On the Thirtieth Anniversary of Soviet Physics

Some Studies in the Field of Nonlinear Oscillations Conducted in the USSR Since 1935

N. D. Papaleksi, A. A. Andronov, G. S. Gorelik,
S. M. Rytov

Over the last ten years the situation in the field of nonlinear oscillations has changed substantially*). If before that time the field of nonlinear oscillations, despite the pioneering works of B. van der Pol, Appleton, and other investigators, was still little developed and little known, then at the present time one may confidently say that the need to apply nonlinear theory and nonlinear treatment to the most varied oscillatory problems arising in different areas of modern technology has gained wide recognition not only in scientific but also in engineering circles. Alongside radio and acoustics, the theory of nonlinear oscillations has acquired the rights of citizenship in electrical engineering, in aviation engineering, and in the technology of automatic control, which will be discussed separately.

It is precisely this expansion of the area of application of the theory of nonlinear oscillations that is the most striking feature of the studies of recent years. Undoubtedly, the theoretical equipment at our disposal has become more perfect and effective in comparison with 1935, but it contains no fundamentally new ideas.

Let us briefly enumerate the main elements of this equipment.

1. The qualitative (topological) theory of differential equations, created by H. Poincaré², and the geometrical images it provides (in phase space) of various types of motion of dynamical systems, such as, for example, a limit cycle representing established oscillations³. The study of self-oscillations with the aid of this theory led to the new mathematical concept of “rough systems”⁴.

2. The theory of expansion in a series with respect to a small parameter, developed in connection with problems of celestial mechanics (Euler,

*) For a survey of works carried out before 1935, see, for example, ¹.

Lagrange, Poisson, Tisserand, Lindstedt, Poincaré, and others), which makes it possible to compute periodic motions, as well as the method of slowly varying coefficients (or the van der Pol method, who first applied it to radio-physical problems).

These quantitative methods are applicable in their simplest form in the case, most important for radiophysics, of almost sinusoidal oscillations.

1. Methods for investigating the stability of motion, based on the works of Poincaré and Lyapunov.^{6,7}

2. The fitting method, which consists in the fact that the nonlinear dependence entering the problem (the characteristic of a tube, a servomotor, etc.) is approximated by a series of rectilinear segments, and the problem is reduced to the “joining,” by means of specified continuity conditions, of solutions of various systems of linear equations valid in different parts of phase space.

This method has proved especially effective for the treatment of systems whose motions cannot be regarded as approximately sinusoidal; such systems are of particular interest for the theory of automatic control.

1. Finally—and this is by no means unimportant—a new physical language, adequate to the properties of nonlinear systems and entirely different from the usual linear language; the new nonlinear language was developed simultaneously with physicists’ mastering of the mathematical methods just enumerated and with their creating the corresponding visual representations.

WEAKLY NONLINEAR SYSTEMS

Let us first recall certain results set forth in the report of 1935.

Proceeding from the existence of Poincaré’s “periodic solutions of the second kind” ^6, with a period that is a multiple of the period of the acting force, L. I. Mandelstam and one of us (N. D. Papaleksi) came to the conclusion that, in a regenerative system operating in a definite regime and subjected to the action of a harmonic e.m.f., it is possible to excite and maintain oscillations of a multiple period corresponding to a solution of the second kind. The theory of these phenomena embraced not only the “resonance of the second kind” proper, which occurs in a non-self-excited regenerative system, but also a whole series of phenomena observed in a self-excited system under the action of one or several harmonic e.m.f.’s.

Included here are: synchronization on an overtone ^7, the phenomenon of quenching of oscillations ^{8,9}, autoparametric or fractional resonances ^10, combi-

national resonances\(^{11,12}\), as well as the phenomenon of so-called “asynchronous excitation”\(^{9,3,14,5}\).

The same authors developed the theory of parametric generation of oscillations (by means of a periodic change of a parameter—capacitance or self-inductance) in systems with a small modulation depth and small nonlinearity\(^{16}\), and clarified many characteristic phenomena that occur in this new method of obtaining electrical energy by mechanical means.

With the aid of a combination of the quantitative and qualitative methods indicated above, the remaining unresolved questions concerning the behavior of a regenerative system, both underexcited and self-excited under the action of an external harmonic e.m.f., were investigated; in particular, the controversial question of the existence, under synchronization, of a threshold for the amplitude of the external harmonic e.m.f.

