, I
Critical threshold, I 3, 11; III
46, 125; IV 26
Critical value of amplication factor, I 102 of amplitude, II 69 of coupling, III 92 of damping, 1 123 of equivalent parameter, II
30 of index of modulation,
64 of negative resistance, III 15 of parameter, I 31-35, 39, 87, 97, 104; II 26-27,
69-70 of resistance,
18, 21 of self-excitation, III
49 of transconductance, III
54 of variable, II
Curve of contacts of Poincar6, I 75
Cylindrical phase space, I 116-130
Cylindrical phase trajectories of conservative system, I
118-119 of non-conservative system, I
Damped oscillator, differential equation of, I 16
Damped oscillatory motion, equation of, I 13
Damping aperiodic, III 14
Coulomb, II 61 negative,
53 positive, IV
53 critical, III123 critical value of, I
123 linear, II 59-62 negative, I 105; II 44-45, 63, 65 definition of, I 21 examples of, I
21-22 non-linear, I 5 non-linear dissipative, II 58-62,
106-107 non-linear variable, I 105; II
44-45, 63-65 positive, I 18, 105; II
44, 63 quadratic,
60-62, 106 variable, I 105; II
47, 82-83 deforest, Lee,
Degenerate differential equation, IV 16-17 systems of, IV 16-18
Degenerate equation, IV 10,
56, 58, 60 absolutely, IV 11-12, 58 of first order, I 112; IV
5, 7, 15, 17
Degenerate limit cycle,
Degenerate phase trajectory, I
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 -
of first order, periodic solution of, IV 19 transition between continuous and discontinuous solutions of, IV 35-36 with one degree of freedom, IV 19-29
Degeneration absolute, IV 64 complete, IV 9-10 condition of, IV 38 of equations, I 4-5; IV 15,
40 of system, IV 61 solutions of differential equations in neighborhood of, IV 8-11
Demultiplication of frequency, III 24-25
Differential equation, I 1-6, 10-11, 14, 22, 24,
49, 51 62, 64, 66-67,
107; III 75 linear, I 1
critical points of, IV 12-15 degeneration of, I 5; IV 15-17
101, i06, 112, 117;
Cauchy's theorem for existence of, I 2 with constant coefficients, I 1 linearized, I 67, 87 non-linear; see Non-linear differential equation of damped oscillator,
16 of dynamical system, I 20 of electromechanical system, I 117-118 of first order,
8-9, 12, 28, 33, 58 degenerate, IV 5, 7, 15, 17 describing system with one degree of freedom,
IV 25-29 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 non-dissipative circuit, III 115 of nth order, I 9 of pendulum, II 56 of second order, 1 10, 17, 24; IV 6-8, 12, 15,
56 full, IV 5 quasi-discontinuous solution of, IV 38 of synchronous motor, I 126 of Van der Pol, III 94-96 quasi-linear 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 8-11 represented by phase trajectories, III 2 systems of degenerate,
IV 16-18 theory of,
4 with periodic coefficients, III 2, 56, 74, 108-112,
Discontinuities, IV 13-15, 32-34, 41, 43, 48, 51 occurrence of, IV 13, 15-16, 27
II 32; IV 6, 12, 15, 21-22, 24,
28, 32-33, 36, 41-42, 45-46
Discontinuous periodic motion, IV 3
Discontinuous stationary relaxation oscillations,
Discontinuous solutions of degenerate system, transition between continuous and, IV 35-36 of differential equations of first order, IV 17
Discontinuous stretches, IV 2, 42-43
Discontinuous theory of relaxation oscillations, IV 3,
8-18, 20-22, 25, 29, 56-57
III 21; IV 16
Dissipative system, II 67, 73; IV 45 heteroparametric excitation of, III 121-123
3, 11, examples of,
72 of common topography,
Dry friction, I
22; IV 47
Durand, Dr. W.F., II 61
Dynamical system, I
51, differential equation of, I 20 equilibrium of, IV 58-60 stationary states of,
I 40; IV 56-65
Electrical oscillations in circuit containing iron core, II 58 in circuit containing saturated core, II 108
Electrodynamical system, I 87
Electromechanical system differential equation of, I 117-118 self-excitation of, I 99,
102-104 self-excited oscillations in, I 99, 102-104
Entrainment of frequency,
I 4; III 1, 93-106 acoustic, III 94, 103-104 artificially produced, III 104 band of, III 104 definition of, III 93 mechanical, III 94 other forms of,
100 state of, III 102 steady state of, III 102 transient state of, III 98, 102 zone of,
93, 100, 103-104,
Equations absolutely degenerate, IV 11-12, 58 autonomous, II 21 canonical, I 45 change of, IV 47-48,
53 characteristic; see Characteristic equation degenerate, IV 10, 56, 58, 60 of first order, I 112; IV 5,
15, 17 degeneration of, I 4-5; IV 15,
40 differential; see Differential equation
126-127 linear, I II 2-3, ,0; approximate, I 100 canonical form of, I 42-44 equivalent,
101-102 harmonic solution of, II 33
Mathieu, III 124
Mathieu-Hill, 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 non-degenerate, IV
59 non-linear, I 1, 118;
II 5, 47-48, 108 of oscillation, II 58 non-linear differential; see Non-linear differential equation of aperiodic damped motion, I 14 of damped oscillatory motion,
I 13 of first approximation, I 3, 49-52, 55-56, 65, 81,
94, 97, 99, 101oi, 103, 114, 120,
45, 52-54, 59, 83, 90, 95,
102, 110; III 11,
29, 31, 33, 37-35,
40-i45, 53-57, 63, 67-69, 74; IV 31, 60-61, 4 linear,
I 49 of first order,
I 9; IV 3, 16-17, 20, 22 of Froude for rolling of ship, II 61, 107 of Hill,
113-114 definition of, III 109 of Hill-Meissner, III 114-115, 117 topology of, III 114-117 trajectories of, III 114, 117 of improved first approximation,
III 40 of isoclines,
121 of Li6nard, I 107, 109-110 of limit cycle, II 46 of Mathieu, III 109-110 112-114, 124, 127-128 definition of, III 108 linear,
124 parameter of, III 127 stable and unstable regions of, III 111-112 of Mathieu-Hill, III 108-109, 113-114, 124-125 linear, III 108 stability of,
113 subharmonic resonance on basis of, III 125, 129 of nth order, IV 16
of phase trajectory, I 66, 88, 128 of Poincar6, II 53 of
Rayleigh, II 44-45, 47-48
65 of second order, IV 16-17 36 of separatrix, I 35-36, 36 of unstable motion, I 13 of Van der Pol, I 79,
85, 97, 103, 105 108
113; II 27,
quasi-degenerate, IV 11-12 quasi-linear; see Quasi-linear equation variational, I 83; II 19; III 35-36, 63-64, 70, 80 with both linear and quadratic damping, II 61
Equilibrium branch point of, I
55, 60, 82; II
IV 65 conditions for stable, II 27 conditions of, I 11, 29, 60, 80, 94; II 71; IV 63 dynamical, III 34 half-stable, 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 non-linear conductor, I
55-61 of conservative system, I 40 of dynamical system, IV 58-60 of electromotive forces in circuit, II 58 of point, I
32 point of, I 11,
III 87, 106; IV
60-61, 63-64 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
54; stability of, I 9 stability of, I
33-35, 40-61,- 71, 85, 87
II 35-36, 68; III 106, 110 IV
21, according to Liapounoff, I 4b-61 criteria for, I 60-61 theorems for, I 29 state of,
22 types of,
Equilibrium phase angle, III 106
Equiphase interval, III
Equivalent Balance of Energy, Principle of, II
99, 102-104 .
