ON SOME CONTEMPORARY PROBLEMS IN THE FIELD OF OSCILLATIONS*

N. D. Papaleksi, Leningrad

I have had the honor of presenting at today’s session a report “On Some Contemporary Problems in the Field of Oscillations.” First of all, I would ask permission to say a few words—not, of course, in order to explain why one of the sessions of the Section of Electrical Oscillations of our congress is devoted to discussion of questions in the field of oscillations; there is obviously no need for that—but in order to clarify precisely which problems will be discussed and why it was decided to make their discussion the subject of the present session.

Everyone is well aware of the enormous role played by oscillatory processes in the most varied branches of physics and its applications, and of the fundamental upheavals in our concepts and representations to which they have already led and continue to lead in various areas of physics. It is enough to point, for example, to the problems of wave mechanics, brought to the fore by oscillatory processes in atoms, to which a number of sessions of our congress have been devoted. But besides these “microscopic” phenomena, for the explanation of which we are in many respects compelled radically to break with established concepts and representations, there also exists another, extremely extensive field of oscillatory processes, likewise representing a significant

* Report at the First All-Union Congress of Physicists in Odessa, August 22–23, 1930.

interest. This includes both the many periodic processes in astronomy, which have a purely “macroscopic” character, and those phenomena that form the basis of the physics of oscillations in the narrower sense of the word—in particular, the essence and content of high-frequency electrical oscillations and their principal application, radio engineering. The periodic process in a tube oscillator, the phenomenon of intermittent current in a glow discharge, relaxation oscillations of the Abraham–Bloch multivibrator type, the problem of generating electrical oscillations in general, oscillations of electromechanical systems, the phenomenon of parametric excitation and resonance, the sounding of a string under the action of a bow, the action of a hydraulic ram, the oscillations of Froude’s pendulum, periodic chemical reactions, the activity of the cardiac muscle, the fluctuation of animal species in the struggle for existence, the problem of variable stars (Cepheids) in astronomy, and so on—these are a number of examples of oscillatory processes in the most varied fields of natural science. Although these processes do not affect the foundations of our worldview to the same degree as the former, they nevertheless present many very interesting and fundamentally important features, the study of which leads to a number of problems that are not always easily accessible to analysis, yet are fundamentally essential. A rigorous solution of these problems—some of which will be discussed below—is important not only for the physics of oscillations and for the physics of electrical oscillations in particular, but also for the most diverse other branches of exact and applied knowledge. From the mathematical point of view, these problems are interesting because they lead to nonlinear differential equations of a special kind, whose properties and peculiarities physicists, and especially those engaged in applied physics, have not yet mastered to the same extent as the linear differential equations that are so close and, as it were, native to us all. Nonlinear differential equations require, for their full discussion, the application of new, distinctive methods and devices. In recent years, some of the problems belonging here have been

as, for example, the problem of self-oscillations of the Thomson and non-Thomson type, the conditions for their occurrence and stability, the problem of the interaction of such systems with one another, especially the problems of frequency “pulling” and “entrainment,” parametric excitation, etc., have been subjected to in-depth theoretical and experimental study both abroad (the works of van der Pol, Hettner, Watanabe, Günther-Winter, and others are especially important) and in our country. In particular, for several years now the whole circle of questions belonging here has been studied from various sides, both with regard to scientific deepening and to practical application, at the Institute of Theoretical Physics of Moscow University, in the Scientific-Research Laboratory of the Central Radio Laboratory, in the Physical Laboratory of the All-Union Electrotechnical Institute, and in the Communications Department of the State Physico-Technical Institute. Since these investigations have already led to extensive material which is in many respects new and of fundamental interest, and since the paths and methods of investigation that have emerged, as well as the mathematical methods that have been developed, have proved quite adequate and very fruitful, it was therefore considered desirable to devote one of the meetings of the section to their presentation and discussion. In view of the abundance and the character of the material itself, naturally dividing into two parts—general and specially mathematical—it was decided to develop it in two reports, which in a certain sense complement each other. One of them I have the honor to present; the other will be given by A. A. Andronov.

PART I. SELF-OSCILLATING SYSTEMS]

From the whole extensive field of oscillatory processes, allow me first of all to dwell on questions connected with the very occurrence and establishment of oscillations, in other words, with their generation. In particular, permit me to consider in somewhat greater detail the basic problems arising in the study of “autonomous closed” systems, in which oscillatory processes can arise and be maintained for a long time, and which have received the name “self-oscillating.”

I shall first dwell a little on very simple and universally well-known things. Let us consider, for example, such an oscillatory system as a load (mass) with a spring or an electrical oscillatory circuit. In order to set this system into oscillation, we must evidently remove it from the position of equilibrium, i.e. impart to it some store of energy from outside and then leave it to itself. The system will then begin to perform oscillations about its position of equilibrium, and the physical essence of this process obviously consists in the fact that, owing to the presence in these systems of two heterogeneous forms of energy accumulation (two reservoirs of energy), energy passes from one (elastic or electrical) form into another (kinetic or magnetic) and back again. However, the oscillations do not remain unchanged, but, owing to inevitable losses (the systems, as we say, are nonconservative), they gradually die out. As is well known, such behavior of these systems can be described with sufficient completeness by a linear differential equation of the second order with constant coefficients, for example:

\[ L\frac{d^{2}q}{dt^{2}}+R\frac{dq}{dt}+\frac{q}{C}=0, \tag{1} \]

and therefore we denote such systems as linear oscillatory systems (of the nonconservative type) with one degree of freedom. The behavior of more complex oscillatory systems of this kind, characterized by the presence of several energy reservoirs, is described not by one but by several linear differential equations. A further generalization leads to continuous oscillatory systems with an infinite number of degrees of freedom, described by linear partial differential equations. Let us note that the behavior of any system near a position of stable equilibrium may, for sufficiently small deviations, be represented by a system of linear differential equations, and thus, under known conditions, it is transformed into a linear oscillatory system. The properties of such linear

of oscillatory systems reduce chiefly to the following:

1) the presence of a stable position of equilibrium;

2) the possibility of carrying out damped oscillations about this position of equilibrium;

3) independence of the period of the oscillations from the amplitude of the oscillations (isochronism).

