Issues of Electrical Oscillatory Systems and Radio Engineering*

L. I. Mandelstam, Moscow

I would like today to draw your attention to certain questions concerning electromagnetic oscillations, and especially to questions connected with the problems posed by modern radio engineering. In doing so, I feel a certain awkwardness, which is due to the fact that, within the framework of that most extensive and highly varied material treated by the theory of oscillations in general, the topic chosen embraces a rather narrow field. Some justification for the choice of this comparatively narrow, at first glance, topic I see in the distinctive character possessed by the problems indicated.

The problems posed by radio engineering are specific. The methods applied here, both experimental and theoretical, bear the imprint of this specificity; and it seems to me that what has already been created, and what is being created and will continue to be created here, will, in its significance, far outgrow those tasks for which these methods were originally conceived.

Allow me to point out at once one characteristic feature that distinguishes our field from a number of other fields of technology in which oscillations also play an important role.

Problems of oscillations in machine building, in engineering practice, have recently—especially with the increase in the size of machines and in speeds—been acquiring ever greater importance. But here the task consists chiefly in avoiding those harmful, and sometimes destructive for structures, effects of oscillations which occur at certain critical speeds or periods, chiefly owing to resonance.

* Report delivered at the opening of the All-Union Conference on Oscillations at the Institute of Physics of Moscow State University on November 12, 1931.

It is necessary to avoid the loosening of a ship’s hull by the operation of the engine, the destruction of a bridge under the action of a periodic load, etc. What catastrophic consequences resonance can have is well known. Its destructive influence must be blamed for the bridge collapses, breakages of shafts of powerful machines, and many other failures observed in earlier times. The task is to prevent these phenomena. Of course, only a profound theory of oscillations can here provide radical means for rendering the influence of oscillations harmless.

The fundamental task of radio engineering is precisely the opposite. Here the task is to create, as far as possible, powerful oscillations of a definite type (they must be undamped, stable, must have a definite spectrum, etc.). And a second, equally important task is to create a device—in this lies the task of reception—that would be “set into oscillation” as strongly as possible even by extraordinarily weak oscillations arriving from a distant transmitter.

In solving the problems of a rational generator of oscillations, on the one hand, and a sensitive receiver of oscillations, on the other, a number of physical and mathematical problems arise. A new era for radio engineering began with the appearance of the three-electrode cathode tube. Devices for generation and reception were created whose basis was formed by new physical phenomena, little studied before and whose study is not complete even now. To master the new radio-engineering problems, the mathematical methods that the theory of oscillations had used chiefly up to that time proved insufficient. It became necessary to turn to a mathematical apparatus that would be adequate to the new tasks.

It is on some of the physical and mathematical questions belonging here that I would like to dwell today. Allow me to begin somewhat from afar.

When we set about studying some field of science, the desire quite naturally arises to give a definition of this field, i.e., to sharply outline the circle of questions and phenomena that enter into it. And each time we here encounter great difficulties. A definition in science is one of the most difficult things. Attempts to give an exact and complete definition of some field of science usually end in failure. In general the question arises whether it is expedient to fence off separate fields of science with the barbed wire of rigid definitions and thereby make their mutual penetration more difficult. I do not think that one should be especially zealous about definitions of this kind. On the other hand, it is highly desirable to single out those guiding points

... points of view that allow us to unite an entire class of problems. Thus, finding such guiding points of view is, in my opinion, an essential matter. They allow us to create a coherent, integral theoretical conception; they allow us to bind into one whole problems that seem heterogeneous, and they make it possible to give a systematic character to further investigations.

And so, if we turn to the problems of oscillations that are under discussion, we shall arrive at the following conclusion. What interests us here, in the overwhelming majority of cases, is not oscillations in themselves, but chiefly the action of oscillatory processes on systems—on oscillatory systems—or, perhaps more generally, we are interested in the interaction of various oscillatory systems with one another. And before us arises the task of finding that point of view which would make it possible to characterize a given oscillatory process (this may be an electromagnetic field, variable mechanical forces) with respect to its influence on a resonator. Or, more concretely: imagine that we have some resonator—electrical or mechanical, it makes no difference—and that a variable force acts on it

$y=f(t)$.

What do we know? We know that in some cases the resonator responds extremely strongly; in other cases it almost does not react to the action—it remains almost deaf. When does the first case occur, and when the second? It is often said: here the matter lies in the periodicity of the acting force $y$. If the period of the acting force coincides with the natural period of the resonator, then a strong action occurs. But this is not correct. Let us take a simple example. Let

$$ y=\sin 2\pi mt+\sin 2\pi(m+1)t, $$

where $m$ is an integer. This function has fundamental period one and does not have periods $\frac{1}{m}$ or $\frac{1}{m+1}$. Meanwhile a resonator tuned to its period, i.e. to the period one, will almost not respond, whereas a resonator tuned to the period $\frac{1}{m}$ or $\frac{1}{m+1}$ will respond very strongly. On the other hand, you may take an altogether nonperiodic force, for example such:

$$ y=\sin 2\pi mt+\sin 2\pi nt, $$

where $m$ and $n$ are incommensurable. And yet our resonator, tuned to the period $\frac{1}{m}$ or $\frac{1}{n}$, will respond very strongly. Hence, the periodicity of the force and the coincidence of its period with

by the period of the resonator and are neither necessary nor always sufficient. What, then, is important? One must evaluate the function according to the following principle. One must represent it as a sum of simple (harmonic) oscillations—a sum of cosines and sines. If in the expansion there is a term with the period of the resonator, then resonance will occur; otherwise it will not. Thus the periods of the individual terms and their amplitudes, the totality of these numbers—this is what characterizes my force with respect to its action on an oscillatory system.

The Fourier series is a special case of such an expansion. A more general case is the expansion of almost periodic functions, with which mathematicians have recently been much occupied.