Equivalent linear equation, II 101-102
Equivalent linear impedance, III 33
Equivalent linear system, II 102; III 33, 38
Equivalent linearization for multiperiodic systems, III 26-30 method of, II 3, 99-102; III 21, 31, 40, 49, 66 applied to steady state of quasi-linear system,
III 10-14 applied to transient state of quasi-linear system, III 14-19 examples of application of, II 105-111 in quasi-linear systems with several frequencies,
37-40 principle of, II 105; III 10, 27, 29, 61
Equivalent linearized equation, II 110
Equivalent linearized system,
66 self-excitation of, III 30
62, 66, 68-69 critical value of, II 100; III 30 definition of, II 100-101; III determination of, II
38 non-linear, III 29 of multiperiodic system,
Euler's identity, III 3
Exchange of stability,
Excitation autoparametric, III 107-108, 126-129 condition for, III 126 autoperiodic, III 49-50, 57 external,
41, absence of, III
89-96 system with, III
41 external periodic, III
46, 65, 69, 72, 87,
94, 128 of quasi-linear systems, III
41-52 system with, III
75 heteroparametric, III 107-108, 112, 115, 125-126 conditions of, III 108, 123 dependence on frequency and phase of parameter variation, III 117-121 of dissipative system, III 121-123 principal features of,
119-121 heteroperiodic, III 50, 107-108 absence of, III 51 non-resonant external, of quasi-linear system,
46-47 parametric, I 4; III 1, 60-65, 107-129 definition of, III 60 of critically damped or overdamped circuit,
123 periodic non-resonant, III 41-44
Existence of autoperiodic oscillation, III 47-52 of closed trajectories, I
21, of critical point, criteria for, IV 13-14 of focal point, criteria for, I 13 of limit cycles, I 99; II 47-48, 64, 66-69, 90-95;
100 condition for, II 35, 39, 42, 45, 65, 73, 93 criteria for, I 75-77 proof of, I 79 of nodal point, criteria for, I 13 of periodic solution, I
90 geometrical analysis of, I 105-115 in proof of, I 105 of principal solution, III 85 conditions for, III 87 of saddle point, criterion for, I 13 of self-excited oscillations, II 47
Expansions of Poincar6, II 7
External excitation, III 41,
87-88, 92 absence of,
89-96 non-resonant, III
46-47 system with, III 41
External frequency, III
100, 103, 105-106
External periodic excitation, III 1, 41, 46, 65, 69,
87, 94, 128 of quasi-linear systems, III 41-52 system with, III
Externally applied frequency, III 88, 92
Faraday-Maxwell theory, I 124
Fleming, I 5
Floquet, theorem of,
Focal point, I 16, 18, 22, 69-70, 75-77, 79, 101-103,
114, 120, 123, 125, 127;
101; IV 6, 8,
17, 27, 35-36, 51, 53-54, 61,
64 criterion for existence of, I 13 definition of, I
13 neighborhood of, I 1 stability of, I 47-48, 57-60, 8-89, 91-92, 97, 99;
11 31-32, 41-43, 68 stable, I 13; II 68 unstable, I 13; III 63
53, 55, 57-61, 125
Frequency acoustic entrainment of, III 94, 103-104 autoperiodic, III 49-51, 77, 94, 101 coalescence of, III 31, 93, 103 combination, III 25-26, 30-31, 38, 42, 48, 53 correction, II 40, 59, 83-85, 106 coupled, III
, 1 iiju
demultiplication, III 24-25 demultiplication network, III 104 entrainment of, see Entrainment of frequency external,
59, 93, 100 103, 105-100 externally applied, III 8, 92 harmonics of, III
48 heteroperiodic, III 49-53, 100-101 of pendulum,
94 of quasi-linear oscillation, II 83 of ripple,
III 115, 117, 119-120 of self-excited non-linear system, III 17 of self-excited oscillations, III 13 of subharmonic oscillation, III 88 of thermionic generator, II 24-25 synchronization of,
Coulomb, IV 46-47, 49-51, 53 existence of, IV 46 negative, IV 50 presence of, IV 47-49 dry,
22; IV 47 non-linear, IV 40 static,
IV 48-49 zone of, IV 49
Froude's equation for rolling of ship,
Froude's pendulum, I
21-22, 67; II 44-46; IV 37
Gaponow, III 100-101
Gauss's theorem, I 77
24, 26-27, 38 47-4,
78 amplitude of,
II 12 definition of, II 7 of Poincar6, III 102
Geometrical analysis of existence of periodic solutions, I
Geometrical definition of stability, I 41
Geometrical theory of limit cycles, I 63-86
Graphical method of topology of trajectories, I 25-
Gylden, II 3 method of, II 75
Half-stable equilibrium, I 51
Half-stable limit cycle, I 70
I 74, 99; II 32, 41, 44,
III 72, 83,
91; IV 46, condition for, II 31-32 definition of, I 71
49 example of, I
87 subharmonic resonance of order one-half for,
Harmonic Balance of Energy, Principle of, II 99,
38, 49, 53, 62
Harmonic oscillator, I 86; III 128; IV 48 differential equation of, I 7 motion of, I
86 non-dissipative, I 23
Harmonic solution of linear equation, II 33
Harmonics, III 28-29, 35, 42, 53-54, 62, 66 combination, III 25-26, 28, 38, 46 of current,
32 of frequencies, III 48
Heegner's circuit, IV 34-36
Hertz, I 5
Heterodyning, III 100
Heteroparametric excitation, III 107-108, 112, 115,
125-126 conditions of, III 108, 123
Heteroparametric excitation - continued dependence on frequency and phase of parameter variation, III 117-121 of dissipative system, III 121-123 principal features of, III 119-121
Heteroparametric generator, III
Heteroparametric machine of Mandelstam and Papalexi,
Heteroparametric self-excitation, III 116
Heteroparametric oscillations, conditions for selfexcitation of, III 120
Heteroperiodic excitation, III 50, 107-108 absence of, III 51