The first property, as you know, is connected with the presence of losses of the dissipative term \(R \frac{dq}{dt}\) in equation (1). The second depends on the relation between the energy imparted to the system and its expenditure during a certain interval of time, determined by the parameters of the system \((L, C)\), and is expressed by the well-known condition for the oscillatory character of the solution:

\[ \frac{1}{LC} > \frac{R^2}{4L^2}. \tag{2} \]

The magnitude of the “period” in this case is given by Thomson’s well-known formula:

\[ \tau=\frac{2\pi}{\sqrt{1/LC-R^2/4L^2}} \tag{3} \]

or, for small logarithmic decrements:

\[ \tau=2\pi\sqrt{LC}, \tag{3_1} \]

and this latter dependence is fulfilled the better, the smaller the damping of the system, i.e., the less the oscillations deviate from sinusoidal ones. Systems, even nonlinear ones, whose oscillations are determined with sufficient approximation by this dependence are called systems of the Thomson type.

If we consider such a linear oscillatory system as some isolated closed autonomous system, then obviously we shall not obtain long-lasting oscillations in it. For this we must either act on it from outside by a periodic force (but then the system will already be governed by a differential equation with a free term—it will, as we say, perform forced oscillations), or we must introduce into it some device permitting oscillations—

to arise and be maintained at the expense of a local store of energy.

In the first case, to be sure, the resonance properties of linear oscillatory systems are especially striking; they offer so much of interest theoretically and experimentally and present a number of interesting problems in themselves. However, this case gives nothing new for the problem of the generation of oscillations that concerns us, since it transfers the difficulty of the problem essentially to the problem of creating a periodic action, i.e., again to the problem of generating oscillations.

As for the second case, here various paths are possible for creating an autonomous system that generates oscillations. Permit me here and in what follows to use electrical examples, since in many respects they have been more fully developed and are more convenient for illustrating the problems under consideration; nevertheless, the conclusions drawn apply equally to other systems as well.

Historically, the first is the classical method of the spark discharge of a capacitor, the method of Fessenden. For a long time it was the only practical method of generating electrical oscillations and it played (and continues, in a certain sense, to play even now) an enormous role both in science and in technical applications. This method consists in the fact that a capacitor charged to a certain voltage is automatically closed through a self-inductance; the role of such an automatic switch is played by the electric spark, which goes out again at the end of the oscillatory process and thereby makes possible the renewal of the charge of the capacitor. If we add to this oscillatory system another device for charging the capacitor, which can be carried out either directly by a direct-current source or, as was done originally, with the aid of an inductor with an interrupter (which in turn constitutes an autonomous oscillatory system of another type,* to which we shall return below), and

* As one example of such autonomous oscillatory systems one may point to the electric interrupter (bell), whose theory

source of current, we obtain a truly autonomous oscillatory system—a generator of oscillations. Here we have an example of a self-oscillatory system, i.e. an autonomous system in which, at the expense of the energy of a source included in it, oscillations are generated, i.e. can arise and be maintained for a long time. Since it is obvious that the complete oscillatory process in this system can no longer be represented by Thomson’s linear differential equation, the question of a theory of such a system naturally arises. I shall note at once that a complete and more or less rigorous mathematical theory, i.e. one that would take into account both the process in the spark and the operation of the interrupter, so far as I know, does not exist even now. Therefore the theory, especially at first, contented itself with giving an account of the most important aspects of the process and, for this purpose, idealized the process itself. Since, on the one hand, the most important phase of the process was undoubtedly the oscillatory discharge of the capacitor, which is represented with sufficient completeness by Thomson’s equation, and, on the other hand, the process of closing (formation of the spark) takes place very rapidly (in comparison with the period of the oscillations, down to the very shortest), it was natural not to consider the process in the spark at all, but, taking into account the losses in it by introducing an additional resistance into the system, to regard its principal role as the instantaneous creation, at certain moments, of new initial conditions. Thus the difficulty of interpreting the very problem of generating oscillations was here bypassed by introducing a discontinuity—a jump in the initial conditions—thereby creating the appearance of a linear interpretation. The fundamental questions of self-oscillations, such as the very occurrence of oscillations, the establishment of a definite amplitude, and so on, remained—and this could not be otherwise—unexamined.

Before proceeding to consider the methods of interpreting both these and other questions, which it is more convenient to do on

which M. A. Leontovich gave (JRFKhO, physical section, 59, pp. 261–268, 1927).

using another self-oscillatory system as an example, namely the vacuum-tube generator, allow me to make several essential, in my opinion, remarks. All that has been said above may be summarized as follows. The concept of an autonomous system, i.e., a system in which external actions are excluded, is expressed mathematically in the fact that time does not enter explicitly into the coefficients of the differential equation. The only type of linear equations satisfying this requirement is differential equations with constant coefficients. But in this case stationary (undamped) oscillations are characterized by the fact that the amplitude of the oscillations depends on the initial conditions, and that the principle of superposition is applicable here. Both of these properties are inseparably connected with “linearity,” whereas the chief interest in the process of generating oscillations is concentrated precisely on the independence of the stationary amplitude from the initial conditions and on the inapplicability of the superposition principle. In other words, the process of oscillation generation, the most essential and important aspects of self-oscillations, cannot be conveyed by linear differential equations. However, despite the fact that already by Poincaré, Kaufmann, Barkhausen, and others there had been analyzed and formulated both the conditions for the transformation of direct current into alternating current and the conditions for the onset of oscillations, and although it would seem that the proposition that it is impossible to describe the process of oscillation generation with the aid of only linear relations should raise no doubts, the linear interpretation of questions of oscillation for a long time almost exclusively prevailed in the theory of oscillations. Even at the present time one very often encounters attempts to apply such a linear interpretation to solving such questions where it is entirely unsuitable (for example, in many questions of the theory of the vacuum-tube generator, etc.). The creation of such a “linear” psychology was undoubtedly influenced by the very simplicity of linear differential equations, their simple and “intelligible” properties (such as the principle of superposition), and the obtaining of a complete solution with the aid of well-known functions.