Thus, in order to evaluate the effectiveness of a given oscillatory action, the corresponding function must be expanded in a series of harmonic functions, after which the amplitudes and periods of the terms give us the required measure. This is, in essence, a very subtle evaluation of the given function. It takes into account, so to speak, the entire form of the process over the whole interval of time, in contrast, for example, to such characteristics as the maximum value, the mean, the root mean square, etc. Those who deal much with oscillations are accustomed to this device, but in essence it is a profound and delicate evaluation. Such an approach, when we evaluate a function by expanding it in a series—in the present case, in harmonic oscillations—we shall call spectral. For us this is the guiding point of view.

The value of the spectral point of view lies in the fact that, knowing the spectrum of a given force, we also know its action on the resonator. All our resonators, since they are linear, are, as it were, reactive to sine-functions. Linear resonance singles out precisely the sine-function, and not any others. It can give each term of such an expansion, as it were, an independent existence. This is joined also by the circumstance that harmonic oscillations are the only oscillations which, when applied to a linear, even a complex system, pass through all its links without distortion of form.

Now the answer to a frequently asked question is clear to us as well. We can expand a function in a series in the most varied functions, and not necessarily in sines and cosines. Why does the expansion precisely in these functions play such a role in physics? The answer is clear: this is not a mathematical question, but a physical one. The expediency of expansion in one or another set of functions, and the evaluation corresponding to this expansion, is determined by physi-

tical problem. In our case, the expediency of expansion into sines is conditioned by the properties of the receiving system, which responds precisely to sine-functions.

Allow me for a minute to digress from our immediate topic. Allow me to point out that the significance of spectral estimation as a principle distinct from ordinary approaches—only formulated, of course, in a much more general form than we need here—is acquiring ever greater and greater importance. One of the founders of wave mechanics, Schrödinger, sees in this principle of evaluating action in general a fundamental characteristic and the distinctive nature of modern wave mechanics. From this point of view, an approach which is very correct for us, but which does not claim high fundamental significance, would in a certain sense be only a special case or an example of a principled approach to the investigation of phenomena; moreover, of an approach that is an inseparable part of a definite worldview which now governs our physical understanding of the world. Unfortunately, I cannot dwell on these matters in greater detail.

But let us return to our modest tasks.

Although what I have said above is well known, I considered it desirable to emphasize the guiding role of the spectral approach for many reasons. First, because this point does not always sufficiently enter into the consciousness of people working in this field. Allow me to give one example. We know that if we modulate a telephone transmitter in the simplest way,

\[ y = a(1 + b \sin 2\pi mt)\sin 2\pi nt, \]

then the spectrum emitted by it consists of three lines—the carrier wave and two side frequencies. It is clear to us what this means. A resonator tuned to the side wave will pick it out just as a resonator tuned to the carrier will pick out the carrier. The spectral approach gives us the possibility of tracing the interaction of the transmitter and receiver; it indicates how they must expediently be constructed, and so on. It brings clarity to the whole problem of radiotelephony. All this is well known. And nevertheless, many of you know that approximately two years ago, in the pages of Nature, a controversy arose, begun by Fleming, and Fleming flatly denied the “physical” existence of the side waves. It apparently took about 15 letters to put the matter back in order. Here is an example showing that even leading people, although of course they know the theory, do not always sufficiently penetrate into “spectral thinking.”

The second circumstance is that the habit

approach to a wide variety of oscillatory phenomena with a spectral estimate helps us bring into relation phenomena which at first glance are extremely different. As an example, allow me to cite one case. Physicists have recently been interested in and have been studying the following phenomenon. If a strong beam of light is passed through a transparent body—for example, a crystal or a liquid—then the molecules of the body scatter a part, to be sure a small part, of this light in all directions, acting as if they were antennas excited by the incident wave. Let our incident light be monochromatic. What then turns out to be the case? If you take the spectrum of the scattered light, then in it, in addition to the carrier wave, side frequencies are also observed, and the general scattering spectrum in its essential features reproduces the spectrum of a modulated telephone radio transmitter. And then the question involuntarily arises: is not the optical phenomenon under consideration a more or less complete analogy of radiotelephone modulation? And so, if this thought is pursued further, it turns out that here, speaking somewhat schematically, we indeed have nothing other than modulation of the incident wave by the proper oscillations of the molecule or of molecular aggregates. And then it is entirely clear that, just as the spectrum of an ordinary telephone transmitter contains your whole conversation, everything that you want to say, so too the spectrum of the scattered light conveys what the molecule says about itself. By studying it, you study the property of the molecule, you study its structure.

Finally, a last remark concerning the spectral point of view. Once we have clarified what lies at the basis of applying this principle, we can readily give ourselves an account in advance of when and by what its applicability is limited. This circumstance, it seems to me, is very important. In the whole spectral approach one must distinguish two aspects. One may seem trivial to you. It consists in the fact that, in order to estimate a function, we decompose it into a sum of separate terms. The second aspect consists in the fact that as the terms we take definite functions, in the present case sines. When is it expedient to estimate an action by decomposing it into a sum? It is expedient to do so insofar as, in our problem, the principle of superposition is valid: knowing how each term acts, we thereby know how the whole sum acts, or otherwise: the decomposition of a whole into terms has meaning when the action of the sum is equal to the sum of the actions of the separate terms. At the moment when we have before us a case where the principle of superposition is not valid, the expediency of any spectral method must be subjected to reconsideration.

As for the second aspect—the choice, as the terms, of ipsa-

required special functions, then the question arises here: are there not such oscillatory problems in which it is more expedient to choose as the fundamental functions other functions of time than sines? It seems to me that such cases are quite real, and we are now concerned precisely with questions where the spectral approach remains advisable, but as the fundamental functions one must take not sines, but other periodic functions of time.

Allow me to finish the general part at this point and turn to the consideration of certain fundamental problems of radio engineering.