Heteroperiodic frequency, III 49-53, 100-101
Heteroperiodic oscillations, III 53, 87, 90-92, 97,
102 definition of, III 47 existence of, III 47-52 possibility of, III 58 stationary, III 100 synchronized with autoperiodic oscillation, III 55
Heteroperiodic solution, III 85-86, 96
Heteroperiodic states, III 96 condition of, III 57 of non-linear systems, III 57 stability of, III 57
Heteroperiodic variables, III 50
Hill equation, III 113-114 definition of, III 109
Hill-Meissner equation, III 114,115, 117 topology of, III 114-117 trajectories of, III 114, 117
"Hunting," I 101-102
Hurwitz; see Routh-Hurwitz
Huygens, III 93
Hysteresis, III 61 of oscillation, II 27-28 of resonance, III 72
Hysteresis cycle, II 28
III 46-47 application of, II 80-85 equations of, III 40
Impulse-excited oscillations, IV 4, 7, 15-16, 43
Impulse-excited oscillator, phase trajectories of,
Indices of Poincar6, I 75-77 theory of, I 75-78
Index of modulation, III 123 critical value of, III 64 of singularities, I 76-77, 117; III 97 of stepwise modulation, III 115 of trajectory, I 76
Instability, regions of, I 3
"Introduction to Non-Linear Mechanics," I 3; II 1;
Island of trajectories, I 30-32, 62, 119, 128-129
Isochronism, condition of, I 42
Isochronous motion, I 42; II 12
Isochronous oscillations, II 55-56, 68, 105
Isochronous system, II 37;
Isoolines application of method of, I 108 definition of, I 20 equation of, I 121 method of, I 20-21, 105, 113
Jacobian, II 6
Jump, III 65, 72-73, 91 discontinuous, II 32; IV
32-33, 36, 41-42, 45-46 quasi-discontinuous, III 72; IV 10, 29
Kaden, III 104
Kaidanowski, IV 40, 42
Kaufmann's criterion for stability of equilibrium,
Kennard, Prof. E.H., I 6
Kryloff and Bogoliuboff, I 2-3, 67; II 1, 3, 47,
75-76, 100 analytical method of, I 4 argument of, III 60 condition of self-excitation, III 64 method of equivalent linearization of, II notation of, III 6, 15
3, quasi-linear method of, III 83 quasi-linear theory of, I 4; II 100, 110; III 1, 10,
83, 94 theory of first approximation of, II 49, 52-54, 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
LeCorbeiller, Ph., I 114; IV
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, 108-109; IV 2
Liapounoff, I 4; II 39 criteria of stability of, I
III 63-64 application of, I 49-55 equation of first approximation of, I 114; II 42,
60, 64 method of, I 49, 56, 105 stability in theorem of, I 3, 29,
78 advantage of, I 55 proof of, I 49, 51-55 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 38-39 equation of, I 107, 109-110 method of, I 106-108 plane of, I 107-115 limit cycles in, I 113-115 qualitative analysis of, IV 38 theory of, I 107
Limit cycle, I 90-92 II 35, 38,
100; 11I 54, 78, 96-97 amplitude of, II 32 analytical examples of,
62-66 bifurcation of,
87 branch points of,
88 coalescence of,
65, 94-95, 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 47-48, 64, 66-69, 90-95;
III 100 condition for, II 35, 39, 42, 45, 65, 73, 93 criteria for, proof of, I 79
1 75-77 geometrical theory of, I 63-86 half-stable, I 70 in case of polynomial characteristic, II 72-74 in clock, 1 67-68
Van der Pol plane, I 113-115 inwardly stable, I 70
motion on, II 37 nature of, I 99 of first kind, I 125,
130 definition of, I
116-117 of Poincar6, I 62-86 of second kind, I 123,
125, 129-130 definition of, I
116-117 of thermionic generator, II 24-25 outwardly stable, I 70 physical examples of, I
66-68 piecewise analytic,
IV 28, 45-46, 49, 55 possibility of, II 63 properties of, I 75, 105 representation of, II 37 stability of, I 62, 64-66,
68-73, 75, 85,
98-99, 102; II
27, 31-32, 35-37, 39, 41,
III 97, 101 stable, IV 35, 37, 55 systems with several, II 66-69
43-45, theorem of, I 72-75; II 71'-72 theorem of stability of, I 70 topology of trajectories in presence of, I 68-75
Lindstedt, II 3 method of, I 5; II 76 method of Gylden and, II 75
1, 22, 100
Linear circuit characteristic equation of, III 21 theory of, III 4-7, 11
Linear damping, II 59-62
Linear differential equation, I 1
Cauchy's theorem of existence of, I 2 with constant coefficients, I
Linear dissipative circuit with constant parameters,
Linear equation, I 80; II
2-3, 5, 45, 52; IV 27, 51 approximate, I 100 canonical form of, I 42-44 equivalent, II 101-102 harmonic solution of, II 33
Mathieu, III 124
Mathieu-Hill, 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;
4; 27, 36 phase trajectories of, I 7-23
Routh-Hurwitz theorem for, I 3 with several degrees of freedom, III 30
Linearized differential equations, I 67, 87
Linear dissipative system with constant parameters,
Linearized equation, I 67, 87, 100 equivalent, II 110
Linearized motion, I 27
Ludeke, III 73
2, 4; III
113 conditions of, IV 15-16, 21, 24, 27, 32, 39, 41-42,
Mandelstam and Papalexi, III 103, 107-108, 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
109-110, 112-114, 124, 127-128 definition of, III 08 linear, III 124 parameters of, III 127 stable and unstable regions of, III 111-112
Ia-hieu function, III 109, 113
108-109, 113-114, 124-125 linear,
III 108 stability of, III 113 subharmonic resonance on basis of,
III 125, 129
Maxwell, theory of,
