Papaleksi-Andronov - Some Research Results in the Field of the Nonlinear Oscillations - 1947.pdf

Papaleksi-Andronov - Some Research Results in the Field of the Nonlinear Oscillations - 1947.pdf
335
Original Title: Nekotoriy Issledsvaniya V Oblastiy Nelineiniy Kolebanii, Provedennyi V SSSR, Nachinaya S 1935
Published In: Uspekhi Fizicheskikh Nauk, Volume 33 Number 3, pp 335 – 352, 1947
Translated, Transcribed and Edited by: Peter J. Pesavento & Philip V. Pesavento
1947 Successes in the Physical Sciences Volume XXXIII Number 3
On The Thirtieth Anniversary of Soviet Physics
Some Research Results in the Field of the Nonlinear Oscillations, with the Lead
Position Going to the USSR, Since 1935
N.D. Papaleksi, A.A. Andronov, G.S. Gorelick, S.M. Rytov
For the last ten years, the leading position in the field of nonlinear oscillations has
essentially fluctuated1 *). It appears that this will remain the present condition until such
time as the research area expands further. Despite the pioneer works of B. van der Pol, E.
Appleton and other contributors, this research arena remains poorly developed and little
known; however, it is now possible to safely say, that the necessity of application of the
nonlinear theory and nonlinear treatment for the diversity of oscillating problems has
been successfully applied in various areas of modern technology, and has received wide
recognition not only in scientific, but also in engineering circles. Alongside wireless and
acoustics research, the theory of nonlinear oscillations has received the rights of
citizenship in the labors of the electrical engineer, the aircraft technician, as well as in the
technology of automatic control, which will be especially emphasized in this speech.
This expansion of the scope of the theory of nonlinear oscillations is the brightest feature
of practical Soviet research over the past several years. Undoubtedly, it is a theoretical
branch that is at our disposal and has become more perfect and effective in comparison
with the initial efforts back in 1935, but it does not in essence contain any new ideas.
Let us briefly enumerate the basic elements of this topic.
1 – The qualitative (topological) theory of the differential equations created by H.
Poincare2, and the geometrical images provided by him (in phase space) of the various
types of movements of dynamic systems, as, for example, the limit cycle representing
established oscillations3. Research on auto-oscillations by means of this theory has led to
new mathematical concepts of « structurally stable systems »4.
2 – The theory of a series expansion based on a small parameter, developed in connection
with problems of celestial mechanics (Euler, Lagrange, Poisson, Tisseran,
*) For a review of the research works completed up to 1935 see for example1
336
N.D. Papaleksi, A.A. Andronov, G.S. Gorelick, S.M. Rytov
Liendshtedt, Poincare and so on), which allows one to calculate a periodic driving
function, together with a method of slowly varying factors (or via van der Pol’s method5,
which was the first to be applied to radio physical problems).
These quantitative methods applied in their most simple form are the most important for a
case in radio physics of almost sinusoidal oscillations.
3 – The methods of research in regards to the stability of mechanically driven systems
based on the works of Poincare and Lyapunov6,7.
4 – The methods of reduction, in which the nonlinear associations entering into a problem
(that consists of the performance of vacuum tubes, servomotors and so on) approximates
a problem that is reduced by a series of rectilinear segments to “seaming”, by which
means that the certain conditions of the continuity, that includes in its solutions various
systems of simple equations, are equal in various parts of its phase space.
This method has appeared especially effective for treatment of systems in which the
mechanical driving cannot be considered approximately sinusoidal; such systems
represent a special interest for the theory and applications of automatic control.
5 – About this last idea we will be discussing is not completely unimportant – it
necessitates new physical language, and adequately describes properties of nonlinear
systems that are absolutely distinct from the usual linear languages; this new nonlinear
language was developed simultaneously when physicists seized upon the just enumerated
mathematical methods and created, through their efforts, obviously new representations
corresponding to their goals.
POORLY DEFINED NONLINEAR SYSTEMS
Let us remind readers first of all about some outcomes stated in the 1935 report.
Emanating from the existence of a book entitled « the periodic solutions of the second
type » by Poincare6, with additional detailed discussions about phase, multiple phases,
and the operating force, authored by L.I. Mandelstam and one of us (N.D. Papaleksi), we
have come to the conclusion that the possibility of excitation and maintaining a certain
working condition in the recycled system while being under an operation with harmonic