All the problems of radio engineering, I think, may be divided into three categories: the creation (generation) of oscillations, the radiation of oscillations and their propagation, and, finally, the reception of oscillations. Permit me today not to touch upon questions relating to the radiation and propagation of oscillations—not because they are less important and interesting than the other two. But, first, I would not have enough time for everything, and second, because our work concerns questions of generation and reception. I think it will be in the spirit of the tasks of the conference if I touch in somewhat more detail upon certain works that are being carried out among us (“among us” means both here, in the Physical Institute, and in TsRL, and partly in GFTI in Leningrad, in the laboratories headed by Prof. N. D. Papaleksi, because the work that is being done here and the work that is being done there are so closely intertwined that it is simply impossible to separate them from one another).

What is the fundamental problem that confronts us when we speak of generation—of the creation of oscillations? It is perfectly clear: we must create a device which makes possible the occurrence of stable undamped oscillations. These oscillations must be powerful; they must be generated with a good efficiency, and so on. There are very many incidental, associated aspects here. In addition, the advantages of particular devices are assessed to a high degree (now we know this) also by the spectrum which they produce. At the present time we consider it, for the most part, desirable that this spectrum be as simple as possible, i.e., that it consist of a single spectral line. This means that we wish, insofar as possible, to approach the generation of sinusoidal oscillations. From the mathematical point of view the matter reduces to setting up and integrating the corresponding differential equations, which would allow us theoretically to investigate the dependence, on the one hand, between the parameters of the system as it is given to us, and, on the other hand, the properties of those oscillatory processes which take place in these systems.

Classification, both of systems and of processes, is conveniently made, first, from the physical point of view, and, second, of course, from the point of view of differential equations. Classifications here may be of the most varied kinds. One may, for example, divide them into systems that have a finite number of degrees of freedom—in the limiting case, one degree of freedom—and continuous systems, or systems having an infinite number of degrees of freedom. Both are extremely important, and both play a very great role both in electrical engineering and in other fields. The former are governed by a system of ordinary differential equations; the latter, by equations in partial derivatives.

But one can and must go further and characterize systems according to the type of differential equations to which they are subject. For example, we speak of linear systems or nonlinear ones, depending on whether the equations by which they are governed are linear or nonlinear.

I must say that linear systems have hitherto played the largest, predominant role in the theory of oscillations. And indeed, an enormous field, both in applications and in physical questions, is connected with such systems, and mathematically this field is extremely interesting. The entire field of classical boundary-value problems essentially pertains precisely to linear systems of differential equations, i.e. is characterized by the fact that their equations in partial derivatives are linear equations.

But in the problems of which I shall speak, we shall have nonlinear equations. In order not to complicate the question and in order to single out the essential points, we shall consider equations with a finite number of degrees of freedom, even, chiefly, with one degree. Under this scheme falls a schematized ordinary cathode transmitter. We shall try to study its equation and to clarify the problem that interests us here.

If I take an ordinary circuit and introduce into it a variable electromotive force, then oscillations will arise in it; and in essence this is an oscillator of oscillations. But if I am to investigate it, I shall not embrace the question of the generation of electromagnetic oscillations, because immediately the question will arise: whence did I take the variable electromotive force? I have only pushed this question aside. Thus, in order to solve the question completely, one must turn to such systems as do not need “foreign” oscillations. Such systems, which produce oscillations, roughly speaking, from themselves, i.e. make use of a constant source of energy and do not need other sources of oscillations, we shall call autonomous oscillatory systems. Their more precise mathematical characte-

characteristic consists in the following: the differential equations by which they are governed must not contain time explicitly. In the mathematical sense this is a very important restriction. In the present case it is dictated precisely by our physical considerations.

Let us see what, from the mathematical point of view, is the simplest physical system that could be an autonomous system. What type of differential equations is the simplest? Undoubtedly, linear equations. If this is an autonomous system, then the coefficients of the equation must not depend on time; consequently, they must be constant. Thus, the simplest type of autonomous oscillatory system is a system satisfying a linear differential equation with constant coefficients. Such systems do indeed exist, and there are very many of them. All systems that are based on (sufficiently small) elastic oscillations, systems based on the use of an electric circuit consisting of a capacitor and a self-inductance—for example, a cable, etc.—all of them are described, to a sufficient approximation, by linear equations with constant coefficients.

If we turn to such a system (we shall confine ourselves to discrete systems), then we know all the mathematics that pertains to it. We know and are able to integrate the equations. The limiting case of undamped oscillations is sinusoidal oscillations or a sum of sinusoidal oscillations—this is all that such a system can give. Then there may be oscillations with a decaying amplitude and with an increasing one, the increase (since the system remains linear) and the damping occurring exponentially. These are the three types possible in systems with constant coefficients. Thus here we know not only the general character of the solutions, but the entire solution to the end.

From this we can conclude that, in order to create devices that will give us the oscillations we need, linear systems are unsuitable, because their fundamental property, directly connected with linearity, is that their amplitude or, in the final analysis, the energy that they deliver is not a property of the system as such, but depends entirely on the initial conditions. If you charge such a system at the initial moment with a certain energy, then the whole process, throughout time, in the sense of energy, depends essentially on these initial conditions.

Meanwhile, if you look at all modern devices from which stability is required, the devices with which we in fact work, they, of course, do not possess this property or, more precisely, this defect. They

possess the property that their oscillations are stable in the sense that, if you start them from some arbitrary state within wide limits, they oscillate with a definite period and with a definite amplitude. They have a tendency, independently of the initial conditions, to settle into a definite regime. Actual devices have this property. I think it is rational to demand this property of good generators. Therefore we can place this property as the basis of our requirements. And other considerations, on which I shall not dwell, lead to a conclusion which, in my opinion, cannot be avoided: the requirements we impose on generator devices are incompatible with linearity, and hence the systems we need must satisfy nonlinear differential equations.