In all the problems listed, the perturbation method, or the small-parameter method, as well as the van der Pol method, are applicable under the assumption that, in the “zeroth approximation,” the systems under consideration are linear and conservative, i.e. that in their differential equations the nonlinear terms, the linear nonconservative terms (damping), and the terms containing periodic parameters are sufficiently small.

Let us consider, as an example, an oscillatory circuit with a variable capacitance changing according to the law

\[ \frac{1}{C}=\frac{1}{C_{0}}(1+m\cos 2\omega t), \]

and with nonlinear resistance \(R=R_{0}(1+\gamma i^{2})\), tuned approximately to the frequency \(\omega\).

We can then write the differential equation describing this system in the form

\[ \frac{d^{2}q}{dt^{2}}+\omega^{2}q=(\omega^{2}-\omega_{0}^{2})q-2\vartheta_{0}\left[\frac{dq}{dt}+\gamma\left(\frac{dq}{dt}\right)^{3}\right]-m\omega^{2}\cos 2\omega t\cdot q \tag{1} \]

or

\[ \ddot q+q=-\mu\Delta q-2\mu\vartheta_{0}(\dot q+\gamma\dot q^{3})-\mu\cos 2\tau\cdot q, \tag{2} \]

where

\[ \mu\Delta=\frac{\omega_{0}^{2}-\omega^{2}}{\omega^{2}};\qquad 2\mu\vartheta_{0}=\frac{R_{0}}{L\omega};\qquad \mu=m,\quad \text{and}\quad \tau=\omega t. \]

We regard these quantities as small—of the order of smallness \(\mu\). Then, according to Poincaré\(^{6}\), the desired periodic solution of equation (2) will differ little from one of the solutions of the equation

\[ \ddot q+q=0, \tag{3} \]

and it can be represented in the form of a series in powers of \(\mu\), where the term with the zero exponent of \(\mu\) will be one of the solutions of equation (3), which we therefore call the “zero” or initial solution.

Such an interpretation made it possible to discover a number of new properties of systems with periodically varying parameters. These include, for example, phenomena in parametrically coupled systems, the investigation of which led to the creation of a new type of motor with smooth regulation of the number of revolutions\(^{17}\); the second parametric resonance for the frequency ratio \(1:1\)\(^{18}\); phenomena of parametric regeneration\(^{19,20,21}\) and combination parametric resonances occurring in coupled systems\(^{22}\).

However, for practically important cases of parametric generation of alternating currents, the developed theory proved quantitatively insufficient. The point is that the magnitude of the power and the efficiency of parametric generators increase with the magnitude of the depth of modulation of the parameter, and in practice neither the modulation depth of the parameter, nor the damping, nor the detuning \(\omega^2-\omega_0^2\) can be regarded as small; consequently, one also cannot regard the original system as a Thomson one, i.e. take as the initial solution a sinusoid for the case of a system with one degree of freedom or a sum of sinusoids for systems with many degrees of freedom.

This led L. I. Mandelstam several years ago to develop, as applied to these cases, a new form of the small-parameter method, where—and this is quite natural—the initial approximation is a periodic solution of a certain linear differential equation with periodic coefficients.

Let us return to the case considered above of an oscillatory system with periodically varying capacitance. The equation of the system (1) can be represented in the following form:

\[ \ddot q+2\vartheta_1 \dot q+\frac{\omega_1^2}{\omega^2}(1+m_1\cos 2\tau)q = \]

\[ =\frac{\omega_1^2-\omega_0^2}{\omega^2}q-2(\vartheta-\vartheta_1)\dot q-2\vartheta q^3+ \left(\frac{\omega_1^2}{\omega^2}m_1-m\right)\cos 2\tau\cdot q . \tag{4} \]

If here, on the one hand, \(\vartheta_1\), \(m_1\), and \(\dfrac{\omega_1^2}{\omega^2}\) are chosen so that the linear equation

\[ \ddot q+2\vartheta_1\dot q+\frac{\omega_1^2}{\omega^2}\left(1+m_1\cos 2\tau\right)q=0 \tag{5} \]

has a periodic solution and, on the other hand, \(\dfrac{\omega_1^2-\omega_0^2}{\omega^2}=-\mu\Delta\),

\(2(\vartheta-\vartheta_1)=\mu\vartheta_0\) and \(m-m_1=\mu\) are small quantities of order \(\mu\),

moreover, if the quantity \(\gamma\), determining the nonlinearity of the system, were also small, then, following Poincaré, it can be shown that the approximate periodic solution of equation (4) will lie near the periodic solution of equation (5).