EMF’s, the multiple-phase oscillations are corresponding to a solution of the second type.
The theory that these appearances have enveloped themselves in are not only
characteristic of « a resonance of the second type », taking place in non-self-excited
recycled systems, but also within many appearances observed in the self-excited system
under one operational EMF or several harmonic EMF’s.
Here, there are also additional concerns: synchronization on the overtone7, an appearance
of the suppression of oscillations,8,9 auto-parametric or fractional resonances,10
337
Some Researches in the Field of Nonlinear Oscillations
combinatory resonances11,12 as well as an appearance of the so-called « asynchronous
excitation » 9,13,14,15.
These same authors had for a lengthy period of time been developing the theory of parametrical
generation of oscillations (by means of a periodic modification of parameters—capacity or selfinduction) in systems with small depths of modulation and small nonlinearity indicators16 and it
also became clear there were further numerous examples with many additional characteristic
appearances that were taking place via ion mode by the mechanical deriving of electrical energy.
By means of a combination specified above that highlighted quantitative and qualitative
problem solving methods, including the concern for the behavior of regenerative systems
both non-excited, and self-excited under an exterior operation with harmonic EMF’s,
these problems and methods have been investigated but there are issues that have
remained that have not become satisfactorily resolved. An example of this in particular,
regards a question at issue on the existence at synchronization on a threshold for an
amplitude exterior to harmonic EMF’s.
In all of the enumerated problems for the application of a method of perturbations or
methods of small parameters as well as the van der Pol method, being applied in this
supposition, one can say that in « a zero approximation » the considered systems are
linear and conservative, i.e., that in their nonlinear terms, for these specific differential
equations the more significant ones are the linear and not the conservative terms
(damping) as well as the terms containing periodic parameters, when these are small
enough.
Let’s consider as an example a tank circuit with a variable capacitance varying under the
law
1
1
(1 + m cos 2ωt )
=
C C0
(
)
And the nonlinear resistance R = R0 1 + γ i 2 , adjusted approximately on frequency with
ω. We can write then the differential equation describing this system, in the form of
3
 dq
d 2q
 dq  
2
2
2
+ ω q = ω − ω0 q − 2δ 0  + γ    − mω 2 cos 2ωt ⋅ q
(1)
2
dt
 dt  
 dt
Or
q&& + q = − µ∆q − 2 µϑ0 q& + γ q& 3 − µ cos 2τ ⋅ q
(2)
Where
ω2 − ω2
R
µ∆ = 0 2 ; 2µϑ0 = 0 ; µ = m ; and τ = ωt
ω
Lω
We consider these magnitudes small – on the order of a smallness µ. According to
Poincare6, a required periodic solution of the equation (2) will differ little from one of the
solutions of the equation
q&& + q = 0
(3)
(
)
(
)
338
N.D. Papaleksi, A.A. Andronov, G.S. Gorelick, S.M. Rytov
and its result will present false indications with regards to an aspect of some on degrees
m, and the term with a zero index that would be one of the solutions of these equations
(3) which we therefore name a « zero » as the initial solution.
Such a treatment has allowed us to discover a series of new system properties with
periodically varying parameters, for example, with appearances that in parametrically
connected systems where the research has led to the creation of a new type of motor with
a slide control for the number of revolutions17; that concern the researchers here; also, the
second parametrical resonance for a spectrum of frequencies 1:1 18; additionally, the
appearances of parametrical combinational regeneration19,20,21 which display the
parametrical resonances that are taking place in connected systems22.
However, for important practical cases, for example the parametrical generation of
alternating currents, the developed theory appeared to be, in a quantitative sense,
insufficient. The fact of the matter is that the magnitude of the potential, or EMF, of
parametrical generators grows with the magnitude of the depth of modulation of the
parameter and practically neither the depth of modulation of the parameter, or damping
detuning ω 2 − ω02 can be measured, as it is impossible to consider such a small number,
and hence it is impossible to consider also Tomsonovskiy’s initial system, i.e., to take the
initial sinusoidal solution for the case of a system with one degree of freedom, or the sum
of sinusoids for a system with many degrees of freedom.
It was condensed several years ago by L.I. Mandelstam, during the development with
reference to these cases of reduced methodical forms of small parameters where it is quite
naturally the initial approximations for the periodic solution of some linear differential
equation with periodic factors.
Let’s return to the case of the oscillating system considered above with periodically
varying capacity. The equation system (1) can be presented in such an appearance
q&& + 2ϑ1q& +
2
2
 2