I think that when Meissner discovered his feedback principle, which lies at the foundation of modern radio engineering, both receiving and transmitting, he was least of all thinking about differential equations. But now, when the point is to deepen and make use of this fundamental idea, to apply it in various ways, when the point is the ever-increasing development of devices, one cannot refer to this, but must turn to the mathematical apparatus adequate to the given problem. And there is no other such mathematical apparatus than the apparatus pertaining to nonlinear differential equations. The physics of oscillations is now faced with the task of becoming seriously acquainted with the theory of nonlinear differential equations.

But tradition and habit are a great force. And it is natural that this should be so. And therefore, especially at first, but even now as well, in applications to systems known in advance to be nonlinear, people nevertheless try to approach them by linear methods. I cannot deny that some results have thereby been achieved. But we have seen that a whole series of fundamental answers could not be obtained in this way. Then additional assumptions were introduced, which were not included in the basic formulation and were made ad hoc, but which led to partial successes, because people knew what ought to be obtained. Sometimes such a method helps, sometimes it does not; but it is always a palliative. This is, first of all. Secondly, we know from experience that linearizing such problems has also led directly to errors. A well-known error, which runs through the literature, consists in the following: the question is the determination of the conditions for the occurrence of oscillations in a generator. Then the procedure is as follows: the oscillations, known in advance to be nonlinear, are considered for very small values of the deviations, expanded in a series, limited to the first term, and

QUESTIONS OF ELECTRICAL OSCILLATORY SYSTEMS

they obtain genuinely linear equations. We can solve them. And so, if the result is an increasing amplitude, we say that we have an unstable state, that we have a growing process. The fact that one can and must, with the help of linear equations, find the conditions necessary for oscillations to arise is correct. But then they sometimes proceed as follows. It is known from the theory of linear equations that if you further increase the “negative” damping, practically further increase the feedback, then the solutions of the linear equation cease to be oscillatory. Then they reason thus: this means that there comes a moment after which there can be no oscillations. This is, in general, erroneous. It is impossible to conclude from the behavior of linear systems what process will become established in nonlinear systems. And yet this reasoning, which is refuted by experiment and by theory, can still be encountered even now.

Thus it is necessary to readjust and to pass over to an apparatus that is adequate to the given problem. This was realized very long ago—soon after the introduction of such nonlinear systems into practice, after such systems had become enormously widespread and had acquired tremendous importance for the actual realization of practical aims. Practice itself prompted that this had to be done in the proper way. Works appeared that quite consciously took a nonlinear point of view. Among these one should especially include the works of van der Pol, who did this, and did it correctly, and whose results still have fundamental significance for us. He proceeded correctly in the sense that he knew this was a nonlinear problem and at once approached it as such. But the initial works, quite naturally, did not possess that generality which is undoubtedly desirable. Very many things were simply postulated. For example, the existence of periodic solutions was very often postulated; the series that were obtained were often not investigated for convergence. In addition, because of the absence of a general approach, each case was treated separately. Nevertheless, the results were often good. But the modern mathematical apparatus that exists was not used; and, incidentally, such an apparatus did exist and had existed for a long time. It finds its foundation, one might say, in the famous works of Poincaré of the eighties.* But the connection that exists between these works of Poincaré and the works that later, especially by Birkhoff, were greatly deepened—the connection

* Poincaré, “Sur les courbes définies par les équations différentielles,” Oeuvres, t. I.

these mathematical works to our physical problems is pointed out by A. A. Andronov. Later, together with A. A. Witt, he adapted this apparatus to our problems, applied it to a whole series of concrete problems, checked against it solutions of problems obtained earlier by other authors, and made it into a more or less workable apparatus in our field. It turned out that the principal results of van der Pol retain their force. And now they are a special case of well-founded general theorems. Van der Pol himself, in one of his recent works, also views the matter in this way. And this, in my opinion, is a very substantial point.

But, in addition, it proved possible to find new things. And then certain questions which, under the former approach, had remained either open or insufficiently illuminated, could thus, with a general theory in hand, be illuminated much more fully. I shall return to these questions later. And now allow me to draw attention to the following.

The equations and mathematical questions that arise in formulating problems of the generation of electrical oscillations, it turns out, have a much broader significance than this may seem at first sight. There is a whole series of questions which, in essence, lead to the very same formulation of the theoretical problem. I shall mention only some of them: a whole series of acoustical problems, for example the sound of a string under the action of a bow, the sound of organ pipes, the sound of most musical instruments, with the exception of percussion and plucked instruments, belong here. These problems, physically, of course, are extremely close to those with which radio engineering is concerned. Then questions relating to the theory of machine regulation (one of the extremely important problems of technology) are closely connected with these same questions. Questions of the general dynamics of flight have very many features in common and are solved by these same means. But certain other, still more remote, fields also belong here. For example, the question of Cepheid-type variable stars, a question of oscillations, apparently reduces to the problem of nonlinear systems: there, apparently, schematically speaking, the same oscillations are established as in a generator. Further, you know that in chemistry there exist so-called periodic reactions. It seems to us that this question belongs to the same category of mathematical problems. Finally, from an entirely different field: several years ago there appeared a mathematical work by Volterra relating to the question of the coexistence of two biological species, depending on the struggle for existence. Apparently, this question too leads to a completely analogous mathematical formulation. Thus you see that

these methods of Poincaré are now acquiring ever greater and greater importance. Quite apart from the fact that they provide an immediate direction for the solution of our problems, in other fields as well they acquire guiding significance. Therefore, I think you will allow me to dwell in a few words on the essence of these methods, in the very simplest case, although I must say that to go into this more or less deeply at our general meeting would hardly make sense. All these questions will be examined much more thoroughly in the sections. Here I shall confine myself to the most superficial indications, and I would ask the mathematicians to forgive me for not striving for any particular rigor of exposition, and for not making the reservations which from the mathematical point of view ought to be made; I shall not make them because this would weigh down the exposition, whereas my aim is only to give the most general idea.