Quite recently L. I. Mandelstam developed, on the basis of the idea just set forth, a theory of an approximate solution of a system of differential equations with periodic coefficients for any depth of modulation and small nonlinearity \(^{23}\). This theory was then applied to concrete cases of parametric generation of alternating currents.

Let us also mention the works of G. S. Gorelik \(^{24}\) and S. M. Rytov \(^{25}\) on nonstationary processes in systems with periodically varying parameters, where the method of slowly varying coefficients (the Van der Pol method) is extended and generalized in application to such systems.

Alongside the cases when the parameters vary with a period comparable in order of magnitude with the (mean) natural periods of the system, the case in which the period of variation of the parameters is very large was also analyzed in depth. This special kind of action gives rise to oscillations which are usually called modulated. Following Rayleigh’s definition of a modulated oscillation (approximately simple) as an oscillation slowly departing from a harmonic one, S. M. Rytov gave a general interpretation of a number of both kinematic and dynamic questions of modulation \(^{26}\).

In this study two points may be noted.

1. The method of perturbations is consistently applied to questions of modulation. In this case the small parameter \(\mu\) is introduced as a factor multiplying the independent variable \(t\). The modulated oscillation is written in the form

\[ s=A(\mu t)e^{i[\omega t+\varphi(\mu t)]}, \qquad \mu \ll \omega . \]

Thus \(\dfrac{dA}{dt}\) and \(\dfrac{d\varphi}{dt}\) are of order \(\mu\), i.e. \(A\) and \(\varphi\) are the closer to constants the smaller \(\mu\) is. Various kinds of problems concerning systems with modulated parameters are reduced to equations whose coefficients depend on \(t\) through \(\mu t\). For the solution of such problems a method of slow perturbations was developed, in which the zeroth approximation is a quasistationary solution. The latter is the solution of the corresponding stationary problem (\(\mu=0\)), but with the arbitrary constants replaced by slow functions of \(t\). The form of these functions is determined from the analysis of the subsequent approximations, and the procedures used for this purpose vary depending on the character of the problem. As applied to

for nonlinear systems close to Thomson systems, the procedure described repeats, of course, van der Pol’s method, but for linear modulated systems the form of the zero approximation is established differently (from orthogonality conditions).

The method of slow (but not necessarily small) perturbations substantially supplements the usual method of small (and otherwise arbitrary) perturbations.

1. A very natural generalization is given of the concept of modulation, i.e. of a slow deviation from sinusoidality, to spatial and space-time (wave) problems. These problems correspond to partial differential equations in which the coefficients or boundary conditions contain parameters depending on \(\mu x\), \(\mu y\), \(\mu z\), and \(\mu t\). Various orders of smallness in one or several coordinates are, of course, also possible.

With respect to the wave equation, this formulation of the problem also embraces the usual transition to the approximation of geometrical optics (if the coefficient in the equation itself, i.e. the propagation velocity, is spatially modulated), and diffraction by sufficiently smooth structures (if the boundary conditions are spatially modulated). The application of the method of slow perturbations to Maxwell’s equations in the case of an inhomogeneous medium made it possible, in the approximation of geometrical optics, along with the law of conservation of the luminous flux, to obtain the law of variation of the polarization along a ray, namely:

\[ \frac{d\varphi}{ds}=\frac{1}{T}, \]

where \(\varphi\) is the angle between the electric vector and the principal normal to the ray, \(s\) is the length of the arc measured along the ray, and \(T\) is the radius of torsion of the ray.

We now turn to another group of theoretical investigations relating to the method of the small parameter.

Recently it has proved possible to expand the range of problems solved by this method by introducing, in the mathematical formulation of certain problems, not only “large” quantities of zero order \((\mu^0=1)\) and “small” quantities of positive orders, but also “very large” quantities of negative orders \((1/\mu,\ 1/\mu^2,\ \text{etc.})\). From the formal point of view this does not change the method of the small parameter, but from the physical point of view the operation with “very large” quantities is nontrivial and in a number of cases proves useful.

The point is that physical considerations for the most part indicate to us the relations between the orders of magnitude. The question of precisely which quantities should be regarded as unchanged as \(\mu \to 0\), i.e. taken as quantities of zero order, is a matter of arbitrariness

agreement. Each of the possible agreements of this kind is equivalent to a definite choice of the form of the system under consideration in the zeroth approximation \((\mu=0)\). Therefore all such choices, while formally equivalent to one another, are not equivalent with respect to the physical model under discussion[^27].

It was precisely along these lines that it proved possible to pose and solve the problem of stabilizing the frequency of a vacuum-tube generator by means of stabilizers (quartz, tuning fork, cavity resonator).