124; III 7-10; IV 37 differential equation of, III 9 parameter of, IV 37 self-excited oscillations of, I 99-102 stability of,
Meissner, III 114 (see also Hill-Meissner equation)
III 128 experiment of,
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,
18-19 asymptotic, I 3, 28,
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,
40 in neighborhood of focal point,
13 in vicinity of equilibrium point, I 29 in vicinity of saddle point, I 26 isochronous,
12 laws of, I
2-3 linearized, I 27
Method of Small, I 1, 66 non-isochronous, II 38 non-stationary, I 40 of conservative system, I 42 of elastically constrained current-carrying conductor, I 34-37 of harmonic oscillator, I 86 of pendulum, I
118, 123 of representative point,
10 of rotating pendulum, I
37-39 of synchronous motor, differential equation of,
I 126 of system with one degree of freedom, I 82 on limit cycle, II 37 oscillatory dan1ped,
48, 106 phase trajectories of,
I 16-18 periodic, I 3, 26, 31, 85,
105 discontinuous, IV 3 in conservative system, I 62-63 stability of,
82-86 periodic stationary, I 63 quasi-isochronous, frequency of, II 83 stable, II 21 stability of, I 24, 41-42, 85; II 21; III 111-112,
125 in neighborhood of singular point, I 40-41 in vicinity of critical value of parameter,
I 33-34 in vicinity of equilibrium point,
I 24, 26 of harmonic oscillator, I 86 stationary, I 23, 40-41; II 14, 24;
54 stability of,
110 stationary periodic, I
61; IV 55 uni-dimensional real, I 8 unstable, I
28-29, 37 equation of,
I 13 phase trajectories of,
M~ltiperiodic system equivalent linearization for, III 26-30 equivalent parameters of,
III 27 resonance in, III 23
Multiply degenerate system, IV 30-36
Multivibrator of Abraham-Bloch, IV 30-34
Negative criterion of Bendixson, I
75, 77-78; III
Negative damping, I 105; II
44-45, 63, 65 definition of,
21 examples of,
I 18-19, 22, 45-46, 59, 69, 76-77, 81-82,
101-103, 114 127;
101; IV 17, 27, definition of, I 13 criterion for existence of, I 13 stability of, I
13, 45, 47-48, 58, 60-61;
III 63 theorem for occurrence of,
17, 23, 106 cylindrical phase trajectories of,
I 119-121 existence of periodic solutions in, II 90-95 with non-linear variable damping, II 63-65
Non-degenerate equation, IV 59
Non-dissipative circuit, differential equation of,
Non-dissipative harmonic oscillator, I 23
Non-isochronous motion, II 38
Non-isochronous system, II 106; III 23
Non-linear circuit, self-excitation of, I 1
Non-linear conservative system, I 23, 128 behavior of, I 32-33 definition of, I 24 examples of,
56-58 general properties of, I 24 phase trajectories of, I 24-39 theory of first approximation applied to,
55-56 with cubic term,
Non-linear coupling, III 127 autoparametric, III 127
Non-linear differential equation, I 1, 24, 63, 66, 75,
1, 3, 7, 67, 82, 100;
24; IV 37, 51 conditions for periodicity of solutions of, II 4-19 exact solutions of,
1 of dissipative type, II
45 of oscillating circuit, II 28 of second order, II 24 of self-excited oscillations, I 66 of torsional oscillation of shaft, II 57
Non-linear dissipative damping, II 58-62, 106-107
Non-linear dissipative system, II 48
Non-linear equation, I
5, 47-48, 108 of oscillation, II 58 stability of, I 50, 52
Non-linear equivalent parameter, III 29
Non-linear mechanical vibrations, I 1
Non-linear non-conservative system, I 23, 40, 63, 105 autonomous, I 62 higher approximations for, II 89-95
Non-linear oscillation, I 100; II 3, 99 self-excited, II 63
Non-linear oscillatory system, III 24
Non-linear resonance, I 4; III 1-129 external,
III 53-74 stability of III
72 internal, III
Non-linear restoring force,
Non-linear self-excited oscillations, II 63
Non-linear self-excited system, III 103
Non-linear 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 self-excited, III 17 heteroperiodic and autoperiodic states of,
III 47-52, 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 30-31
Non-linear variable damping, 1 105; II 44-45, 63-65 non-conservative systems with, II 63-65
Non-resonance, III 23, 42
Non-resonant oscillations, III 27
Non-resonant self-excitation of quasi-linear system,
Non-resonant system, III 29, 33, 55
Non-stationary motion, I 40
Ordinary point, I 12-14, 20 definition of, I 11
Ordinary value of parameter, I 32
Oscillating system, experiments with, III 60
Oscillation hysteresis, II 27-28
Oscillations autoperiodic, III 53, 97, 100, 102 amplitude of, III 102 conditions for stability of, III 58 definition of, III 47 existence of, III 47-52 self-excitation of, III 52 stability of, III 58 stability of stationary, III 56 synchronized with heteroperiodic oscillation,
synchronous, III 56 combination, III 47 electrical circuit containing iron core, II 58 in circuit containing saturated core, II 108 heteroparametric, conditions for self-excitation of,
III 120 heteroperiodic, III 53, 87, 90-92, 97, 102 definition of, III 47 existence of, III 47-52 possibility of, III
53 stationary, III
100 synchronized with autoperiodic oscillation,
55 impulse-excited, IV 4, 7, 15-16, 43 in stationary state, III 11 isochronous, II 55-56, 68,
105 maintained by periodic impulses, IV 43-55 non-isochronous, II 57-58 non-linear, I 100; II 3, 99 self-excited, II 63 non-linear equation of, II 58 non-resonant, III 27 of conservative system, II 80 of limit-cycle type, IV 43 of pendulum, I 1 of synchronous motor, 1 124-130 quasi-discontinuous, IV 28,