ω12
(1 + m1 cos 2τ )q = ω1 −2ω0 q − 2(ϑ − ϑ1 )q& − 2ϑγ q& 3 +  ω12 m1 − m  cos 2τ ⋅ q
2
ω
ω
ω

If, on the one hand, one here chooses ϑ1 , m1 , and
ω12
so that the linear equation
ω2
ω12
q&& + 2ϑ1q& + 2 (1 + m1 cos 2τ )q = 0
ω
2
2
has a periodic solution, and on the other hand, ω1 −2ω0
small and of the order of µ,
(4)
ω
= − µ∆, 2(ϑ − ϑ1 ) = µϑ0 and m − m1 = µ
(5)
as
339
Some Researches in the Field of Nonlinear Oscillations
and the magnitude γ, defining also the nonlinearity of the system, which is also small,
then, following Poincare, it is possible to show, that the effort to approach the periodic
solution of the equation (4) will lie near a straightforward periodic solution of the
equation (5).
During L.I. Mandelstam’s final months, he developed the theory of the approximate
solution of a system of differential equations grounded on just this stated idea of periodic
factors for any depths of modulation and small nonlinearity23. This theory was then
applied to concrete cases of parametrically generating alternating currents.
Let us also mention the works of G.S. Gorelick24 and S.M. Rytov25 about non-stationary
processes in systems with periodically varying parameters, where it has also spread to
general applications in such systems where a method of slowly varying factors, such as
the method of van der Pol, can be applied.
Along with those cases where parameters vary with phase, the comparisons are basically
under an order of magnitude with the average characteristic phases of the system, in
particular those cases when the phase of the modification of the parameters is very great
and that has been deeply analyzed also. This special aspect of action gives rise to
oscillations which are usually identified as modulated. Following closely the definition of
modulated oscillation simply approximated as the oscillation slowly drifting away from
the harmonic, S.M. Rytov has given some common treatments of both kinematic as well
as dynamic modulation problems26.
In this research it is possible to note two moments.
1. The method of perturbations is applied sequentially to problems of modulation. The
small parameter µ is introduced thus as a factor at independent variable t. Modulated
oscillation, is noted in the form of
s = A(µ t )ei [ωt +ϕ (µt )], µ << ω
dA
dϕ
which are on the order of µ, i.e. A and ϕ are especially close to
and
dt
dt
a constant rather than being less than µ. Various problems in regards to systems with
modulated parameters have led to the equations whose factors vary from t through µt. For
a solution of such problems the method of slow perturbations was developed, in which
the zero approximation is similar to a quasi-stationary result that relates to an
approximate conjugate solution. This last example represents a solution of the stationary
case that is set with (µ = 0), but additionally with the replacement of arbitrary constants
of the slow functions t. This particular aspect of these functions is defined from the
analysis of the subsequent approximations, and the modes serving for this purpose, are
various, depending on the character of a problem. With reference to
Thus we have
340
N.D. Papaleksi, A.A. Andronov, G.S. Gorelick, S.M. Rytov
the nonlinear systems close to Tomasonovsky, the described procedure repeats, certainly,
if one applies the van der Pol method, but for the linear modulated systems the aspect of a
zero approximation is established differently (from the conditions of an orthogonality).
The method of measuring (slow but not necessarily small) perturbations essentially
supplements the usual method of measuring (small and remaining arbitrary)
perturbations.
2. There is a rather natural generalization given about the concept of modulation, i.e., the
slow deviation from sinusoidality, relating to spatial and existential (wave) problems. As
to a problem there that corresponds to the differential equations, quotients and
derivatives, in which the factors or the boundary conditions that hold parameters that are
dependent upon µx, µy, µz and µt. All of these outcomes are possible, certainly, and
would be dependent upon the various orders of smallness on the one or several of these
coordinates.
In relation to the wave equation, this statement of the question also envelops the usual
passage of the approximation of geometrical optics (if the factor in most of the equations
is spatially modulated, i.e., a velocity distribution), and with its diffraction impacting on
enough smooth structures (if boundary conditions are modulated spatially). Application
of the slow perturbations method to the Maxwell’s equations (in the case of an
inhomogeneous medium) has allowed us to obtain an approximation for the geometrical
optics, alongside with the conservation law of a light beam, as well as the law of a
modification of polarization along a ray, namely:
dϕ 1
=
ds T
where ϕ - is the angle between an electric vector and a principal normal to a ray, s – is a
length of an arc measured along the ray, and T – is the radius of torsion of the ray.
Let us now pass on to another group of theoretical research concerning applications of the
method of the small parameter.
Recently it has appeared possible to expand the circle problem, solved by this method,
with the introduction of specific mathematical formulas to apply to the statement of some
problems of not only « greater » magnitudes of the zero order (µ0 = 1) and « small »
magnitudes of the positive order, but also « very much greater » magnitudes of negative
orders (1/µ, 1/µ2, etc.). From a formal aspect it does not change the method for a small
parameter, but from the physical point of view of the operation of « very much greater »
magnitudes, it is not trivial and in some cases renders useful results.
The fact of the matter is that the physical reasons of greater frequency suggest to us that
the interconnectedness of the relations is on the order of several magnitudes. It is
certainly a question about which magnitudes to consider constant as µ → 0, i.e. to accept
for magnitudes of zero order is a matter of consequence for any
341
Some Researches in the Field of Nonlinear Oscillations
agreements. Each agreement of such kind is possible to have a certain equivalent choice
regarding a specific considered aspect of the system and also have a zero approximation
(µ = 0). Therefore all similar samplings, being formally equivalent to each other, are not
equivalent where discussing a physical model is concerned27.
In this way it was possible to supply data points and resolve a problem about stabilization
of frequencies of a vacuum tube generator by means of stabilizers (using a quartz
oscillator, a tuning fork, as well as a volumetric resonator).
The vacuum tube generator connected with the quartz oscillator behaves essentially
differently, than the generator with its two usual contours, both at the strong and at the
weak connection. Thus already at most the frequency control in a method of small
parameter it is necessary for a statement of the problem to reflect from the very beginning
prominent features of quartz (a contour of the 2nd order), distinguishing it from the usual
contour (1st order). Such a singularity of the stabilizer is of extremely great
value L2 / C2 . If one is to accept L2 / C2 ~ µ−1, in view of L2 / C2 ~ 1, it turns out:
L2 ∼ µ−1, C2 ∼ µ, i.e. the inductance of quartz appears to have a « very big » magnitude.
Owing to the equations for currents I1 and I2 (accordingly in a contour of the generator
(and, in an equivalent contour for quartz) becomes*)
 I&&1 + I1 = − µϑ1I&1 − µ x1I&&2 + µ I a − µ ∆ I1 
(6)
 &&