A vacuum-tube generator coupled with quartz behaves in an essentially different way from a generator with two ordinary circuits, both for strong and for weak coupling. Thus, already in the very formulation of the problem of frequency stabilization by the small-parameter method, it is necessary from the outset to reflect the characteristic features of quartz (the 2nd circuit), which distinguish it from an ordinary circuit (the 1st). Such a feature of the stabilizer is the exceedingly large value of \(\sqrt{L_2/C_2}\). If one takes \(\sqrt{L_2/C_2}\sim \mu^{-1}\), then, since \(\sqrt{L_2 C_2}\sim 1\), one obtains: \(L_2\sim \mu^{-1}\), \(C_2\sim \mu\), i.e. the inductance of the quartz turns out to be a “very large” quantity. Owing to this, the equations for the currents \(I_1\) and \(I_2\) (respectively in the generator circuit and in the equivalent quartz circuit) take the form1

\[ \left. \begin{aligned} \ddot I_1 + I_1 &= -\mu \theta_1 \dot I_1 - \mu x_1 \dot I_2 + \mu I_a - \mu \Delta I_1,\\ \ddot I_2 + I_2 &= -\mu^2 \theta_2 \dot I_1 - \mu^2 x_2 \ddot I_1 . \end{aligned} \right\} \tag{6} \]

A characteristic feature of equations (6), due to the introduction of \(L_2=1/\mu\), is the asymmetry in the orders of smallness not only of the decrements but also of the coupling coefficients. This circumstance makes it possible to construct, by means of the small-parameter method, a rigorous nonlinear theory of frequency stabilization2. In addition, one obtains a certain visual picture of the processes occurring during stabilization. Namely, if one takes into account the terms of equations (6) that do not contain \(\mu\) (the zeroth approximation), then one obtains two independent linear conservative oscillators oscillating with the same frequency. When the terms with \(\mu\) are taken into account (the first approximation), we have a self-excited generator on which, almost in resonance (detuning \(\mu\Delta\)), there acts a force \(-\mu x_1 \ddot I_2\), i.e. a problem of capture is obtained. The source

the force is the second oscillator, still independent and conservative. Finally, when terms of the second order are taken into account, the damping of the quartz and the action of the generator, maintaining the oscillations in the quartz, enter simultaneously. Thus the interaction is asymmetric: the generator acts on the quartz very weakly \((\sim \mu^2)\), in accordance with the very small decrement of the quartz, while the quartz acts on the generator simply weakly \((\sim \mu)\), and therefore can entrain it near resonance.

The very concept of a regime with frequency stabilization acquires a precise meaning in the language of the small parameter, namely: it is such an auto-oscillatory regime in which, as a result of detuning of the circuit of order \(\mu\), the frequency deviates from a constant value by no more than an amount of order \(\mu^2\).

The extension of the small-parameter method briefly described here proved useful not only in the problem of frequency stabilization. As an example one may point to the theory of alternating-current generators, where the assumption of a “very large” inductance of the stator also makes it possible to carry the whole calculation through to the end, while preserving all the features of interest in the phenomenon under consideration.

The small-parameter method has also been applied to the solution of certain problems concerning systems with distributed parameters (nonlinear problems in partial derivatives), which have acquired special importance in connection with the successes achieved in the field of very-high-frequency technology ^30, ^31, ^32.

STRONGLY NONLINEAR SYSTEMS

One of the main tendencies in the development of the investigations set forth here was that, beginning with questions of radiophysics, they spread into a field which, at first sight, seems very remote from it—the field of the theory of automatic control. This theory presents a broad arena for the application and development of physical ideas and mathematical methods that have become customary for radiophysicists engaged in the study of self-oscillations. In spite of the great importance which relaxation oscillations have acquired, radiophysics, for quite obvious reasons, is more interested in nearly sinusoidal oscillations generated by weakly nonlinear systems. As was already said above, the theory of automatic control for the most part deals with strongly nonlinear systems, in which self-oscillations, if they exist, are essentially nonsinusoidal.

Let us first recall, by means of a simple example taken from radiophysics ^33, the method of “matching” trajectories in phase space, which we have already mentioned in the introduction. After this we shall be able briefly to consider some recent works relating to nonlinear problems of automatic control.