(\(e^x\) and \(\sin x\)) and so on, which found extensive application in the theory of oscillations with small amplitudes, leading to the “linearization” of the equations of motion, especially in the phenomena of forced oscillations (the phenomenon of resonance). It should be noted that of great importance in creating such a prejudice—especially in questions of the generation of oscillations—was also, in essence, the “linear” treatment of alternating processes, as it is reflected in the so-called “courses of alternating current.” This linear psychology, this bias of thought that seeks to explain all oscillatory processes in a “linear” manner, played an enormous role in the development of our discipline, and it is undoubtedly very valuable even at the present time, since, of course, a very important tendency is to describe phenomena as simply as possible. However, it can—and often already does—exert a great negative influence, becoming a brake on the path of the development of theory.

Allow me to clarify my thought by the following analogy from the history of the development of physics, which many of us have lived through. After the creation by Maxwell of the electromagnetic theory of light and Hertz’s brilliant experiments, the development of the theory of light was for a long time determined by Maxwell’s differential equations. The very generation of light and its transformations—the processes of emission and absorption—were simply postulated, and only the processes of propagation, reflection, and so forth were subjected to investigation. Lorentz’s electron theory, which introduced into the theory the concept of the electron, foreign to the Maxwellian conception, made it possible to approach more closely the treatment of the processes of emission and absorption of light. However, the dipole model he created and its further generalization and development, despite a number of brilliant successes in the description of known phenomena and in the prediction of new ones, proved powerless to convey very many highly essential properties of the generation of light and its transformations. A radical breaking-up of the concepts and representations connected with the classical theory of light was required in order to obtain, in quantum theory and wave mechanics, a more adequate instrument for treating the processes of the generation of light. I am, of course, far from

in order to compare this grand scientific revolution with the necessity of changing the “linear” orientation in our modest field. It seems to me only that, in seeking adequate means for overcoming the difficulties encountered along the way, we should keep this very instructive example in mind and not persist in the linear deviation.

Allow me, after this digression, to return again to the problem of self-oscillations. The problem of transforming an oscillatory circuit into a self-oscillatory system was solved by Meissner in the following manner, which has already become classical (Fig. 1). Here the discontinuity of the restoration of losses is achieved by means of the feedback principle and a new element—a three-electrode electron tube. From the physical point of view, the action of this whole device may be regarded roughly as a certain continuously operating relay, replenishing the energy expended from a local constant source of energy.

Fig. 1.

The mathematical formulation of the relations occurring here (Fig. 1) leads to the following system of equations:

\[ i_A=i+CR\frac{di}{dt}+CL\frac{d^2i}{dt^2}; \tag{4} \]

\[ i_A=f(V_g+DV_A)=f\!\left[E_g+DE_A+(M-DL)\frac{di}{dt}\right]^* = f_1\!\left(\frac{di}{dt}\right), \tag{5} \]

where equation (5) expresses the so-called static

\[ {}^*\ \text{Since } V_A=E_A-L\frac{di}{dt},\ \text{and } V_g \cong E_g+M\frac{di}{dt}, \]
and, for simplicity, it is assumed that the grid current is absent; at the same time, the influence of the interelectrode capacitances is neglected here.

characteristic of the electron tube. The equation obtained from equations (4) and (5),

\[ \frac{d^2 i}{dt^2}+2\delta \frac{di}{dt}+\omega_0^2 i=\omega_0^2 f_1\!\left(\frac{di}{dt}\right) \tag{6} \]

is the equation governing the entire process of self-oscillation in the system under consideration. If it is compared with equation (1) for a simple oscillatory circuit, it is seen that it differs by the presence of the expression

\[ \omega_0^2 f_1\!\left(\frac{di}{dt}\right), \]

which determines the action of the feedback and of the electron tube. This expression, representing a nonlinear dependence on \(\frac{di}{dt}\), thus determines the character and properties of equation (6) that make this equation capable of describing the behavior of a self-oscillatory system.

What questions, then, must this equation answer; what principal aspects of the oscillatory process must it describe? First of all it must, of course, express the first fundamental property of a self-oscillatory system, namely the possibility of the occurrence of oscillations. Indeed, at first glance at Fig. 1 the question naturally arises: why is such a state of the system impossible in which \(i_A\) would be constant and would be determined by the equation

\[ i_{A0}=f(E_g+DE_A), \tag{5_1} \]

since then equations (4) and (5) would both be satisfied? Such a state of the system would obviously be an equilibrium state \(\left(\frac{di}{dt}=0\right)\), and consequently the answer to the question posed reduces to clarifying whether an accidental small deviation from the equilibrium position can carry the system away from it or not. The investigation of the stability of the state of a system, whether a state of motion or of rest, is one of the important problems in the theory of oscillations. For the case of the onset of oscillations, when the initial state corresponds to rest, this problem can be

completely resolved by using methods and techniques developed already by Routh and Hurwitz for mechanical systems. These methods reduce to considering a possible infinitely small change (variation) of the initial state, whereby the nonlinear differential equation is transformed into a linear differential equation for the variation \(a\).