37 quasi-harmonic, II 49 quasi-linear, I 128 II 24; IV 13 frequency of, II
relaxation; see Relaxation oscillations self-excitation of,
I 58-59, 66 self-excited; see Self-excited oscillations stability of, III
36-37, 84, 124; IV 52 stationary,
71; III 15,
21, on limit cycle, I166 subharmonic, III
67, 90-92 frequency of, III 88 self-excitation of,
107 synchronized, stable condition of, III
Oscillator coupled electronic, stability of, II 24 damped, differential equation of, I 16
Oscillator - continued electron-tube, III 48 harmonic, I 86; II 128; IV 48 differential equations of, I
7 motion of, I 86 non-dissipative, I 23 impulse-excited, phase trajectories of, IV 53-55
Oscillatory damped motion, I 48 106 phase trajectories of, I 16-16
Oscillatory parameter, III 60; IV 5, 56, 61
Oscillatory system, I 68; III 23 non-linear, III 24 resonance of, III 23
Papalexi, I 2, 4 (see
Mandelstam and Papalexi)
Parameter, III 36, 86-87, 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
31-35, 39, 87, 97, 104;
II 26-27, 31,
69-70 dissipative, III
21; IV 16 equivalent, II
102-105, 107; III
33-35, 49, 62, 66,
68-69 critical value of, II 100; III 30 definition of, II 100-101; III 38 determination of, II 99-102 non-linear, III
29 of multiperiodic system, III 27 finite, IV 60 fixed, III 81 large values of, I 4, 111-113; IV 2 linear, III 17 method of small, II 1-32 non-linear, 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,
existence of, IV 60 influence on equilibrium of dynamical systems,
IV 58-60 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
36, 62, 107, 117, 121,
124-125, 128-129; IV
Parametric excitation, I
III 1, 60-65, 107-129 definition of, III 60 of critically damped or overdamped circuit,
Parasitic capacity, IV 57
Parasitic parameter, IV
7 effect on stationary states of dynaical systems,
existence of, IV 60 influence on equilibrium of dynamical systems,
III 112; IV 65 as example of non-linear conservative system,
11 56-57 asymptotic motion of, I 16; IV 62 differential equation of,
56 elastic, III 126 energy of, I 26 frequency of, III 94
Froude's, I 21-22, 67; II 44-46; IV
37 mathematical, IV 46 mechanical, III 94 motion of, I 29-30, 100, 118, 123 in neighborhood of unstable equilibrium, I 14-15 of clock, IV
44 oscillations of,
Periodic coefficients, III 81, 114 differential equations with, III 2, 56, 74, 108-112,
i i3iI lIIIE II
Periodic impulses, oscillation maintained by, IV
Periodic motion, I 3, 26, 31,
39, 41-42, 62-63, 77-78,
85, 105 discontinuous, IV 3 in conservative system, I 62-63 stability of,
Periodic non-resonant excitation, III 41-44
Periodic solution, I 66, 79, 82, 84, 86, 122-123;
II 1-2, 7, 12, 18-19, 25-26,
III 125 analytical, IV 3 condition for stability of, II 80-82 existence of,
II 10-11, 86,
geometrical analysis of, I 105-115 proof of, I 105 of degenerate system of first order, IV 19 of non-linear problem, II
6 of non-linear system,
II 4, 6 of quasi-linear equation,
II 87; III 78-80 stability of,
2, 13 condition of,
I 122, 129;
Perturbation, II 50; III 70, 80, 125 in phase angle, III 63 of amplitude,
Perturbation method, II 15, 19
Perturbation variables, III 64
Phase angle, I 7; II 4, 25, 38, 82; III 62-65, 120 critical, III 120 equilibrium, III 106 of ripple,
III 118-119 perturbation in, III 63
8, 17, 21,
16-17, 21-22, 24-30,
32, 36, 43, 45-47
, 2-63, 68-69, 71-72
11 6-7, 1 ,
40-41, 64; III
7 , 96,
100-101, 114, 116, 118, 122; IV 2, 5-8, 12,
14-15, 17, 25-26, 28-29, 33, 43-46, 53,
62 definition of, I 7-8
2, cylindrical, I 116-130 uni-dimensional, IV
Phase trajectories, I 2-3, 7-23, 40-41, 43,
90, 94, 97, 102,
105-107, 109-110, 113-114, 117-11 , 125,
II 33, 35-36, 64; III 78, 96, 101, 112, 116-118,
122; IV 2, 4, 6, 12-14, 17, 19-20, 22, 26-27,
42-44, 46, 48-49, 55, 62-63 analytical, IV 3, 31, analytical method of topology of,
I 26-29 behavior of, in neighborhood of singularities,
Cauchy's theorem of,
11, continuum of,
62, 128; IV 27 continuum of closed,
6-7 cylindrical of conservative system,
118-119 of non-conservative system,
119-121 definition of, I 8 degenerate,
12 equations of, I 66, 88, 128 existence of closed, I 21, 108-113, 129 graphical method of topology of, I 25-26 index of,
islands of, I 30-32, 62, 119, 128-129 of aperiodic damped motion, I 18-19 of Hill-Meissner equation, III 114, 117 of impulse-excited oscillator, IV 53-55 of limit-cycle type, IV 22 of linear system, I 7-23 of non-linear conservative system, I 24-39
.of oscillatory damped motion,
Phase trajectories - continued of second kind, I 121-123, 129 of unstable motion,
14-16 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,
100; IV 26 analytical method of, I 26-29 graphical method of, I 25-26 in neighborhood of singular points,
25-29 in phase plane,
29-32 in presence of singularities and limit cycles,
I 9, 16-17, 24 definition of, I 9
Piecewise analytic curves,
Piecewise analytic cycle, IV 22, 24, 42, 45
Piecewise analytic limit cycle, IV 28, 45-46, 49, 55
Piecewise analytic representation of phenomenon,
Piecewise analytic spiral,