In order not to distort the historical perspective, it is necessary first to make the following remark. Ten years before the advent of radio, the French engineer Leauté\({}^{34}\), studying self-oscillations in a certain automatic-control device, investigated the phase space of this device and drew for it integral curves and limit cycles (without giving them this name: he apparently was not familiar with Poincaré’s work, published somewhat earlier, in which limit cycles first appeared in mathematics). For reasons of which we shall not speak here, Leauté’s remarkable works were almost completely forgotten. The studies to be discussed here, while constituting new applications of methods used in radiophysics, are at the same time a continuation of Leauté’s work.

Fig. 1.

Fig. 2.

If the characteristic of the tube is idealized as indicated in Fig. 1, then the differential equation of the system whose circuit is shown in Fig. 2 will have the form

\[ L\ddot I+R\dot I+\frac{I}{C}= \begin{cases} I_s/C, & \text{if } \dot I>0,\\ 0, & \text{if } \dot I<0. \end{cases} \tag{7} \]

The phase space is the plane \(I,\dot I\). The half-plane \(\dot I>0\) is filled with segments of trajectories representing oscillations that decay about the equilibrium position \(I=I_s,\dot I=0\); the half-plane \(\dot I<0\) contains segments of trajectories corresponding to oscillations decaying about the equilibrium position \(I=0,\dot I=0\). The segments of trajectories must be joined on the axis \(\dot I=0\) in such a way that \(I\) and \(\dot I\) remain continuous.

Let \(I_n\) be the abscissa of the \(n\)-th intersection of the trajectory with the half-line \(D(\dot I=0,\ I>0)\). The abscissa of its \((n+1)\)-st intersection with the same half-line will be

\[ I_{n+1}=I_n e^{-hT}+I_s\left(e^{-\frac{hT}{2}}+1\right); \quad h=\frac{R}{2L}, \quad T=\frac{2\pi}{\sqrt{\frac{1}{LC}-\left(\frac{R}{2L}\right)^2}}. \tag{8} \]

Equation (8) can be discussed by Leauté’s method (Fig. 3). This equation expresses a point transformation of a straight line into itself. The heavy straight line corresponds to equation (8), while the thin one makes an angle of \(45^\circ\) with the coordinate axes. It is easy to see that for any \(I_0\) and \(n \to \infty\), \(I_n\) tends to the finite limit

\[ I^*=\frac{I_s}{1-e^{-hT/L}}. \]

Such is the “amplitude” of the limit cycle representing the established oscillations in the phase plane\(^*\).

Self-oscillations arising in many automatic-control devices can be studied in an analogous way. We shall consider, as an example, the work of Andronov, Bautin and Gorelik\(^{35}\), concerning a task which, although greatly simplified, is nevertheless a quite modern one and differs little from Leauté’s problem. The question concerns a propeller with an automatically controlled variable pitch.

Fig. 3.

The motor rotates with angular velocity \(\omega\) according to the equation

\[ I\dot{\omega}=P(\omega,\lambda)-Q(\omega,\varphi); \tag{9} \]

here \(I\) is the moment of inertia, \(P\) is the driving torque, \(Q\) is the resisting torque, \(\lambda\) is the parameter characterizing the gas feed, and \(\varphi\) is the angle of rotation of the propeller blades, while the servomotor, controlled by a centrifugal tachometer, changes the propeller pitch according to the law

\[ \dot{\varphi}=F(\xi), \tag{10} \]

where \(\xi\) is the displacement of the tachometer sleeve. Ideally, one would make \(\xi\) a single-valued function of \(\omega\), and \(F(\xi)\) such that it would be zero only for \(\omega=\omega_N\), where \(\omega_N\) is the required value of the angular velocity, which must remain constant. In such a case we would have

\(^*\) The periodic solutions of the equation \(L\ddot I+I/C=0\) are very “delicate” formations: they disappear as soon as we introduce, for example, the term \(R\dot I\), however small \(R\) may be. On the contrary, the periodic solution (7) continues to exist under (not too large) changes of the differential equation. The concept of a “rough system” (see the introduction) generalizes this property, which may serve as a basis for the mathematical definition of self-oscillations.

SOME INVESTIGATIONS IN THE FIELD OF NONLINEAR OSCILLATIONS

so that in the stationary regime \((\dot{\omega}=0,\dot{\varphi}=0)\), by virtue of (10), \(F(\xi)=0\) for \(\omega=\omega_N\), whereas by virtue of (9) \(\varphi\) (and consequently also the thrust force) becomes a function of \(\lambda\). A corresponding choice of the parameter values guarantees in this case the stability of this stationary regime.

This ideal is never realized. The tachometer possesses inertia and friction. The servomotor possesses a dead zone. The function \(F(\xi)\) may have, for example, the form indicated in Fig. 4. These circumstances can completely change the properties of the system. Its behavior was investigated in the work under consideration under the following assumption: the inertia, but not the friction, of the tachometer may be neglected.