In our case, for the variation \(a\) of the current we obviously have the equation

\[ \frac{d^2 a}{dt^2}+2\delta \frac{da}{dt}+\omega_0^2(i_{\lambda 0}+a) =\omega_0^2 f_i\left[E_g+D E_\lambda+(M-DL)\frac{da}{dt}\right]. \tag{6_1} \]

and since, by assumption, \(i_{\lambda 0}=f(E_g+D E_\lambda)\), then, restricting ourselves to terms of the first order of smallness (assuming that \(\frac{da}{dt}\) is continuous), we have:

\[ f\left[E_g+D E_\lambda+(M-DL)\frac{da}{dt}\right] = f\left[E_g+D E_\lambda\right] \]
\[ +(M-DL)\frac{da}{dt}\, f'\left[E_g+D E_\lambda\right] \]

and thus for \(a\) we obtain the following linear differential equation:

\[ \frac{d^2 a}{dt^2} +\left[2\delta-\omega_0^2(M-DL)S\right]\frac{da}{dt} +\omega_0^2 a=0, \tag{7} \]

where \(S=f'[E_g+D E_\lambda]\) is the steepness of the characteristic at the initial point. The question of the stability or instability of the state reduces to investigating the roots of the characteristic equation

\[ x^2+\left[2\delta-\omega_0^2(M-DL)S\right]x+\omega_0^2=0, \tag{7_1} \]

by the sign of whose real part the character of the solution of equation (7) is determined. But this linear equation, which governs the process only at its very initial stage, differs substantially from the differential equation of a linear oscillatory system of the usual dissipative-

type. The difference consists in the fact that, depending on the sign of the expression:

\[ 2\delta-\omega_0^2(M-MD)S=0 \]

the system has directly opposite properties and, from stable, may become unstable. It should be noted that, besides the question considered here of stability under small changes of state, the so-called stability in the small, there may be other practically important cases when a system, being stable under small changes, becomes unstable, or, more precisely, becomes capable of performing oscillations about another position of equilibrium under a sufficiently large initial change. Here we encounter a new problem of stability under large changes (Stabilität im Grossen*), to which we shall return below.

Fig. 2.

Let us turn again to the condition of instability:

\[ 2\delta-\omega_0^2(M-DL)S<0 \tag{8_1} \]

or

\[ RC-s(M-DL)<0, \tag{8_2} \]

If it is satisfied, then a new question arises: what will happen to the system? If the conditions to which the system is subject are such that they do not allow another position of stable equilibrium and the state cannot change all the time in one and the same direction (for example, the current cannot increase to infinity), while the equations remain in force, then the system has no other possibility than to perform oscillations. However, in order to carry out further analysis, general considerations alone are insufficient. One cannot, for example, assert whether the oscillations will be periodic or not. Nor, in the case of periodicity, can one indicate the magnitude of the amplitude and of the fundamental period of the oscillations—

* Such cases occur, however, not only in self-oscillating systems. For example, the case of a ball lying on a stand that is a body of revolution with the cross-section shown in Fig. 2.

oscillations. In order to obtain answers to all these questions, it is necessary to carry out a rigorous analysis of the equation of the system. However, equation (6), even for the simplest forms of the function \(f_1\!\left(\dfrac{di}{dt}\right)\), cannot be solved in a general form, as is the case with linear equations. Therefore, in analyzing the equation one has for the most part confined oneself to approximate methods of solution, while the character of the oscillation (for example, periodicity, etc.) was simply postulated. These methods (van der Pol, Ollendorff, and others) have in a number of cases (for example, for systems of the Thomson type) led to practically quite satisfactory results. Such a way of treating the problem, however, cannot give a rigorous justification of the results obtained. It is also insufficient for the solution of a number of questions of a more delicate nature. For this, other methods are necessary, methods which penetrate more deeply into the nature of these differential equations. The need for this had already long been felt, and by Appleton, van der Pol, and others attempts were made at a more profound approach to the theory. Such methods, however, are available among mathematicians. They were developed by Poincaré and other mathematicians in application to questions of celestial mechanics—to problems of periodic processes in the macrocosm—and are based on an analysis of the topography of integral curves. The merit of applying these very adequate methods to cases of self-oscillations belongs to the school of L. I. Mandelstam. By his student A. A. Andronov, separately and jointly with A. A. Vitt, in a number of interesting works, the theory of Poincaré limit cycles was developed and applied by them to problems of oscillations. Since in today’s lecture A. A. Andronov will speak in greater detail about this theory, allow me not to dwell on it and to say only that little without which it will be difficult for us to proceed further.

Before turning to this, I would like to make one small remark in connection with the problem of the origin of oscillations. Let us consider the following scheme (Fig. 8).

In this system the oscillatory circuit is connected with the grid circuit not directly, but through a certain aperiodic circuit. If the mutual inductances \(M_1\) and \(M_2\) and the self-inductances \(L_1\) and \(L_2\) of the aperiodic circuit are chosen in an appropriate manner, then it is possible to arrange that the system, stable when \(L_1\) and \(L_2\) are short-circuited, will become unstable when section 1—2 is opened. Since the act of opening, in view of the absence of any current in the intermediate circuit (in the stable static regime the currents in the system are only constant), cannot produce any induction impulses, the indicated circuit makes it possible to study the behavior of an auto-oscillatory system situated arbitrarily close to a position of equilibrium. In particular there arises the question of the nature of the forces (changes) that take the system out of the state of unstable equilibrium and bring about the occurrence of oscillations.

Fig. 3.