2
2
&
&&
 I 2 + I 2 = − µ ϑ2 I1 − µ x2 I1

The prominent feature exhibited by equation (6) is caused by the introduction of L2 = 1/µ,
where this is an asymmetry of orders of smallness not only by decrements, but also via
connection factors. This circumstance enables the possibility to construct by means of a
small parameter method, the strict nonlinear theory of frequency control**). Besides, this
turns out to be the somewhat obvious picture of a stabilization process. Namely, if we
consider the terms of the equations in (6) which are not containing µ, (thus having a zero
approximation) we receive two independent conservative linear oscillators, both with
identical frequency. At the accounting of terms with µ (to the first approximation) we
have the self excited generator, which is almost on a resonance (detuning µ∆) operates
with a driving force - µχ1I&&2 , i.e. this turns out to be a problem about coherence. A radiant
*) Here too we introduce dimensionless time τ = ωt, where ω = 1 / L2C2 is the quartz frequency.
Decrements µϑ1 , µ 2ϑ2 and the factors of connection µx1 = µ/L1, µx2 = M/L2 have different orders, as do
resistances R1, R2 where the order µ for a coefficient for mutual induction is M ~ µ, (L1 ∼ 1, L2 ∼ 1/µ).
∗∗)This theory is developed and quantitatively confirmed from experimental experience for the
basic schemes of stabilization (tightenings and oscillators) in28. Except for that in29, the small parameter
method it is developed in a general view for systems with two degrees of freedom and with those equations
that contain terms of various orders of smallness.
342
N.D. Papaleksi, A.A. Andronov, G.S. Gorelick, S.M. Rytov
with this force impacts the second oscillator as before even if there are independent and
conservative factors indicated. At last, to account for the terms of the second order,
simultaneously, we would enter some damping into the quartz, as well as supporting
oscillations in the quartz through action from the generator. Thus this interaction is
asymmetric: the generator acts on the quartz very poorly (~ µ2) according to a small
decrement of quartz, and quartz acts on the generator equally as poorly (~µ) and
consequently seizes (Editor’s note: phase locks) when near a resonance level point.
The precise language sense of small parameter gains is also a concept that is conditional
with frequency control, namely: it is in such a self-oscillatory condition at which, as a
result de-tunings of a contour by way of µ, the frequency deviates from a constant value
not more strongly, than by way of µ2.
The expansion of the method of small parameters briefly described here has appeared
useful, but not only with problems about stabilization of frequencies. As an example, it is
possible to specify the alternating current theory of generators where the assumption of «
very big » inductance stators also allows us to finish all the calculations, thus preserving
all of the features of interest of this considered example.
The method of small parameters also has been applied to a solution of some problems,
especially those concerning distributed parameter systems (nonlinear problems in partial
derivatives) which have gained special value in connection with such successes, reached
in the field of technology in regards to very high frequencies30,31,32.
STRONG NONLINEAR SYSTEMS
One of the basic tendencies of the research developments discussed here consists of,
having begun with problems of radio-physics, those achievements and application
protocols that were spread to an area of endeavor which at first sight seems rather far
from it like the field theory of automatic control. This theory represents a wide field for
the application and developments of physical ideas as well as the mathematical methods
– which have become usual for radio-physics – to be engaged in by research into autooscillations. In spite of the great value which has been gained with efforts on relaxation
oscillations, the radio-physicist, for abundantly clear reasons, saw infinitely more in its
purpose: he is interested in almost sinusoidal oscillations generated by poorly defined
nonlinear systems. As already mentioned above, the theory of automatic control mostly
deals with strongly nonlinear systems in which auto-oscillations, if they exist, are as
essential as they are non-sinusoidal.
Let us remind ourselves all over again of the simple example taken from the area of
radio-physics33, a method of « seaming » the trajectories in a phase space, about which
we had already made mention of in the introduction. After that we can briefly consider
some final works that are not concerned with linear problems for automatic regulating.
343
Some Researches in the Field of Nonlinear Oscillations
To avoid distorting the historical perspective, the following is a necessary criticism
beforehand. Ten years prior to the origin of wireless, the French engineer Leaute34, was
studying auto-oscillations in some device of automatic control, and actually investigated
the phase space of this device from which he had traced its integral curves and limit
cycles (but not having given it this labeling: although it was extant, Leaute was not
familiar with the work of Poincare, who, a little bit earlier, had published about limit
cycles for the first time it ever appeared in mathematics). For reasons about which we
here will not speak, the remarkable works of Leaute have almost been completely
forgotten. Research about this topic will be brief in this speech, representing itself a new
application of methods used in radio-physics, which are at the same time a demonstration
of the persistence of the works of Leaute.
Figure 1
Figure 2
If we idealize the performance of a vacuum tube such that it is specified as shown in
Fig.1, the differential equation of the system whose schematic is shown in Fig. 2, will
look like:
 I / C , when I& > 0
(7)
LI&& + RI& + I / C =  s