Fig. 4.

Fig. 5.

Let us first describe the dynamics of the tachometer, assuming that the friction in it obeys Coulomb’s law. Since, by assumption, the tachometer has no inertia, there exists an equilibrium between the friction force and the resultant \(R\) of the centrifugal and restoring forces,

\[ R+F(\xi)= \begin{cases} F=-k, & \text{if } \dot{\xi}>0,\\ 0<|F|<k, & \text{if } \dot{\xi}=0 \quad (k>0),\\ F=+k, & \text{if } \dot{\xi}<0. \end{cases} \]

Assuming that the motions under consideration are so small that \(R\) can be linearized \((R=b\eta-a\xi;\ \eta=\omega-\omega_N;\ a>0,\ b>0)\), we obtain:

\[ \begin{aligned} b\eta-a\xi&=k, & \dot{\xi}&=\dot{\eta}, & \text{if } \dot{\xi}&>0,\\ |b\eta-a\xi|&<k, &&& \text{if } \dot{\xi}&=0,\\ b\eta-a\xi&=-k, & \dot{\xi}&=\dot{\eta}, & \text{if } \dot{\xi}&<0. \end{aligned} \tag{11} \]

When the angular velocity performs an oscillation, the representing point describes in the \(\xi,\eta\)-plane the cycle shown in Fig. 5. The “phase shift” between \(\xi\) and \(\eta\) may be regarded as the physical cause of the occurrence of self-oscillations.

Linearizing (9), we obtain:

\[ \dot{\eta}=-M\eta-N\varphi \qquad (M>0,\ N>0). \]

Equation (10) may, according to Fig. 4, be written in the form

\[ F(\xi)=-F(-\xi)= \begin{cases} 0, & \text{if } |\xi|<\overline{\psi}_0,\\ \dfrac{2}{a}(\xi-\overline{\psi}_0), & \text{if } \overline{\psi}_0<\xi<\overline{\psi}_1,\\ +\overline{A}, & \text{if } \xi>\overline{\psi}_1. \end{cases} \]

Let us put:

\[ \frac{a\xi}{2k}=x,\qquad \frac{b\eta}{2k}=y,\qquad -\frac{b}{2k}\left(\eta+\frac{N}{M}\right)=z,\qquad Mt=\tau, \]

\[ \frac{a\overline{\psi}_0}{2k}=\psi_0,\qquad \frac{a\overline{\psi}_1}{2k}=\psi_1,\qquad \frac{Nb}{2kM^2}\,\overline{A}=A \left(\alpha=\frac{A}{\psi_1-\psi_0}\right). \]

We can now write the differential equations

\[ \frac{dx}{d\tau}=f(x,y,z);\qquad \frac{dy}{d\tau}=z;\qquad \frac{dz}{d\tau}+z=g(x), \tag{I} \]

where the functions \(f\) and \(g\) are defined as follows in the three-dimensional “degenerate” phase space formed by the half-planes

\[ H\left(x-y=\frac{1}{2},\ z<0\right) \quad \text{and} \quad H'\left(x-y=-\frac{1}{2},\ z>0\right) \quad \text{and the layer} \]

\[ G\left(|x-y|<\frac{1}{2}\right): \]

\[ f(x,y,z)= \begin{cases} z \text{ in } H \text{ and } H',\\ 0 \text{ in } G, \end{cases} \tag{II} \]

\[ g(x)=-g(-x)= \begin{cases} 0, & \text{if } |x|<\psi_0,\\ \alpha(x-\psi_0), & \text{if } \psi_0<x<\psi_1,\\ A, & \text{if } x>\psi_1. \end{cases} \]

By virtue of (II), the nonlinear system (I) reduces to six systems of linear equations, which are valid in different regions of the phase space. The system (I) and the condition of continuity of \(x,y,z\) at the boundaries of these regions completely determine the motion of the representative point.

In Fig. 6 a segment of a typical trajectory is shown by a thin line. It has the form of a spiral; here we have oscillatory motion. While the tachometer is being displaced, the representative point lies on \(H\) or \(H'\); when the servomotor stops, it moves along a straight line, and when the servomotor is in action, along a curve. When the angular acceleration changes sign, the tachometer, because of friction, stops for an instant, the representative point enters the layer \(G\) and describes an arc in the plane \(x=\mathrm{const}\). Then the tachometer again begins to act, and the representative point again moves along \(H'\) or \(H\).