122; IV 28, 54-55
Piecewise analytic trajectory, III 117; IV 17, 33, 37,
43, 49, 54
4, 34, 38;
1, 54, 79,
96, 107-108 analytical method of, I 4 87, 99, 104-105; II 4-32 bifurcation theory of, I 7-104 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 63-64 curve of contacts of, I
75 equations of, II 53 expansions of,
7 functions of, II 13, 35 generating solution of, III 102 indices of, I 75-77 limit cycles of, I 62-86 method of, I 5; II 1-32, 38, 50, 96; III
41, applied to systems with several degrees of freedom,
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 33-34 theorems of indices, I 76-77 theory of,
66, 75-92, 113, 125-126;
44, topological methods of, I 2; II 69 variational equations of, III 80
49 method of, II 50 application of, II 52
Proper trajectory, I 12
II 60-62, 106
Qualitative analysis of
Lienard, IV 38
Qualitative methods, II 1
Quantitative method of approximations, I 2-3;
II 1-2 advantage of, I
Quasi-degenerate equation, IV 11-12
Quasi-degenerate system, IV 10-11, 65
Quasi-discontinuity, IV 39, 43
Quasi-discontinuous jump, III 72; IV 10, 29
Quasi-discontinuous oscillation, IV 28, 37
Quasi-discontinuous relaxation oscillation, IV 35
Quasi-discontinuous stationary relaxation oscillation,
Quasi-discontinuous solution of differential equations of second order, IV 38
Quasi-discontinuous timing of electronic switch, IV 52
Quasi-harmonic oscillation, II 49
Quasi-isochronous motion, frequency of, II 83
Quasi-isochronous system, III
Quasi-linear equation, II 33, 49, 52, 55 58, 66, 80,
89, 96, 99-102, 105; III
IV 2, 37 definition of, II 2 of system with external excitation III 41
87; III 76-80 with forcing term, periodic solutions of, III 78-80
Quasi-linear method of Kryloff and Bogoliuboff,
24; IV 13 frequency of, II 83
Quasi-linear system, II 26-27, 95, 99, 103; III 27,
30, 53 autonomous, III 41 bifurcation theory for, II
26-27 condition of resonance of, III 23 external periodic excitation of, III
Kryloff-Bogoliuboff theory of, III 41 method of equivalent linearization applied to steady state'of, III 10-14 applied to transient state of, III 14-19 non-resonant external excitation of, III 46-47 non-resonant self-excitation of, III 19-21 of differential equations, II 14 resonance in, III 23 resonant self-excitation of, III 21-23 self-excitation of,
41 with several degrees of freedom, III
3-23, 26 with several frequendies, III 37-40
Quasi-linear theory of Kryloff and Bogoliuboff,
III 1, 10, 83, 94
Quasi-linearity, condition of, II 29
93, 103, 107 equation of, II
44-45, 47-48, 65 experiments with oscillating systems, III 60
RC-multivibrator, IV 22-24
RC oscillations in thermionic circuits,
References, I 131-133; II 112-113; III 130-132;
Reich, H.J., III 25
Relaxation oscillations, I 68, 128, 130; IV 1-65 definition of, IV 1, 4 discontinuous stationary, IV 12 discontinuous theory of, IV 3, 8-18, 20-22, 25, 29,
7 examples of,
97, 115 mechanical, IV 37-42 quasi-discontinuous, IV 35 quasi-discontinuous stationary, IV 12 stability of, IV 63-65 theory of, I 4
62, 69, 73, 94, 112, 115, 125; II 38, 40;
III 116-117, 119;
IV 6, 21-22, 24, 26-28,
32-33, 38, 41-42, 48, 54, 62-63, 65 motion of, I 10
Resonance external, III 53-74 definition of, III 1 fractional-order, III 53, 55, 57-61, 1-25 in in quasi-linear system, III 23 internal, III 33, 36, 53 definition of, III
1, 31 of order one, III 36-37 linear, III 1, 69 non-linear, I 4; III 1-129 external, III 53-74 internal,
36 undamped, III
73 of order n, III 75 of oscillatory system, III 23 of quasi-linear system, condition of, III 23 subharmonic;
Resonance hysteresis, III 72
Resonant self-excitation of quasi-linear system,
Resonant system, III 33, 53-57
Restitution, coefficient of, IV
Richardson, Dean R.G.D., I 5-6
111-112, 121-123 capacity, III 116 frequency of,
115, 117, 119-120 phase angle of,
III 118-119 rectangular, III
114-115, 117; IV 2 timing of, III 114
Routh-Hurwitz criteria of, II 22; IV 58, 60 theorem for linear system, I 3
Saddle point, 1 23,
38-39, 41, 46-48,
55-59, 61-62, 76-77, 81-
III 97, 99, 101; IV 17, 27, 31, 35-36, 62,
64 criterion for existence of, I 13 definition of, I 12 example of,
I 16 motion in negative, IV 64-65 positive, IV 64-65 theorem for occurrence of,
Saturation voltage, II 29; III 95
Savart; see Biot-Savart law
13, 25, 51,
87 appearance of, II
13 condition for absence of, II 88-89, 92-93 condition for elimination of, II 97 definition of, II 2 effect of, II 49 elimination of, II
97-98 in solutions by series expansion,
49-52 presence of, II 52
Self-excitation, I 58-59, 73; II
25, III 30-31, 60,
112, 126 asynchronous, III 51 autoparametric, III 127, 129 autoperiodic, III 50, 52 condition of, I
39, 70-71; III
108, 123 critical value of, III 49 disappearance of, I 74 existence of,
44, 120-121 hard, I 74, 99; II
32, 41, 44, 69;
72, 83, 91;
IV 46, 49 condition for, II 31-32 definition of, I 71 example of, I 87 subharmonic resonance of order one-half for,
111 88-91 heteroparametric, III
116 lack of, III
37, 43, 46, 90 non-resonant, of quasi-linear system,