Here we come very close to questions concerning spontaneous changes of currents and voltages, i.e. to those statistical fluctuations by which, in principle, the degree of constancy of a whole series of physical quantities, conceived by us as the result of averaging, is limited. The circuit considered by us makes it possible to clarify this question, at least in principle, by direct experiment. Indeed, by measuring a large number of times, after the opening has occurred, the amplitude of the random initial impulse which, in view of the linearity of the variational equation, grows according to an exponential law, we shall be able, after statistical processing of the experimental data, to draw a conclusion about the probable magnitude of the random initial amplitudes. Some experiments in this direction have already been carried out in the physics laboratory of the VЭI, and it is proposed to set them up more systematically at the GFTI. In connection with what has just been said it is interesting

It should also be noted as follows. As we see from conditions (8), the boundary of instability represents a certain relation between definite parameters of the system. Remarkable here is the fact that this relation holds at the moment when oscillations arise, i.e., when infinitesimally small alternating currents begin to flow through the circuit—in other words, practically in the absence of current. This circumstance deserves attention, since it may be used for various measurements or investigations in which appreciable currents should practically not pass through the quantities being measured. It may also prove useful for various practical applications, as, for example, has already been proposed by S. E. Khaikin.

Let us now return to the theory of self-oscillating systems. Equation (6) can, by substituting \(y=\dfrac{di}{dt}\) and \(i=x\), easily be replaced by a system of two first-order differential equations:

\[ \frac{dx}{dt}=y=P(x,y) \tag{9_1} \]

\[ \frac{dy}{dt}=\omega_0^2 f(y)-2\delta y-\omega_0^2 x=Q(x,y). \tag{9_2} \]

We can evidently regard our problem as solved if, for example, we know the dependence of \(\dfrac{di}{dt}\) on \(i\), i.e. \(y\) as a function of \(x\). For if \(\dfrac{di}{dt}=\Phi(i)\), then a simple quadrature gives

\[ t=\int \frac{di}{\Phi(i)}, \tag{10} \]

i.e. \(t\) as a function of \(i\), whence it is easy to derive the properties of \(i\) as a function of \(t\). In particular, if such a dependence is represented graphically, in the form of curves in the plane \(x(=i)\) and \(y\left(=\dfrac{di}{dt}\right)\), then from the properties of these curves one can derive a number of very important conclusions concerning the character of the solution of equation (10), and consequently also of the oscillatory system. For example, if it turns out that there exists a closed curve

... curve \(y=\Phi(x)\), not passing through a singular point, then this means that the equation has a periodic solution. In fact, it follows from this that the integral \(\int \frac{dx}{\Phi(x)}\), taken along the closed curve, has some definite value \(\tau_0\), independent of the initial point on the curve, and this, according to equation (10), means that after definite intervals of time equal to \(t\), the values of \(x\) and \(y\), i.e. \(i\) and \(\frac{di}{dt}\), are repeated again. In other words, \(i\) and \(\frac{di}{dt}\) are periodic functions of time with period \(\tau_0\). Such closed curves, in the case where they are isolated, are called cycles or, more precisely, Poincaré limit cycles, and the establishment of their presence or absence in the topography of the integral curves is extremely important, since their presence may be regarded as the principal characteristic of a self-oscillatory periodic system.

What characterizes the equilibrium positions of the system in such a topographical picture? Since in them \(\frac{di}{dt}\), i.e. \(y=0\), and \(i=f(0)\), it follows from equations \((9_1)\) and \((9_2)\) that their right-hand sides vanish at such points, and consequently \(\frac{dy}{dx}\) becomes indeterminate. Thus, equilibrium states correspond to those points at which the direction of the curve is not defined, i.e. to those points through which not just a single curve may pass, as is the case for any other point of the topographical system \(x, y\), but an indefinite number of them. Such points in the theory of differential equations are called “singular” points. In them the further course of the process is not uniquely determined by the initial conditions. Investigation of the character of singular points, namely the behavior of integral curves near them, shows (A. A. Andronov will speak of this in more detail) that there exist both points toward which integral curves converge and also points from which the curves diverge. Finally, there is a third type of singular points, toward which some of the integral curves converge, while another part diverges from them.

Obviously, only points of the first type correspond to positions of stable equilibrium, while the others, by their nature, are unstable. It is easy to show that the condition for instability of singular points is identical with the previously derived condition (8), which of course is not at all surprising, since the method of investigation is in essence one and the same—the method of infinitely small variations, applied to equation (6) and leading to a linear characteristic equation.

Singular points and limit cycles are the basic elements of Poincaré’s topographic theory of differential equations, and their analysis makes it possible to obtain a clear answer to the most essential questions of periodic self-oscillations. From the point of view of this theory, points in the plane \(x, y\) denote the state of the system. If the point corresponding to the initial state is unstable, this means that the system cannot remain in this state. The point corresponding to it will therefore move, depending on the kind of random initial impulse, along one or another integral curve which, if a limit cycle is present, will gradually approach it, as if winding itself onto it, tending in the limit to merge with it. Finding such a cycle solves the problem of determining the stationary amplitude (since it gives \(i_{\max}\) and \(i_{\min}\)), and its time

\[ t=\oint \frac{ds}{\sqrt{P^2+Q^2}} \tag{11} \]

gives the fundamental period. Finally, analysis of the form makes it possible to determine the presence and magnitude of harmonics. Here it should be noted that if the basis of a self-oscillatory system is a sufficiently weakly damped circuit, then, owing to the resonant properties of such a circuit, the form of the oscillation curve is close to sinusoidal, and the period, as was already noted above, is expressed with great accuracy by Thomson’s formula—we are here dealing with oscillations of the Thomson type. Finding a continuous periodic solution to the problem, in the case of a self-oscillatory system with one degree of freedom, of the type of the vacuum-tube generator considered by us, still does not exhaust it completely. How, for example, are such phenomena as “soft” and “hard”