It is necessary to know whether the spiral winds up or unwinds, and whether limiting cycles exist.

This question is solved with the aid of calculations analogous to those that we carried out for the vacuum-tube generator. Although here

the phase space is three-dimensional, but it is so degenerate that all trajectories intersect, in the proper way, a chosen half-line in the plane \(H\) or \(H'\), for example the half-line \(L\left(x-y=\frac12,\ z=0,\ x<-\psi_0\right)\). Thus it is sufficient to study the point transformation of this half-line into itself, determined by (I) and (II) and by the “matching” conditions for the trajectories.

The calculation shows that either the oscillations die out for any initial conditions, or (Fig. 6) there exist two limit cycles—one stable and the other, of smaller size, unstable (they are shown by thick lines). In the second case only sufficiently weak disturbances die out. In Fig. 7 there is shown, in the parameter space \((a,\ \psi_0,\ A)\), the boundary of the region in which self-oscillations are possible. It is enclosed between the plane \(\psi_0=0\), the cylindrical surface \([A]\), whose generators are parallel to the axis \(A\), and the surface \([a]\), which contains the axis \(A\). Self-oscillations can always be avoided either by decreasing \(k\) and \(N\), or by increasing the zone of insensitivity.

Fig. 6.

In another paper by Andronov, Bautin, and Gorelik \(^{36}\), a problem is analyzed which pertains to a large class of automatic-control devices and generalizes the case just considered. It concerns systems in which the speed of the servomotor is controlled not only by an instrument measuring the regulated quantity (for example, a tachometer), but, as was first proposed by the other French engineer Farcot, by a linear combination of the displacement of the measuring instrument and of the servomotor itself, which makes it possible to combat more effectively the causes of instability. Two cases were investigated for which it proved possible to draw, in the parameter space, boundaries between the regions of stability of the stationary regime, on the one hand, and the regions in which self-oscillations are possible, on the other: the case of a servomotor with a linear characte-

characteristic and the case of a servomotor possessing a constant velocity in absolute magnitude and a zone of insensitivity.

Up to now we have been speaking of problems in which the phase space, although three-dimensional, had such a simple form that its investigation was reduced to a point transformation of a line into a line. However, a large number of very important problems lead to “complete” three-dimensional phase spaces, and in order to understand the course of the integral curves it is necessary to subject to investigation a point transformation of a surface into a surface.

[Figure labels visible in the diagram: \(A\); \(\alpha\); \(3.04\); \(\psi_0\); \(0.1, 0.2, 0.3, 0.4, 0.5\); \(1, 3, 5, 7, 9\); “Decrease of \(N\)”; “Decrease of \(k\)”; “Region of self-oscillations”; \([\alpha]\); \([A]\).]

Fig. 7.

Such, for example, is the classical problem of Vyshnegradskii, who, together with Maxwell, is the founder of the dynamical theory of automatic regulation. Here it is necessary to make one more brief historical digression.

As early as 1841 the famous astronomer Airy studied the self-oscillations to which a clock mechanism was subjected, intended for maintaining the uniform rotation of an equatorial. However, he was unable to give a satisfactory theory of this phenomenon. Maxwell, in his famous memoir “On Governors,” published in 18683, investigated the roots of the characteristic equation of the linearized equations of motion of a system of direct regulation*) and for the first time

formulated what we now call the conditions for self-excitation of oscillations. There is a direct continuity between Maxwell’s investigations of the conditions for self-excitation of self-oscillations and the later, widely known investigations of Routh^38 on the stability of motion.

Vyshnegradsky was interested in problems of regulation not only as a scientist, but also as an engineer. Unlike Maxwell, whose memoir was not known to Vyshnegradsky,^39 the latter in 1886 posed the problem of direct regulation in the form in which it alone interested technology at that time. Vyshnegradsky’s problem remains to this day one of the fundamental problems of the theory of regulation.

Let us consider a machine, for example a steam engine, in which the driving torque \(P\) depends on the position \(y\) of the regulating element, while the resisting torque \(Q\) is constant:

\[ \dot{\omega}=P(y)-Q. \tag{12} \]

The regulating element is controlled by a tachometer whose inertia and friction cannot be neglected:

\[ y=\alpha x, \]

\[ m\ddot{x}+kx+f(\dot{x})=\beta(\omega-\omega_N). \tag{13} \]

The friction \(f(\dot{x})\) is composed of viscous friction and friction obeying Coulomb’s law.