0,
when I& < 0 
The phase space represents a plane I , I& . The half plane I& > 0 is filled by segments of the
trajectories representing oscillations, damping around a position of
equilibrium I = I s , I& = 0 ; the half plane I& < 0 contains segments of the trajectories
corresponding to the oscillations damping around the position of an equilibrium
I = 0, I& = 0 . Segments of the trajectories should be incorporated on an axis I = 0 so that
I and I& remain continuous.
Take I n so there will be an n-th abscissa intersection of a trajectory with a
ray D (I& = 0, I > 0 ) . An abscissa (n+1)-th that intersects with that ray will be
 − kT

R
I n +1 = I n e − kT + I s  e 2 + 1; k =
,T =
2L


2π
1  R 
− 
LC  2 L 
2
(8)
344
N.D. Papaleksi, A.A. Andronov, G.S. Gorelick, S.M. Rytov
The equation (8) can be interpreted as a mathematical expression of the discussion of the
Leaute method (Fig 3). This equation expresses the point-wise transformation of a
straight line. The thick straight line corresponds to the equation (8) and the thin one
makes an angle of 45 degrees with the axes of the coordinates. It is easy to see that any
I 0 and n → ∞ of any of the I n series that aspire to a final limit will have the form:
Is
I* =
1 − e − kT / 2
It is those « amplitudes » of the limit cycle that represent the established oscillations on a
phase plane *).
The auto-oscillations arising in many automatic control devices can be studied similarly.
Let us consider an example from the
work of Andronov, Bautin and
Gorelick35, concerning, though strongly
simplified, the quite modern problem
although with a few differing aspects
that echoes the concepts described in
Leaute’s problem. Atmospheric matter
(Editor’s note: air) passes through the
airscrew, which has an automatically
controlled variable pitch. The
servomotor rotates with an angular
velocity ω according to the equation
Figure 3
Iω& = P(ω , λ ) − Q(ω ,ϕ ) ;
(9)
Here I – is a moment of inertia, and P – is a driving moment, Q – is a moment of
resistance, λ − is a parameter describing the feeding of a gas, ϕ - is the angle of the turn
of the airscrew’s blades while the servomotor operates via a centrifugal tachometer, this
all results in that the airscrew’s pitch changes under the law
ϕ& = F (ξ )
(10)
Where ξ - is the displacement clutches of the tachometer. An ideal would be to make ξ a
single valued function about ω, and F(ξ) such that it was equaled to zero only at ω = ωn
where ωn − is a demanded value of an angular velocity which should remain constant. Ιn
that case we had
*) Periodic solutions of the equation LI&& + I / C = 0 are very delicate formations: they disappear as soon as
we introduce, for example a captive RI& as though we have a very small R. On the contrary, the periodic
solution (7) continues to exist with not much greater modifications of the differential equation. The concept
a structurally stable system (see introduction) generalizes this property which can form the foundation for
the mathematical definition of auto-oscillations.
345
Some Researches in the Field of Nonlinear Oscillations
at a stationary condition (ω& = 0, ϕ& = 0 ) by virtue of equation (10) the function F(ξ) = 0 or
ω = ωn by virtue of equation (9) where ϕ consequently and by force of draft becomes a
function of λ. The corresponding choice of the parameter’s values guarantees in this case
a stability in regards to this stationary condition.
This ideal is never carried out however. First, the tachometer possesses inertia and
friction. Second, the servomotor possesses a dead zone. Function F(ξ) can have, for
example, an aspect as specified in Figure 4. These circumstances can change the
properties of the system completely. Its behavior has undergone extensive research in
considered work at the following supposition: it is possible to neglect the inertia of, but
not the friction of, a tachometer.
Figure 4
Figure 5
Let’s describe first of all the dynamics of a tachometer, including friction which follows
Coulomb’s law. As under the supposition, the tachometer has no inertia, although there is
an equilibrium between the force of friction R and the equally effective centrifugal and
restoring forces:
 F = −k ,

if ξ& > 0


R + F (ξ ) = 0 < F < k , if ξ& = 0 (k > 0 ),


if ξ& < 0
F = +k ,

Assuming, that the factors being considered in this driving function are so small, that R
can be a linear function (R = bη − aξ ; η = ω − ω N ; a > 0, b > 0 ) as we can see below:
bη − aξ = + k ,
ξ& = η& if
ξ& > 0
bη − aξ < k ,
ξ& = 0
if
(11)
bη − aξ = − k , ξ& = η& if
ξ& < 0
When the angular velocity makes the oscillation, the representing point on plane ξ ,η the
cycle that results is shown in Figure 5. With the « shift in phase » between ξ and η, it is
possible to consider this as a physical reason for the origin of the auto-oscillations.
Linearizing (9), we obtain:
η& = − Mη − Nϕ
(M
> 0, N > 0 )
346
N.D. Papaleksi, A.A. Andronov, G.S. Gorelick, S.M. Rytov
The equation (10) can be, according to Figure 4, noted in the form of