…“soft” excitation, the phenomenon of “dragging” and “quenching”? I do not think that in our age of radio broadcasting it is necessary to explain these terms in detail. I shall only recall that by “soft” excitation we mean such a regime of an auto-oscillatory system in which, when one of the parameters is smoothly varied, for example the mutual inductance \(M\), the stationary amplitude, starting from zero, increases smoothly, and this change is reversible. In “hard” excitation the amplitude of the oscillations, on passing through the point of self-excitation, at once, as if by a jump, acquires a certain definite value. Usually “hard” excitation is associated with the phenomenon of “dragging,” which consists in the fact that the oscillatory regime does not “follow” when the parameter is changed back; for example, the oscillations do not cease after passing through the point of self-excitation, but are preserved for some further time and again suddenly cease (Fig. 4), but already at such a value of the parameter at which they cannot be excited.

Fig. 4.

These phenomena, whose theory A. A. Andronov will describe in greater detail, bring to the fore a number of questions, of which the most important in principle is the problem of the dynamic stability of the state. In the case of periodic processes this problem reduces to the question of the stability of limit cycles.

Questions of dynamic stability have long been the subject of investigation by mechanicians. In Lyapunov’s works this question received, for small variations of states, its complete solution. The methods developed by him can also be applied to our problem. Since for an equilibrium dynamic state we have \(i=\Phi(t)\), which must satisfy equation (6)

\[ i_A=i+CR\frac{di}{dt}+LC\frac{d^2 i}{dt^2}. \]

\[ i_A=f(Z), \]

where

\[ Z=(M-DL)\frac{di}{dt}+Eg+DE_\Lambda, \]

then, assuming that the current has undergone a variation \(\xi\), we obtain

\[ \delta Z=(M-DL)\frac{d\xi}{dt};\quad \delta i_\Lambda=f'(z)\delta z; \]

\[ \delta i_\Lambda=\xi+CR\frac{d\xi}{dt}+LC\frac{d^2\xi}{dt^2}, \]

whence for \(\xi\) we obtain the equation

\[ LC\frac{d^2\xi}{dt^2}+\left[RC-(M-DL)F(t)\right]\frac{d\xi}{dt}+\xi=0, \tag{12} \]

where

\[ F(t)=f'\left[(M-DL)\frac{di}{dt}+Eg+DE_\Lambda\right] =f'\left[(M-DL)\varphi(t)+Eg+DE_\Lambda\right], \]

is a certain periodic function of time.

Thus, the question of dynamic stability is reduced to the investigation of the solution of a linear differential equation with periodic coefficients. Such equations occur in celestial mechanics and were developed by Mathieu, Hill, and others.

The solution of these equations always has the form

\[ e^{-\psi(t)}\left[Ae^{\beta t}\varphi(t)+Be^{-\beta t}\varphi(-t)\right], \]

where \(\varphi(t)\) is a periodic function. Depending on the relation between the constant and variable parts of the coefficients of the equation, the real part of the quantity \(\beta\) may be either greater than, equal to, or less than the constant part of the function \(\psi(t)\).

In the first case we obviously have to do with an unstable equilibrium dynamic state: oscillations on such a cycle cannot be maintained.

Let us note that equations with periodic coefficients, besides their importance for questions of dynamic stability, play a large role in the phenomena of parametric excitation, which we shall encounter below, and also for the theory of modulation, phenomena of entrainment in the theory of the complex tube transmitter, etc.

Let us return again to our self-oscillating system. Instead of the circuit of Fig. 1, such a circuit can also be realized in another, somewhat more complicated way; for example, by means of the circuit shown in Fig. 5.*

Fig. 5.

In fact, here tube 2 plays only the role of a linear amplifier and phase inverter, so that the equations governing the process take the form:

\[ i_{a1}=i_1+i_2=i_1+C\frac{dV}{dt}=f[E_{g1}-\rho i_{a2}] \tag{13_1} \]

\[ R_1 i_1=R_2 i_2+L\frac{di_2}{dt}+V, \tag{13_2} \]

or

\[ R_1 i_{a1}=V+C(R_1+R_2)\frac{dV}{dt}+CL\frac{d^2V}{dt^2}. \tag{14} \]

Since further, by assumption,

\[ i_{a2}=S_2\psi_{g2}=-S_2R_2i_2, \]

whence

\[ i_{a1}=f_1\!\left[S_2\rho\,CR_2\frac{dV}{dt}\right], \tag{15} \]

it follows, therefore:

\[ CL\frac{d^2V}{dt^2}+C(R_1+R_2)\frac{dV}{dt}+V = R_1 f_1\!\left[S_2R_2\rho C\frac{dV}{dt}\right] \tag{16} \]

or

\[ \frac{d^2V}{dt^2}+2\delta\frac{dV}{dt}+\omega_0^2V = R_1\omega_0^2 f_1\!\left[S_2R_2\rho C\frac{dV}{dt}\right]. \tag{16_1} \]

Thus we arrive at the same equation as equation (6). In particular, we obtain the condition for self-

* This circuit was proposed by L. I. Mandelstam and the author, as one which very clearly illustrates the transition from self-oscillations of the Thomson type to relaxation self-oscillations.

excitation, if instead of \(M-DL\) we substitute the quantity \(R_1R_2\rho CS_2\).

\[ R_1+R_2-S_{10}S_2R_1R_2\rho<0, \tag{17} \]

where \(S_{10}\) is the slope of the characteristic of tube 1 at the initial point.