The stationary regime \((\dot{\omega}=0,\ \dot{x}=0)\) may be stable or unstable. It is unstable, in particular, when oscillations due to the inertia and elasticity of the tachometer grow, despite friction, as a result of the interaction between the tachometer and the machine. It is required to find the conditions of stability. Such is Vyshnegradsky’s problem. This problem is nonlinear.

Vyshnegradsky himself, in his famous memoir, gave a solution for the linear case, when \(f(\dot{x})=\gamma\dot{x}\). For the special case when viscous friction is absent, a number of results were obtained by Léauté,^4 Zhukovsky^41 and Mises.^42 We call this case the Mises problem. As for Vyshnegradsky’s problem, it remained unsolved.

By reducing the investigation of a three-dimensional phase space to the investigation of point transformations of the plane into itself, Andronov and Maier have recently not only obtained the possibility of giving a solution of the Mises problem,^43 but also of completely solving Vyshnegradsky’s problem.^44

The space of parameters, which, with a proper choice of the latter, can be reduced to a two-dimensional one, is divided into three regions, the boundaries of which have been calculated: 1) a region in which the system tends to a stationary regime for any initial

conditions; 2) the region in which the system recedes without bound from the stationary regime for any initial conditions; 3) the region in which, depending on the initial conditions, the system either approaches the stationary regime or recedes from it. It also proved possible to compute, for each value of the parameters belonging to this region, the maximum perturbation with which the automatic-control device can still cope.

Let us indicate one more three-dimensional nonlinear problem investigated by the method of transformation of the plane into the plane: the problem of stabilization of an airplane’s course by an autopilot[^45].

Let the angles characterizing the position of the airplane and of its rudder be \(\varphi\) and \(\eta\); then we can write the equations in the form

\[ \ddot{\varphi}+M\varphi=-N\eta;\qquad \dot{\eta}=F(\psi), \]

where \(M\) and \(N\) are positive constants and \(F(\psi)\) is the characteristic of the servomotor. It is assumed that this latter is controlled by the linear combination

\[ \psi=\varphi-\alpha\eta+\beta\dot{\varphi} \]

(\(\alpha,\ \beta\) are positive constants). The servomotor is distinguished by constancy of speed and by a dead zone. The solution of the problem showed that the parameter space is divided into a region in which there are no self-oscillations of the airplane about its course, a second region with weak self-oscillations, and a third with strong self-oscillations.

Let us cite still other analogous works relating to problems of automatic control[^46],[^47].

The investigation of iterated transformations of the form

\[ x_n=f(x_{n-1})\qquad (n=1,\ 2,\ 3\ldots), \]

which play such a large role in this part of our survey, made it possible to study other problems on nonlinear oscillations[^48]–[^52], in particular on self-oscillations of systems with distributed constants, for example, of a violin string excited by a bow, or of a Lecher system excited by an electron tube[^51]. (The problem is reduced to the investigation of successive transformations of the values of a certain function satisfying the wave equation, this transformation being specified by certain nonlinear boundary conditions.) The systems with so-called “delayed feedback” investigated by Bosheverov[^53] lead to an analogous problem; examples of these may be the echo sounder of Jacques-Bodin[^54] and the Kulikov-Shilovskii scheme for measuring distances by means of radio waves[^55].

Recently, by an analogous method, Andronov and Gorelik considered the nonlinear problem of resonance of a relativistic particle moving in a cyclotron[^56]. Here the nonlinearity is due to the depen-

mass on velocity. Undoubtedly, the methods of the theory of nonlinear oscillations are destined to find wide application in studies of the motions of electrons and ions in accelerators, which play such an important role in the modern techniques of physical experiment, as well as of the motions of electrons in devices intended for the generation of ultrahigh frequencies.

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1. Here a dimensionless time \(\tau=\omega t\) has also been introduced, where \(\omega=1/\sqrt{L_2 C_2}\) is the frequency of the quartz. The decrements \(\mu \theta_1,\mu^2 \theta_2\) and the coupling coefficients \(\mu x_1=\mu M/L_1,\ \mu^2 x_2=M/L_2\) have different orders, since the resistances \(R_1,R_2\) are of order \(\mu\), and the coefficient of mutual inductance \(M\sim \mu\,(L_1\sim 1,\ L_2\sim 1/\mu)\). ↩

2. This theory has been developed and quantitatively confirmed experimentally for the principal stabilization circuits (pulling and oscillator) in [^28]. In addition, in [^29] the small-parameter method is developed in a general form for systems with two degrees of freedom whose equations contain terms of different orders of smallness. ↩

3. *) That is, without a servomotor; the measuring device acts directly on the regulating member. ↩