0,
if
ξ <ψ 0


F (ξ ) = − F (− ξ ) = a (ξ − ψ 0 ), if
ψ 0 < ξ <ψ1 

+ A,
if
ξ >ψ1


Let us suppose:
aξ
= x,
2k
bη
= y,
2k
−b
N
η +  = z , Mt = τ ,
2k 
M

A 
aψ 0
aψ 1
Nb
 α =

=ψ 0,
= ψ 1,
A = A,
2
2k
2k
2kM
−
ψ
ψ
1
0 

We can now write the differential equations
dz
dy
dx
(I)
= f (x, y, z );
= z;
+ z = g (x )
dτ
dτ
dτ
Where functions f and g are also certain as follows in a three-dimensional « degenerate »
1
1




phase space organized by half-planes H  x − y = , z < 0  and H ′ x − y = , z > 0  and a
2
2




1

stratum G x − y <  :
2

 z in H and H ′
f ( x, y , z ) = 
(II)

0 in G

x <ψ 0


if
ψ 0 < x <ψ1

if
x >ψ1

By virtue of the equation series in (II) the nonlinear system as described in (I) has led to
linear versions via the previously described equations, which are real in the various areas
of phase space. System (I) and the condition of the x, y, z, coordinates on the continuity
boundaries of these areas completely define the driving function via the represented
points.
0,

g ( x ) = − g (− x ) = α ( x − ψ 0 ),
 A,

if
In Figure 6 a thin line segment of a typical trajectory is shown. It looks like a spiral; we have
illustrated here a driving oscillation. While the tachometer is displaced, the representing point is
on H or H’; at the stopping of the servomotor it moves on a straight line when the servomotor is
in operation on a curve. When angular acceleration changes sign, the tachometer – because of
friction – for an instant stops, the representing point interferes in a stratum G and describes an arc
in a plane x=const. Then the tachometer again comes into action, and the representing point again
moves on H’ or H.
It is necessary to know, whether the spiral is displaced or torn and whether there are also limit
cycles. This problem is solved by means of the calculations, which have similar approaches to
ones that have been made by us for the vacuum tube generator. Through this means
347
Some Researches in the Field of Nonlinear Oscillations
the phase space – becomes three-dimensional, but it is so deteriorated that all trajectories
intersect properly with the chosen ray plane H or H’, for example
1


ray L x − y = , z = 0, x < −ψ 0  . Thus, it is enough to investigate point-wise the
2