Let us now consider what will happen if we gradually decrease \(L\). The condition for self-excitation will obviously continue to be satisfied all the time, down to \(L=0\). As for the circuit, its natural oscillations, weakly damped at first, will become more and more strongly damped, and then, after the boundary of oscillatory behavior has been crossed, the process will become aperiodic. How will such a change be reflected in the character of the self-oscillations? The limiting cycle, which at first had approximately the form of a circle, will gradually depart from it more and more strongly, while the value of the period will deviate more and more from the value given by Thomson’s formula. However, the oscillations will remain periodic and, moreover, continuously periodic; i.e., the oscillations will be such that the quantities determining them (for example, currents and voltages) will remain continuous at all times.

What, then, will happen if we put \(L=0\), in other words, idealize the circuit so that we completely disregard the self-inductances that are always present, although extremely small (the so-called “parasitic” ones)? Let us turn to equation (16). In this case it will take the following form:

\[ C(R_1+R_2)\frac{dV}{dt}+V=R_1 f_1\!\left[S_2\rho\,CR_2\frac{dV}{dt}\right] \tag{16₂} \]

and consequently it represents in the plane \(\left(x=V,\ y=\frac{dV}{dt}\right)\) a certain curve which, however, as is easy to see from the graphical construction (Fig. 6), can under no circumstances be closed. Hence it follows that equation \((16_2)\) has no Poincaré cycle and therefore cannot represent a continuous periodic oscillation. The question naturally arises: is such an idealization of the problem admissible?

For a physical oscillatory process must undoubtedly take place, since the possible equilibrium position is unstable, while the quantities \(\left(V,\frac{dV}{dt}\right)\), by the condition of the problem (the form of the characteristic), cannot grow without bound. Such a doubt did in fact arise; and some investigators (for example van der Pol) were inclined to think that such an idealization is physically inadmissible and that it is necessary to take \(L\) (or \(C\)) into account, however small they may be. However, a more profound consideration of the question shows that there is, in a certain sense, a natural way out of the difficulty. The point is that from equation \((16_2)\) there follows only the impossibility of a continuous periodic process, i.e. one in which the motions and their velocities remain finite and continuous all the time; the process determined by equation \((16_2)\) does not possess such properties.

Fig. 6.

Indeed, for the rate of change \(y=\dfrac{dy}{dt}\) we have from equation \((16_2)\):

\[ C(R_1+R_2)\frac{dY}{dt}+y = S_2R_1R_2\rho Cf_1'\!\left[S_2R_2\rho C\cdot y\right]\frac{dy}{dt}; \]

whence, since \(f_1'[S_2R_2\rho Cy]=S_1\) (the steepness of the characteristic of lamp 1 at the point \((x,y)\)), and \(R_1+R_2=S_0S_2R_1R_2\rho\), where \(S_0\) is the critical steepness of characteristic 1 at which self-excitation is just still possible (cf. equation 17), we obtain:

\[ \frac{dy}{dt} = \frac{y}{S_2R_1R_2\rho C(S_1-S_0)}. \tag{18} \]

It follows from this that \(\dfrac{dy}{dt}\) (or \(\dfrac{dy}{dx}\)) becomes \(\sim\infty\), and conse-

therefore undergoes a jump discontinuity at those points at which the steepness of the traversed characteristic is equal to the critical one. As is seen from Fig. 6, such a discontinuity must take place at \(A\) and \(B\). If in this way a continuous periodic process is impossible, then one asks how to conceive a discontinuous periodic process. In particular, the question arises: what determines the continuation of the process after any jump discontinuity? Since the charge of the capacitor obviously cannot change by a jump, it follows that \(V\) (or \(x\)) must remain unchanged at every discontinuity. On the other hand, equation \((16_2)\) must naturally remain valid throughout the entire process. These two requirements make it possible to find the position of the system (the points \(x,y\)) after the jump, while the sign of \(\dfrac{dy}{dx}\) gives the subsequent direction of the process.

Fig. 7.

In the case considered by us (Fig. 7), the system, from the state determined by the point \(A\), passes by a jump to the point \(C\), and from the point \(B\) by a jump to the point \(D\). Thus, a discontinuous periodic process consisting in motion along the segments \(DA\) and \(CB\) of the integral curve, with jumps at \(A\) and \(B\), can indeed be described by equation \((16_2)\) in connection with the indicated continuity conditions, and consequently such an idealization of the system \((L=0)\) is quite admissible. Let us note that we encounter a similar idealization of processes in other domains of physics as well; moreover, in some cases we have become so accustomed to the discontinuous treatment of the problem that we often cease to notice its unusual character. Perhaps the most striking example of this is the treatment of the reflection of an elastic ball from an elastic plane or from another ball,

as, for example, is usually done in elementary kinetic theory of gases. Without considering in detail a problem essentially different in nature, we usually treat the act of reflection as discontinuous, assuming that the direction of the normal component of the velocity changes its sign by a jump. The magnitude of the velocity itself, as determining the store of kinetic energy, remains unchanged.

Such an idealization, as in our case, substantially simplifies the problem and at the same time gives, with sufficient approximation, an answer to the main questions that interest us: the value of the period, the stationary amplitude, etc. But this is achieved, of course, at the cost of abandoning consideration of the process during the “jump” itself. For example, the time of the actual contact in an impact cannot be determined, etc.

Such cases of discontinuity occur often when considering quasistationary electrical processes. I shall point, for example, to the discharge of a capacitor through an ohmic resistance \((R)\): here the current at the initial moment changes by a jump from zero to the value \(i = \dfrac{V_0}{R}\), where \(V_0\) is the initial voltage of the capacitor.* It should be noted that in all such cases it is quantities whose change is not connected with a change in the store of energy that experience a discontinuity (for example, the voltage at the ends of a resistance or of a self-inductance, the current through a resistance or a capacitance). On the other hand, it is physically necessary that precisely the quantities which determine the store of energy remain unchanged under all discontinuities.