If one writes the equation to which the simplest generator is subject—one which is schematized, but preserves the typical properties of an actual generator—then one obtains the following expressions for the current or for the voltage on its capacitor:

\[ \frac{d^2 y}{dt^2}+\omega^2 y=f(\dot y,y). \]

The form of the function \(f\) is determined by the characteristic of the cathode tubes, and the matter here is the integration of an equation of this type. In this, the autonomy of our system is expressed by the fact that the time does not enter explicitly into this function here.

Put \(\dot y=x\). Then

\[ \frac{dx}{dy}=\frac{-\omega^2 y+f(y,x)}{x}. \tag{1} \]

If we somewhat generalize the problem, then the solution of our problem will reduce to the solution of a system of such equations:

\[ \frac{dx}{dt}=P(x,y),\quad \frac{dy}{dt}=Q(x,y). \]

A system of two such equations is, obviously, equivalent to the first one if we put

\[ P(x,y)=-\omega^2 y+f(y,x),\quad Q(x,y)=x. \]

If I divide one by the other, I obtain one differential equation between \(x\) and \(y\), namely the above-written equation 1. Poincaré’s works are concerned with the question of integrating such an equation.

What does it mean to integrate such an equation? If by integration we mean finding functions known to us (this is a somewhat vague concept) which satisfy this equation, then this problem is in general insol—

solution—there are, generally speaking, no such “known” functions. If we were to pose the question in such a way that \(x\) must be expressed in known functions in explicit form or in quadratures of such functions, then the majority of problems of physics and engineering would be excluded, because the functions defined by this equation are defined precisely by this equation and have no other definition known to us. Thus the task, as Poincaré says, is to derive from the equation itself the principal properties of the function that interest us, the function being defined by it. This is the task he sets himself, and first of all—to reveal the qualitative properties of the solutions of this equation. We now call such a mode of approach the qualitative integration of differential equations. Physically this means the following. If we are able to integrate qualitatively, then we obtain a picture of the qualitative course of the process, i.e. we can know such things, for example, as: can the system be in equilibrium, can the system produce oscillations, be in an oscillatory state, how do the oscillations depend on one or another property? We shall not, generally speaking, be able to give the numerical value of the amplitudes, the numerical value of the periods. But even such, rather approximate knowledge would give extremely much.

Unfortunately, the problems of nonlinear oscillations are in essence far more complicated than the problems of linear oscillations. And therefore we probably cannot at present demand that here we obtain the complete knowledge that we have there. Therefore we often have to be content with such a qualitative solution of problems. But these qualitative results, too, as has been said, play an enormous role for us.

Let us turn to the mathematical side of the question. We wish to investigate \(y\) as a function of \(x\), with the dependence of \(y\) on \(x\) given by the equation

\[ \frac{dx}{dy}=\frac{P(x,y)}{Q(x,y)}. \]

Geometrically this means: to find the behavior of the integral curves in the plane \(XY\). I shall recall the physical meaning of the quantities \(x\) and \(y\) in our simple cases: if \(y\) is the voltage on the capacitor, then \(x\) is the current. We shall call the plane \(XY\) the phase plane. The geometrical interpretation of the solution of a differential equation on the phase plane is very convenient and essentially helps one express and interpret analytic relations.

The following is known: let \(P\) and \(Q\) be regular functions; in our physical problems this is so. Then, generally speaking, through each point there passes one and only one integral curve. But what does a point on the phase plane mean? It means a simultaneous value of the current and vol-

tension. You have specified the current and voltage, and on the phase diagram this corresponds to one of the points. Thus, for a given initial current and voltage, the behavior of the system is determined uniquely. But there are also exceptions in the behavior of the integral curves. The exceptions are precisely those points at which both \(P\) and \(Q\) vanish. These points are called singular points. And it turns out that these singular points correspond to positions of equilibrium of our physical system. In other words, this means the following: our transmitter, generally speaking, when switched on, will oscillate, i.e., will not be in equilibrium. But we want to know whether there are also such states of the transmitter, i.e., such values of voltage and current, for which there are no oscillations and which can remain unchanged. And now we know how to find this out. If there are singular points in the phase plane, then positions of equilibrium exist, and the coordinates of these points give the values of current and voltage for which this equilibrium takes place. Analytically, this reduces to the solution of two simultaneous algebraic or transcendental equations:

\[ P(x,y)=0,\quad Q(x,y)=0. \]

The form of the functions \(P\) and \(Q\) is known as soon as the arrangement of the transmitter is given.

Now here is what is interesting. Poincaré drew attention to the fact that there are (if we disregard quite exceptional cases) four types of equilibrium of our system.

This is a very interesting study, which in general divides all possible positions of equilibrium into four classes (Figs. 1 and 2). If we now turn to equation (1), then we may say: one class is always unstable, i.e., corresponds to unstable equilibrium; another corresponds to stable equilibrium; and the two remaining ones may correspond to either stable or unstable equilibrium. But in Poincaré’s investigations the most interesting point lies in the following. The number and properties of the singular points, and their arrangement, in a certain sense predetermine the behavior of the integral curves. Thus, the mere existence of equilibrium conditions already limits the possibilities of motion. It says, on the one hand, that in such-and-such a system, possessing such-and-such equilibrium conditions, oscillations are altogether impossible; in another they may in general occur—not always, but they are possible.

Which integral curves are especially important for us? Closed integral curves are of quite special interest to us. Why are we interested in closed integral curves? It is easy to see that a closed

the curve in the plane \(XY\) represents the physical process of oscillatory motion, and, specifically, an established periodic process. Indeed, suppose that we have such a closed curve. What does this mean? \(x\) and \(y\) are a representation of current and voltage, or of coordinate and velocity. If we trace the process in time, then the representative point moves along the integral curve. If the integral curve is closed, then, moving along it, it will return to its initial position after some finite time (we assume that there is no singular point on the curve). From the general theory of differential equations it follows that if the point has returned to its initial position, i.e., if both coordinates have simultaneously returned to their initial value, then the point continues thereafter to move in exactly the same way as during the interval of time in which it traversed the closed curve for the first time; i.e., the process is repeated a second time, and also a third time, and so on. A closed integral curve represents a periodic solution of the equations. It represents an undamped oscillatory process. Therefore the search for closed curves is the principal task in the question of solving our oscillatory problems in linear and nonlinear systems.