19-21 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, 102-104 of electronic circuits, I 66 of electron-tube circuit, III 107 of equivalent linearized system, III 30 of heteroparametric oscillation, III 120 of non-linear circuit, I 1 of oscillation, I 58-59, 66 of quasi-linear systems,
41 of shunt generator, II 71 of simple circuit, III 14-19 of subharmonic oscillations, III 107 of system, II 69; III 86, 90 of thermionic circuits, I 87; II 38 of thermionic generators, I
45 parametric, III 113 point of, III 92 possibility of, III 59,
122; IV 23, 35 prevention of, II 43-44; III 51 resonant, of quasi-linear system,
21-23 condition for, II 30-31; III 51-52 definition of, I 71 example of, I 92, 102 subharmonic resonance of order one-half for,
III 83-87 zone of, III 49
Self-excited oscillations, 1 66-68, 97, 103; II 41,
44-45, 48; II11
28; IV 1, 35, 55 amplitude of, 1 104 existence of, II 47 frequency of, III 13 non-linear, II 63 non-linear differential equation of, I 66 of electromechanical system, I 99, 102-104 of mechanical system, I 99-102 of non-linear circuit, I 1 stationary, III 13-14; IV 43
Self-excited state, III
Self-excited system, III
92 asynchronous action on, III
52 non-linear, III 103 frequency of, III 17
Self-excited thermionic generator, III 45
29, 31-32, 35-36, 38-39, 62, 119,
123, 128-129; III 101; IV 62 bifurcation of limit cycles from, 1 87 equation of, I 35-36, 38 of second kind, I 123 stable,
41; IV 63, 65
Series generator, I 102-104 parallel operation of, I 79-82
Shohat, J.A., I 6; 38
Shunt generator, self-excitation of, II 71
97, 116-117, 119-120, 123, 127;
54, 96-98; IV 14, 26, 31, 34 classification of, I 42, 47 coalescence of, I 127;
101-102 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,
I 25-29, 68-75
2, 41, 61,
121; almost unstable, I
62 classification of, I 42, 47 coalescence of, I 29,
45-46; distribution of, III 98-102; IV 64 index of, I
76-77, nature of, III 98-102 of differential equation, I
3, 11 simple, I
29 stability of, I
117; II 97
91-92, 103; II 32,
39, 41; II 100-101; 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,
Sommerfeld, IV 40
Source, I 87; III 100 of trajectories, I 69, 78
I 3, 125, 127 absolute, zone of, I 61 analytical definition of, I 41-42 conditions of, 1 55, 99; II 14; 22-24; III 69-72,
IV 60 conditional,
61 criteria of, I 22-23, 33; III 85 applied to non-linear system, I 3, 55 of equilibrium, I
60-61 of singularities, I 47 defined for motion in neighborhood of singularities,
140 definition of, I 40-41 exchange of, I 34, 38 geometrical definition of, I 41 in sense of Liapounoff, I
40-42, 48-61; II 21;
Kaufmann's criteria of, I 61
Liapounoff's criteria of, I 48-49; III 63-64 application of, I 49-55 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 60-63 of equilibrium, I 33-35, 40-61, 71, 85, 87, 93, 127;
106, 110 IV 21, 58-65 according to Liapounoff, I
Kaufmann's criteria for, I
61 theorems for, I 29 of focal point, I 47-48,
99; 1 31-32,
41-43, 69 of heteroperiodic state, III 57 of limit cycle, I 62, 64-66, 68-73, 75, 85, 93-94,
98-99, 102; II 27, 31-32, 35-37, 39, 41, 43-45,
69-72; III 97, 101 theorem of, I 70 of Mathieu-Hill equation, III 113 of mechanical system, I 102 of motion, I 24, 41-42, 85; II 21; III 111-'12, 125 in neighborhood of singular point, I 40-41 in vicinity of critical value of parameter,
I 33-34 in vicinity of equilibrium points, I 24, 26 of harmonic oscillator, I 86 of nodal oint, I 13, 45, 47-48, 58, 60-61; II 68;
11 63 of non-linear equation, I 50, 52 of non-linear external resonance, III 65, 72 of non-linear system, I
52 of oscillations, III 36-37, 84, 124; IV 52 of periodic motion, I 82-86 of periodic solution, II 19-24 conditions for, III 80-82 of point of equilibrium, III
106 of position of equilibrium, I
9 of relaxation oscillations, IV 63-65 of singular point, I 38, 91; III 97, 102 of singularity, I 38, 41, 69, 87,
91-92, 103; II 32,
of stationary autoperiodic oscillation, III 56 of stationary motion, II 27; III 110 of stationary state, III 44, 70-71
Poincar6-Liapounoff criteria of, III 63-64 regions of, I 3
Routh-Hurwitz criteria for, II 22; IV 58, 60 threshold of, 1 127
Static friction, IV 48-49 zone of, IV 49
Stationary motion, I 23, 40-41;
III 54 stability of,
Stationary periodic motion, I 61; IV 55
Stationary self-excited oscillation, III 13-14; IV 43
Stationary oscillation, I 66-67, 71; III 15, 21, 35,
49, 55, 96 on limit cycle,
Stationary solution, I 64; II 14; III
Stationary state, III 70; IV 43-44 effect of parasitic parameters on, IV 56-65 of dynamical system, I 40; IV 56-65 of motion, I 23, 67, 69 oscillation in, III 11 stability of, III 44,
Stationary value, IV 52-53
"Struggle for Life," I 68
Strutt, III 111-114
67, 90-92 frequency of, III 88 self-excitation of, III 107
Subharmonic resonance, III 75 external, I 4; III 125 for underexcited system, III
order, III 129 of order one-half for hard self-excitation,
III 88-91 for soft self-excitation,
II 83-87 for underexcited system, III 87-88 of order one-third, III 91-92 on basis of Mathieu-Hill equation, III 125, 129
Subharmonic resonance - continued on basis of theory of Poincar6, III 75-92 phenomenon of, III 129
Subharmonic solutions, III 60
Subharmonics, III 24-25
Superregenerative circuit, III 50
"Sur les courbes d6finies par une 6quation diff6rentielle," I 2