transformation of this ray to its utmost, certainly (I) and (II) and conditions of the «
seaming » of trajectories
being satisfied.
The calculations show that
the oscillations damp out at
any entry conditions, or as in
(Figure 6) there are two limit
cycles – one steady and the
second, of smaller size that is
unstable (they are shown by
the thick lines). In the second
case it would damp out only
weak enough perturbations.
In Figure 7 it is shown in the
parameter space (α ,ψ 0 , A) in
which such auto-oscillations
are possible. It is concluded
between a plane of ψ 0 = 0
and a cylindrical surface [A],
Figure 6
forming which axes are
parallel to A and a surface [α] containing an axis A. It is always possible to save autooscillations or a diminution of k or N or the magnification of a dead zone.
Andronov, Bautin, and Gorelick’s other work,36 in the main, highlighted the problem
concerning the large class of devices relating to automatic control and the general
understanding of the case just considered. Of the key points to know about these systems,
here are a few to note: to which the velocity of the servomotor copes not only with the
device measuring the governed magnitude (for example a tachometer), but also, as
presented for the first time by the French engineer Farko, a linear combination of both the
displacement of the measuring device and the servomotor that allows the equipment to
more effectively contend with occasions of instability. Two cases for which it has
appeared possible to plot inside these boundary space parameters between areas of
stability of a stationary condition, on the one hand, and areas in which it is possible to
have auto-oscillations with another, have undergone further research: a case where the
servomotor has linear performance
348
N.D. Papaleksi, A.A. Andronov, G.S. Gorelick, S.M. Rytov
and a case of the servomotor possessing a constant velocity via an absolute value and a
dead zone.
Till now, it was a question of problems in which a phase space had so simple an
appearance, though also three-dimensional, that its research was reduced to a point-wise
transformation of a line and to a line. However, the large number of very important
problems leads to « full » three-dimensional phase spaces and that to understand during
integral curves, it is necessary to subject
Diminution N
Region of
Auto
Oscillations
Diminution k
Figure 7
to research the point-wise surface to surface transformation. Wischnegradski’s classical
problem, which together with Maxwell is the founder of the dynamic theory or automatic
control, is an example. Here it is necessary to make still one further short historical
detour.
In 1841, the well-known astronomer Airy had studied auto-oscillations, which the
clockwork intended for maintaining a uniform equatorial motion in a telescope had been
his subject of interest. He could not however provide a satisfactory theory for this
application. Maxwell, in the well-known memoir « On Governors », published in 1868,37
investigated radicals of a secular equation linearizing the equations of motion for systems
of direct regulation *) for the first time
*) That is, without the servomotor: the measuring device operates immediately off of the adjusting
mechanism.
349
Some Researches in the Field of Nonlinear Oscillations
having formulated what we now call the oscillation conditions of self-excitation. There exists an
immediate continuity between the research efforts of Maxwell about the conditions of selfexcitation of auto-oscillations and the later, widely known researches of Routh38 about the
stability of driving functions to achieve such oscillations.
Wischnegradski was interested in the problems of regulating oscillations not only from the
vantage point of the scientist, but also that of the engineer. Wischnegradski, unlike Maxwell,
(who was not familiar with Wischnegradski’s memoir39), finally, in 1886, described the problem
of direct regulating in that aspect, a technique in which he was singularly interested in at the time.
Wischnegradski’s problem remains, even to this day, one of the primary problems of regulatory
theory.
Let’s consider the machine, for example one driven by steam, in which the moving moment P
depends on position y. A governing mechanism and the moment of resistance Q is a constant:
I&ω = P( y ) − Q
(12)
The governing mechanism driven by a tachometer, where inertia and friction cannot be neglected:
y = α x,
m&x& + kx + f ( x& ) = β (ω − ω N )
(13)
Friction f (x& ) develops from viscous friction and general friction following Coloumb’s law. The
stationary condition (ω& = 0, x& = 0 ) can be steady or unstable. It is unstable, in particular, when
oscillations are caused by inertia and the elasticity of the tachometer, increase contrary to friction
owing to interaction between the tachometer and the machine. It is required to discover the
conditions of stability in connection with these pieces of equipment. Wischnegradski’s problem is
succinctly described and presented by these equations. This problem is definitely nonlinear- in
both description and solution.
Wischnegradski in his well known memoir has himself given a solution for the linear case,
when f ( x& ) = γ x& . For a special case, when viscous friction is absent, a series of outcomes has
been described by Lecornu40, Zhukovskiy41, and Mises42. We name this case the Mises’ problem.
As to Wischnegradski’s problem, it remains unsolved.
This subsequently led to the examination of the three-dimensional phase space with special
emphasis on the examination of dot transformation of a plane in a plane, where Andronov and
Maier had a new opportunity to provide the solutions to Mises’ problems43, but also have
provided a pathway mathematically to completely solve Wischnegradski’s problem as well44.
The parameter space which at an appropriate selective point can be led to be two dimensional,
and broken into three fields whose boundaries were possible to be calculated: 1) a field in which
the system spires to be stationary as to its condition, it can be at any initial
350
N.D. Papaleksi, A.A. Andronov, G.S. Gorelick, S.M. Rytov
condition; 2) a field, in which the system is unlimited leaves from a stationary condition
at any starting condition; 3) a field, depending on the staring conditions, in which the
system comes nearer to a stationary condition, or from it leaves. It has appeared also
possible to calculate for each value of parameters belonging to this peak field
perturbation, with which else the automatic control device can consult.