Fig. 1. Characteristic types of singular points and a scheme of the behavior of integral curves near these points. A stable node, for example, corresponds to the aperiodic discharge of a capacitor. A stable focus corresponds, respectively, to the oscillatory discharge of a capacitor.

Fig. 2. Characteristic types of singular points: \(a\)—saddle—always unstable. Example: an ordinary pendulum in the upper equilibrium position. \(b\)—center—stable. Corresponds to an ideal (frictionless) pendulum near its lower equilibrium position.

Thus, the second essential element of Poincaré’s theory, playing, along with singular points, a particularly important role for us, is closed integral curves, because they represent periodic oscillatory processes, which we wish to study. And it turns out, as I

I have already indicated (partly, of course; I am speaking somewhat summarily), depending on the properties of the singular points, and the arrangement of these closed integral curves varies.

Allow me to dwell somewhat more fully on one question related to this which is extremely important for us.

We distinguish, as is known, between conservative and nonconservative systems. The former are characterized by the fact that they are governed by equations of Hamiltonian type (or, if you like, the corresponding equations admit an energy integral). It turns out that for conservative systems, if we exclude singular points of higher orders, only two kinds of singular points are possible: the so-called center and saddle. Let us consider the simplest picture: there is one singular point. Then the following is true: if we have a saddle, then there are no closed integral curves; steady oscillations are impossible. If the singular point is a center, then through every point of the plane there passes one closed curve, i.e. an infinite multitude of undamped oscillatory processes is possible. Indeed, if all integral curves are closed, then wherever you place your point, i.e. whatever initial values of the current and voltage you give your transmitter, it will at once perform undamped oscillations. And under different initial conditions all these processes will be different; i.e. for a nonlinear but conservative system we again have the fact that the steady oscillatory process depends entirely on the initial conditions. Both its period and its energy are not inherent, so to speak, in the system itself, but are determined by the initial position. Yet we require that, from whatever initial positions we may start, our system should settle into definite stable oscillations. Conservative systems are of no use to us. What is the situation with nonconservative systems? Here, it turns out, there exist closed integral curves, but of an entirely different type (Fig. 2,b). They occur in isolation. Other integral curves, not closed, approach them spirally and wind onto them from the inside and from the outside. Systems possessing such isolated closed integral curves are precisely what we need. This means that if you start from some point, the system runs along this spiral curve and arrives at a certain closed curve and remains there. This indeed represents the fact that our generators possess a tendency toward one definite, sharply expressed oscillatory process. Such curves were called by Poin—

caré calls limit cycles (Fig. 3), and the importance of these curves, along with singular points, is now obvious.

Thus, in formulating one or another physical problem, our task consists in the following: we must set up the differential equations and try to determine how the singular points are arranged and of what type they are, how the limit cycles are arranged, what their type is, and what sort of connection there is between the singular points and the limit cycles. This problem is, in essence, not particularly easy. I am speaking about types of limit cycles. I would like to point out that there are

Fig. 3. Limit cycles and the scheme of the course of integral curves near the cycles: a — stable cycle, b — unstable cycle.

limit cycles toward which the integral curves wind—these are the ones we need; these are stable systems. And there are limit cycles from which the integral curves unwind. This is of no use to us; these are unstable systems. What is mainly important to us are stable limit cycles. And so Poincaré’s theory gives, in qualitative terms, indications of how singular points and cycles coexist, i.e., how positions of equilibrium and periodic solutions are situated relative to one another. From the arrangement of some, from their form and their character, you can predict a great deal about others. And this is extremely valuable for us. For illustration I shall indicate one admittedly very simple example: inside a closed curve there must always be at least one singular point. Translated into our language, this means that oscillation can occur only on both sides of the position of equilibrium. Furthermore, it turns out that not all positions of equilibrium can give rise to such cycles. In mechanical examples these simple propositions, by which, of course, the theory is not limited (this goes without saying), are trivial. We often see these co-

relations without any special theory; but in electricity this obviousness fails us, and in many cases we can draw an important, far from obvious conclusion. I shall point to certain questions: the question of breakdown, of hard and soft excitation—questions extremely important for us, which find clear expression in the language of limit cycles and singular points. I cannot dwell on this further here; I shall only say that I am convinced that this visual geometric representation, giving, to be sure, a qualitative but very good picture of the possible processes, will within a short time enter into the everyday practice of the physicist and, perhaps, the engineer.

Perhaps all these arguments will become clearer if I show you an experimental limit cycle (Fig. 4).

Fig. 4. Photograph of an experimental limit cycle, after the work of G. Ostroumov.

One can make the generator trace such a limit cycle. If you act in the appropriate way on a Braun tube, on a cathode oscillograph, it is possible to arrange matters so that one deflection will be proportional to the current, the other proportional to the voltage, and then such a point will simply trace out the limit cycle.

One of the first tasks that arises when you investigate new devices is the question of whether a cycle exists at all. Often, by comparatively simple reasoning, one can prove the existence of a cycle. I do not know where it lies; I know neither the amplitude nor the form of the oscillation, but I know that it exists. This is already very much; but, unfortunately, in many cases we cannot even do this. And I must characterize this qualitative method of integration

differential equations in such a way: it provides an apparatus that allows one to think in images that are extraordinarily adequate, and often allows one to draw valuable conclusions about the behavior of a system. But here there is still very much that needs to be done and, apparently, can be done. This theory is imperfect, and in the form in which it exists now, both from the mathematical and from the physical side, it is still subject to further deepening. I shall point, for example, to one mathematical problem which, as far as I know, has not yet been solved, although it would seem to be rather simple. As early as 1900 the famous German mathematician Hilbert, at the mathematical congress in Paris, posed a number of fundamental problems, in his opinion, from various mathematical disciplines. One of them directly concerns us. Let \(P\) and \(Q\) be polynomials of some degree. This approximates rather well, when their degree is not especially high, our practical case of a generator. It is necessary to indicate the greatest possible number of limit cycles for given \(P\) and \(Q\). Physically this means finding the greatest possible number of oscillatory states of the given system. As far as I know (I am, to be sure, not a mathematician), this problem has not yet found a solution.