Synchronization, III 35-36 of autoperiodic with heteroperiodic oscillation,
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 124-130
System absolutely degenerate, IV 11, 62 autonomous, IV 44 definition of, I 10 non-linear non-conservative, I 62 quasi-linear, 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, 127-126 cylindrical phase trajectories of, I 118-119 equilibrium of, I 40 motion of, I 42 oscillations of, II 80
I 62-63 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 35-36 with one degree of freedom, IV 19-29 degeneration of, IV 61 dissipative, II 67, 73; IV 45 heteroparametric excitation of, III 121-123 doubly degenerate, IV 17, 33-34 dynamical, I 10-11, 32, 51, 69, 82, 87-88, 107 differential equation of, I 20 equilibrium of, IV 58-60 stationary states of,
I 40; IV 56-65 electrical,
87 electromechanical differential equation of, I 117-118 self-excitation of, I 99, 102-104 self-excited oscillations in, I 99 102-104 equivalent linear, II 102;
33, 36 equivalent linearized, III 66 self-excitation of, III 30 isochronous, II 37; III 17 linear, I 23, 28, 67; II 4; III 27,
36 phase trajectories of, I 7-23
Routh-Hurwitz theorem for, I 3 with several degrees of freedom, III 30 linear dissipative with constant parameters, III 10 mechanical, I 99, 124; III 7-10; IV 37 differential equation of, III 9 parameter of, IV 37 self-excited Oscillations of, I 99-102 stability of, I
102 multiperiodic equivalent linearization for,
26-30 equivalent parameters of,
27 resonance in,
23 multiply degenerate, IV 30-36 non-conservative, I
17, 23, 106 closed trajectories of second kind in,
121 cylindrical phase trajectories of,
I 119-121 existence of periodic solutions in,
II 90-95 with non-linear variable damping, II 63-65 non-degenerate, IV 63-64 non-dissipative, III 20 non-isochronous, II 106; III 23 non-linear; see Non-linear system non-linear conservative; see Non-linear conservative system non-linear dissipative,
23, 40, 63, 105 autonomous,
I 62 higher approximations for,
29, 33, 35 of degenerate differential equations, IV 16-18 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 non-linear, III 24 resonance of, III 23 parameter of, III 23; IV 46 quasi-degenerate, IV 10-11, 65 quasi-isochronous, III 21 quasi-linear; see Quasi-linear system resonant, III 33, 53-57 self-excitation of, II 69;
86, 90 self-excited, III 46,
92 asynchronous action on, III
52 non-linear, III
103 triply degenerate, IV 17, 34 underexcited, III 87 subharmonic resonance for, III 87-88 with external excitation,
III 41 with external periodic excitation, III 75 with internal resonance, III 33 with more than one degree of freedom,
with non-linear variable damping,
63-65 with one degree of freedom,
10, 82; IV 47 conditions of periodicity for, II
4-14 differential equation describing, IV 25-29 motion of, I 82 with several degrees of freedom, II 14
Poincar6 method applied to, II 14 with several limit cycles, II 66-69 with two degrees of freedom, II 14-19; IV 30 with variable damping, I 105
Theodorchik, K., III 103
"Theory of Oscillations,"
I 3-4, 75; II
1, 3; IV 1
Thermionic circuits, I 101
RC oscillations in, I 111 self-excitation of, I 87; II 38
Thermionic emission, I 5
Thermionic generator, I 5, 73;
II 3, 109-111; IV 44 amplitude of oscillation in, I 67 frequency of, II 24-25 limit cycle of, II 24-25 self-excitation of, I 95-99; II 27-32; III 45 condition for hard,
II 31-32 condition for soft,
30-31 self-excited, III
14, 27 critical, I 3, 11;
46, 125; IV 26 of stability, I 127
Threshold condition, E 82
Topological methods, I 7-130 of Andronow and Witt, III 94 of Poincar6, II 69 of qualitative integration,
2-4 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 Hill-Meissner equation, III 114-117 of phase trajectories, II 31; III 100; IV 26 analytical method of, I 26-29 graphical method of, I 25-26 in in phase plane, I 29-32 in presence of singularities and limit cycles,
I 68-75 of Van der Pol plane,
II 35-38, 41
Trajectories; see Phase trajectories
Transition of singularities,
Underexcited system, III 87 subharmonic resonance for, III 87-88
Uni-dimensional phase space, IV 8, 21
Uni-dimensional real motion, I 8
Uniqueness, Cauchy's theorem of, I 24
Van der Mark, I 68
Van der Pol, I
1, 4, 20-21, II
93-94, 97, 104,
111-113; IV 1 abbreviated equation of, II 54; III 44 analytical method of, I 4 differential equation of, III 94-96 equation of, I 79, 85, 97, 103,
47, 53, 63-64, 66, 83, 85; III 45-46,
100; IV 2-3 phase trajectories of, III 100 method of, II 33-48 plane,
41 limit cycles in, I 113-115 solution, II 63; III 100-102 theory of, I
44, 66; III
Van der Pol - continued theory of performance of heart, I 68 variables of, II 35, 37-38
Variable damping, I
Variation of constants, method of, II 3, 49, 99
Variational equations, I 83; II 19; III 35-36, 63-64,
70 of Poincar6, III 80
Vincent, III 93
124-125, 130 theory of, I 125
Volterra, V., examples of limit cycles,
Vortex point, I 23 26-31
33-36, 38-39, 41, 62-63,
76, 119; IV definition of, I 12
Weaver, Dr. W.W., I 6
Witt, I 4; II 14; III 94, 126-127 (see also Andronow and Witt)
Wronskian, III 109
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