Let’s specify one more three-dimensional nonlinear problem investigated via the method
of transformation of a plane in a plane; the problem about stabilization on a course by an
aircraft via auto-pilot45.
Let the angles describing the position of the aircraft and its rudder, be ϕ and η then we
can write the equations in the form of
ϕ&& + Mϕ = − Nη ;
η& = F (ψ )
Where M and N –positive constants and F(ψ)−the characteristic of the servomotor. It is
supposed, that this last function is operated in a linear combination:
ψ = ϕ − αη + βϕ&
where α, β − are positive constants. The servomotor differs in its constancy of velocity
and the dead band region. The solution of this problem has shown that the parameter
space is divided into a field in which there are auto-oscillations of the aircraft around its
course, with the second field having feeble auto-oscillations and with the third area with
strong auto-oscillations.
Let’s give still other works concerning problems of automatic control46,47
Research of the iterated transformations of the aspect
(n = 1,2,3....),
xn = f (xn − 1)
which is indeed playing a much greater role in this part of our review, has allowed us to
investigate other problems about nonlinear oscillations48,49,50,51,52, and in particular, about
those auto-oscillations of systems with distributed constants such as for example a violin
string driven by a bow, or the Laherovskiy system driven by a vacuum tube51. (The
problem is reduced to research of sequential transformations of values of some function,
satisfying the wave equation, and this transformation is set by some nonlinear boundary
conditions). To a similar given problem where Bovscherverov systems53 were
investigated with the so-called « late feedback » of which these examples can serve: the
echo-location of Jaques Bodin54 and the Koulikoff-Chilowsk scheme for the
measurement of distances by means of radiowaves55. Recently a similar method was
developed by Andronov and Gorelick that had considered a nonlinear problem about a
resonance relating to a relativistic particle, moving inside of a cyclotron56.
351
Some Researches in the Field of Nonlinear Oscillations
Here the nonlinearity is caused by the dependence of the particle’s mass upon the
(Editor’s note: relativistic) velocity. Undoubtedly, the methods of applying the theory of
nonlinear oscillations, are fated to find a wide application in the examination of the
movements of electrons and ions in particle accelerators, as well as playing such an
important role in the up-to-date techniques of physical experiments, such as the electron’s
motion in devices intended for ultra-high frequency oscillations (Editor’s note:
magnetrons).
The Literature
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Oscillations; Radiopublishers, Moscow, 1936
2
H. Poincare. Sur les courbes definies par une equation differentielle. Ocuvres, Vol. 1, 1920
3
A. Andronov, C.R. 189, 559, (1929)
4
A. Andronov and L. Pontriyagin; Dokladiy Akademik Nauk USSR, 4, 247 (1937)
5
B. van der Pol, Philosophical Magazine Vol. 7, No. 3, pg 65, (1927)
6
H. Poincare, Methods nouvelles de la mecanique celeste, 1892
7
A. Lyapanov, A common problem about the stability of mechanical driving, 2nd edition, ONTI, M.-L.,
1935
8
L. Mandelstam and N. Papaleksi, Zeits. Fur Physik, 72, 273, (1931)
9
L. Mandelstam and N. Papaleksi, Zhurnal Tekhnicheskoi Fiziki, 2, 775, (1932)
10
V. Migulin, Tech Phys. USSR, 3, 841, (1937)
11
Yuri Kobzarev, Zhurnal Tekhnicheskoi Fiziki, 5, 627, (1935)
12
V. Migulin, Tech. Phys. USSR, 4, 850, (1937)
13
Yuri Kobzarev, Zhurnal Tekhnicheskoi Fiziki 3, 138, (1933)
14
L. Mandelstam and N. Papaleksi Zhurnal Tekhnicheskoi Fiziki, 4, 5, (1934)
15
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16
L. Mandelstam and N. Papaleksi, Zhurnal Tekhnicheskoi Fiziki, 4, 5, (1934)
17
N. Papaleksi, Journal of Phys. (USSR), 1, 373, (1939)
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N. Papaleksi and A. Martinov, Dokladiy Akademik Nauk USSR, No. 1, 25, (1946)
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20
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21
M. Divilkovsky and S. Rytov, Zhurnal Tekhnicheskoi Fiziki, 6, 474, (1936)
22
V. Lazarev, Zhurnal Tekhnicheskoi Fiziki, 10, 918, (1940)
23
L. Mandelstam, Zhurnal Electro-Tekhnicheskoi Fiziki, 15, 604, (1945)
24
G. Gorelick, Tech. Phys. USSR, 5, 320, (1938)
25
S. Rytov, Tech. Phys. USSR, 2, 215, (1935)
26
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27
S. Rytov, Dokladiy Akademik Nauk USSR, 47, 181, (1945)
28
S. Rytov, M. Zhabotinsky, A. Prohorov, Zhurnal Electro-Tekhnicheskoi Fiziki, 15, 557, and 613, (1945)
29
M. Zhabotinsky, Zhurnal Electro-Tekhnicheskoi Fiziki, 15, 573, (1945)
30
S. Strelkov, Tech. Phys. USSR, 2, 235, (1935)
31
A. Witt, Tech. Phys. USSR, 4, 261, (1937)
32
G. Gorelick, Journal of Phys. (USSR), 1, 465, (1939)
33
A. Andronov and S. Chaikin, ”Theory of Oscillations” Section 3,M. (1937)
34
H. Leaute, J de l’Ecole Polytechnique, 55, 1, (1885)
35
A. Andronov, N. Bautin, and G. Gorelick, C.R. Academies of Science USSR, 47, 263 (1945)
36
A. Andronov, N. Bautin, and G. Gorelick, Zhurnal of Automata and Telemechanica, Book VII, No.1, 15,
(1946)
37
J. Maxwell, Proceedings of the Royal Society, No. 100 (1868)
352
38
E. Routh, A treatise on stability of a given state of motion (1877)
I. Wischnegradski, C.R., 83, (1876)
40
L. Lecornu, Regularisation du movement dans les machines, Paris, 1897
41
N. Zhukovsky, The theory of the regulating of machines, M. 1909
42
R. von Mises, E.u.M., 26, 783, (1908), Enzykl. D. math. Wiss., 4, T.2 (1908)
43
A. Andronov, and A. Maier, Dokl. Akad.Nauk, 43, 54, (1944)
44
A. Andronov, and A. Maier, DAN, 47, 340, (1945)
45
A. Andronov and N. Bautin, DAN, 46,143, (1945)
46
A. Andronov and N. Bautin, DAN, 43, 189, (1944)
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A. Andronov and N. Bautin, DAN, 46, 277, (1945)
48
L. Holodenko, Zhurnal Tekhnicheskoi Fiziki, 10, 112, (1940)
49
L. Holodenko, Zhurnal Tekhnicheskoi Fiziki, 11, 276, (1941)
50
V. Gulijaiev, Journal of Physics, (USSR), 9, 21, (1940)
51
V. Vitkevich, Zhurnal Tekhnicheskoi Fiziki, 14, 70, (1944); 15, 33, 777, and 793 (1945)
52
K. Tehodorchik, Journal of Physics (USSR), 9, 139, (1945)
53
V. Bovscheverov, Soviet Physics, 9, 529 (1936)
54
Jaquet Bodin, L’ Aeronautique, No. 170, (1933)
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56
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39
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