What we would need is the following. We are given a characteristic, for example graphically. A question of this kind arises: can one, from certain properties of it, without going into great detail about this function—for example, from the existence of asymptotic values (physically this means the existence of a saturation current)—conclude anything about those oscillatory possibilities which the given system possesses? In many cases we cannot do this; in particular cases we can do it. A solution of this problem would be very useful.

But I am compelled to leave this question and turn to the next one. The approach to our questions set forth above is qualitative. But, of course, a physicist, and especially an engineer, generally speaking, cannot be satisfied with qualitative answers. He needs to obtain numerical data; he needs to represent, of course with approximation, processes by means of functions known to him or, more concretely speaking, by means of functions for which tables exist. This is what he needs in order to calculate and predict what will happen, in order to construct his devices in the corresponding manner.

What, then, can we say regarding quantitative calculations even in the simplest case of one degree of freedom? Unfortunately, not much. We would need to find a possibly simple algorithm that would allow us approximately to calculate these limit cycles for general cases—

of them in the form of a Fourier series or something of that kind. But for the time being we cannot do this. In one case, namely, if the nonlinear terms are sufficiently small (and this case occurs quite often in practice), using the work of Poincaré (which he developed for entirely different purposes, namely for the purposes of celestial mechanics), it proved possible to carry the solution through to the end. Here one can not only solve the question of whether a limit cycle exists or does not exist, but also carry the quantitative solution of the problem to such a degree that it makes it possible to systematize the experimental material obtained and serves as a guide for setting up new experiments, for evaluating them, and for assessing what must be done in order to obtain one effect or another. In a word, for these, I repeat, practically important cases, we can obtain what we need. There is also another limiting case that can likewise be approached quantitatively. I have in mind relaxation oscillations, but, unfortunately, I cannot dwell on them.

I have already indicated that these questions pertain not only to our problems of electromagnetic oscillations. I have listed a whole series of questions where limit cycles are of just as great significance. To give one more interesting example, I shall point out the following: if the matter concerns the dynamics of flight, specifically an airplane flying in a vertical plane, with a constant angle of attack and uncompensated propeller thrust, then the question of dead loops is reduced, with suitable initial equations, to the investigation of limit cycles. This was shown by Dulac; the same was reproduced by Andronov and Witt.

Fig. 5. Schematic drawing of Froude’s pendulum.

In addition to the cases of which I have spoken, there is one very simple mechanical model which, in the first approximation, obeys the same equation to which our generator also obeys. This is the device that we call Froude’s pendulum. Its arrangement is as follows (Fig. 5). Let us imagine a rigid pendulum not on a point support, but provided with a sleeve. Into the sleeve there enters an axle to which a constant rotation can be imparted by an external force, for example by setting it in rotation with a motor. If you write the equation for small deviations of this pendulum, then—

L. I. MANDELSHTAM

obtain with sufficient approximation the following equation:

\[ \ddot{\varphi}+n^2\varphi=f(\omega-\dot{\varphi}). \]

On the right there will be the torque which acts on the pendulum from the rotating axis owing to friction. This torque will be a function of the constant angular velocity of rotation of the shaft \(\omega\) and of the angular velocity of the pendulum \(\dot{\varphi}\). You see that this is nothing other than our original equation. Under suitable conditions the pendulum should begin to swing with its own period (the shaft is rotated by a motor with constant speed), and then there should occur, and in any case it is possible, the establishment of undamped oscillations.

We were attracted by the idea of investigating such a pendulum for the following reasons: on such a mechanical model, the phenomena that interest us, for example in a generator—the instability of the equilibrium position, the growth of oscillations, the establishment of stationary oscillations—can be traced and studied extremely simply. Here we simply see how everything takes place.

Such a pendulum was constructed and investigated by S. P. Strelkov. Strelkov did indeed, at any rate qualitatively, observe the phenomena predicted by the theory and with which we are familiar from the field of electromagnetic oscillations. The instability of the equilibrium position, growth, i.e. spiral integral curves, the established regime—the limit cycle. True, here there was not the constancy that would have been desirable for quantitative investigations, but this can probably be achieved. I note that a pendulum of this sort was constructed in his time by N. E. Zhukovsky. But Zhukovsky, so far as I know, was interested in creating a suspension without friction. In our language, the difference between Zhukovsky’s interest and ours is expressed as follows: Zhukovsky was interested in the case of a center and in the integral curves corresponding to this case. We, however, are interested in limit cycles, which, apparently, Zhukovsky was not interested in at that time. These are two different problems, and therefore the conditions must be chosen differently.

But besides this, this model, if it is developed appropriately, may prove useful in another sense. The point is that the whole process is connected with the form of this function \(f\), and the form of this function depends on the characteristics of friction, i.e. on how the coefficients of friction depend on velocity. Thus, by studying this model, we may perhaps obtain a method for studying the dependence of friction on velocity. This question, after all, plays a major role in technology in general, and this method may turn out to be more useful than those available up to now.

because it is direct, because it can give not only the friction itself, but also directly the derivative of friction with respect to velocity, whereas now this derivative is constructed from the friction curve. Such a direct method may, of course, have advantages.