Let us return again to our idealized scheme. As is seen from Fig. 8, the principal circuit of the system now consists only of \(C\) and the resistances \(R_1\) and \(R_2\), so that the action of the whole scheme may be regarded as the regeneration of an aperiodic discharge of the capacitor. Instead of two heterogeneous reservoirs \((L, C)\) of energy, we are dealing here only with one \((C)\), and therefore there naturally can be no question of

* Another example: the jump of the voltage between \(A\) and \(B\) (Fig. 9) from null to the breakdown voltage of the spark at the moment the discharge begins.

of the oscillations in the circuit itself as such, and consequently also the computation of the period as a function only of the parameters of the system itself. It is therefore customary to say that here we are dealing with oscillations of a non-Thomson type. In particular, since the time of the aperiodic discharge of the capacitor is determined by the so-called relaxation time, such oscillations are also called relaxation oscillations (van der Pol). The system considered by us is one of the simplest examples of the broad class of self-oscillatory systems of the relaxation type, among whose best-known representatives is the Abraham–Bloch multivibrator, which has acquired such great importance in measuring technology. To this class also belong, besides numerous other tube circuits proposed by Friedländer, Hegner, Watanabe, and others, such self-oscillatory systems as, for example, the circuit of the “flashing” neon lamp, the hydraulic ram, and others. Apparently one should also include here such oscillatory processes as the activity of the heart muscle (van der Pol), the fluctuation of animal species in the struggle for existence (Volterra), and a number of others. What is common to all these self-oscillatory systems is that the basic systems entering into their composition, in the absence of regeneration, are incapable of performing oscillations. This is in a certain sense equivalent to saying that in them there exist only homogeneous forms of energy accumulation (homogeneous reservoirs of it), and the course of the process in the absence of regeneration is determined by a certain relaxation time.

Fig. 8.

Fig. 9.

Self-oscillatory relaxation systems have been considered—

of the type we have just described occupy the last place in the series of self-oscillatory systems, at the beginning of which one may place the tube generator with a circuit possessing a very small natural damping. The oscillatory process in them has, as we have seen, a discontinuous periodic character. On the other hand, from Poincaré’s topographical theory of differential equations it follows that the basic characteristic of a self-oscillatory system is the presence of limit cycles, i.e. closed continuous curves. Naturally, therefore, the question arises as to how matters stand here from the mathematical point of view and, in particular, whether the concept of limit cycles can be combined with discontinuous periodic solutions. As A. A. Andronov and A. A. Witt* have shown, an analogy can be drawn between cases of discontinuous periodic solutions and limit cycles. Since S. E. Khaikin will set forth this theory in greater detail in his report, permit me not to dwell on it here and to pass on to another question connected with relaxation oscillations.

Self-oscillatory relaxation systems, like the idealized circuit of Fig. 6 (this also includes, incidentally, the Abraham–Bloch multivibrator), lead, as we have seen, to discontinuous periodic oscillations, i.e. to oscillations very strongly differing from sinusoidal ones and extremely rich in intense harmonics of higher orders. These properties, which make relaxation circuits extremely suitable for various measurement purposes (for example, for absolute measurements of frequency), were at first considered an indispensable characteristic of all relaxation oscillations in general. However, as Hegner, Hegner–Watanabe** and others have shown, relaxation circuits are possible consisting either only of capacitances and ohmic resistances, or only of self-inductances

* A. Andronov and A. Witt. Discontinuous periodic solutions and the theory of the Abraham–Bloch multivibrator. Reports of the Academy of Sciences of the USSR, 1930, p. 189.

** For example, K. Hegner u. Y. Watanabe. Jahrbuch für drahtlose Telegraphie und Telephonie, B. 34, Heft 2, 1929.

N. D. Papaleksi

and resistances, in which the self-oscillations have an approximately sinusoidal character (Fig. 10). It was therefore very interesting to consider this question from the point of view of the topographic theory of self-oscillations. In particular, it was important to determine the dependence of one or another kind of oscillation on the presence and character of limiting cycles. Consideration of these questions led S. E. Khaikin to a number of interesting conclusions, about which he will speak in more detail in his report. I would only like to point out that, as S. E. Khaikin has shown, such relaxation circuits are possible which, under a smooth change of one definite parameter, can be transformed from a state characterized by the presence of a closed Poincaré cycle into a state corresponding to discontinuous periodic solutions. One such circuit is shown in Fig. 11.

Fig. 10.

Self-oscillatory systems of the Thomson and relaxation-discontinuous type occupy extreme positions in the series of self-oscillatory systems. As we saw in the example of the circuit in Fig. 5, one can pass, through a number of intermediate systems, from a system of the Thomson type to, in the limit, a relaxation system of the discontinuous type. It is also possible to pass from a discontinuous-relaxation circuit to a relaxation circuit of continuous type (for example, Khaikin’s circuit). All these systems are characterized by the presence of an unstable equilibrium point and of a stable continuous or gener-

Fig. 11.

…of a limiting cycle. In other words, the stationary oscillatory process in all the self-oscillatory systems we have considered was periodic (continuously or discontinuously periodic). Naturally, questions arise: is periodicity an invariable attribute of self-oscillations? Are, in other words, nonperiodic self-oscillations possible? What kind of process, for example, occurs in such a self-oscillatory system as an intermittent vacuum-tube generator, studied by B. Vedensky and S. N. Rzhevkin as early as 1921? These questions raise a number of new problems for the theory of oscillations and impose new requirements on their mathematical treatment. With this indication of that considerably more difficult and still little-developed area, allow me today to conclude this rather protracted first part of my report.