What we have spoken about so far concerns the search for periodic solutions, the clarification of the question of their number, of what form they have, and so on. But the physicist requires still additional investigation. The physicist (and, of course, the engineer as well) requires an answer to the following. Suppose a periodic solution has been found. The question arises: is it stable or unstable? That is, here there arises the same question as in the investigation of an equilibrium position. If it turns out that your solution is unstable—for example, let it be a limit cycle, but one from which the curves unwind—then, of course, it cannot serve for the generator. The slightest displacement will at once lead to the disturbance of the stationary oscillations, for example to a breakdown. And so the question arises of the stability of periodic solutions that occur in nonlinear nonconservative systems.

How do matters stand here? It turns out that a complete mathematical apparatus for this exists, namely in the works of Lyapunov. Lyapunov solved, with great completeness, the question both of the stability of equilibrium and of the stability of periodic solutions in nonconservative systems.

It is interesting to note the following here: when Lyapunov was doing his work, these technical problems had not yet arisen, and radio did not yet know of them. He created this theory chiefly out of abstract mathematical interest. And then the situation changed in a well-known way: when the technical and physical devices of which I have spoken were being created, people did not know about Lyapunov. And physical practice required methods that would make it possible to solve the problem of stability in relation to concrete physical questions. And it turned out that, thanks to the fact that Lyapunov had taken an interest in a mathematical question, we have the apparatus, and we need do nothing else than simply apply this apparatus. This is what is being done, and the necessary answers are obtained.

Here is what results. The question of the stability and instability of an equilibrium position of nonlinear systems leads to the analysis of a special linear equation (we know how to construct it): this is a linear equation with constant coefficients.

The question, however, of the stability or instability of periodic solutions can also be reduced to linear equations, but with periodic coefficients.

And thus a new class of differential equations must now enter our field of vision—linear equations with periodic coefficients. The theory of such a linear equation is known. I would only like to indicate its distinctive features. The simplest such differential equation has the following form:

\[ \ddot{y}+a^{2}(1+q\cos 2t)y=0. \]

(I write it in canonical form, having chosen definite units for the period of variation of the coefficient.) This is the so-called Mathieu equation. And the question of the stability and instability of periodic solutions reduces to this type of equation. And here is what turns out. These equations generally have solutions of various types. Depending on the relation between the quantity \(q\), which gives the amplitude of oscillation of the coefficient, and the quantity \(a\), which gives the mean frequency, the situation may be such that the oscillations do not leave the initial bounds, i.e., if they (the solutions or oscillations) were small at first, then they remain small. If the situation is such, then we say that the solution is stable. If, on the contrary, the relations are different, then it happens that the solution of this equation has the property that, however small you may prescribe it at the beginning, it grows exponentially upward, continually increasing and increasing. In this case our equation, which we derived as a criterion for stability and instability, shows that our initial solution is unstable. Thus we can decide, by investigating the corresponding linear equations with periodic coefficients, the question of the stability of the oscillatory process given by our original nonlinear equation.

For us, the solution of these problems, equations with periodic coefficients, plays an auxiliary role. They are intended to decide the question of the stability or instability of solutions of equations which are themselves nonlinear. But physics begins to approach this from another point of view as well. There is a whole series of major physical problems which lead to the same type of equations; for example, the question of the oscillations of an elliptical membrane, and also the oscillations of armatures in electric carts.

I shall also point to problems from an entirely different field. One of the most essential tasks of modern wave mechanics is the question of metals. And so, questions of metallic conductivity and many other questions relating to metal reduce directly to such linear equations with periodic coefficients.

There they play an enormous role; there the question of stable and unstable solutions of these equations as such is of primary importance, though, to be sure, the matter is complicated there by the fact that there is not one degree of freedom, but three, because there is a spatial lattice and the equation is more complex; but in principle it is one and the same. A whole series of problems from other fields also leads to this same type of equations. Permit me to dwell on one of them in somewhat greater detail.

Indeed, imagine that you have a circuit in which you take a capacitance or a self-inductance and periodically vary its magnitude. Then the equation for the oscillation in this circuit is an equation with periodic coefficients, and the period of the coefficient is given, since the variation of the capacitance or self-inductance in time is given.

What, then, should we expect if we apply what has just been said to this physical case? Under certain relations (namely, those under which the solution of the equations is stable), nothing will happen. You will charge the capacitor a little, but nothing interesting will occur; there will be a small change, and it will soon die out. But we know that there is such a relation between $\alpha$ and $q$ under which, however small the initial state may be, however small the initial currents may be, they grow more and more. Hence, what should be expected? If you take a circuit with capacitance and self-inductance and, let us say, vary the capacitance periodically at a definite tempo, with a definite amplitude (and the tempo can always be chosen so as to satisfy the condition that must be satisfied), what will happen? Since in every circuit small currents always exist, if only by virtue of statistical fluctuations, it follows that if you do not introduce any electromotive force, but simply vary the capacitance, then oscillations must arise spontaneously in such a circuit. An entirely analogous conclusion is also valid for the case when the self-inductance of the circuit is varied periodically. We can thus say that we create electrical oscillations out of nothing, since initially we have no need to create either electric or magnetic fields. Here, of course, we have the conversion of mechanical oscillations directly into electrical ones. We have a special type of alternating-current generator. We now know this. It must be said that Rayleigh drew attention to this possibility as far back as fifty years ago. He says little about it, but he does indicate the possibility of such a phenomenon.

Now such a generator of a special, very simple constr—

of the construction is being carried out by us in the GFTI laboratory, directed by Prof. Papaleksi.

It is very easy to understand physically how, in such a generator, the process of the growth of oscillations takes place.

Allow me briefly to dwell on this and, for simplicity, to take a schematic case. I shall suppose that we periodically change the capacitance. In the model actually realized it is, as we shall see below, the self-inductance that changes. But the case with a changing capacitance seems to me somewhat more transparent. And in essence one may, of course, transfer almost without change all the reasoning to a change of self-inductance. I shall speak very briefly, so as not to delay, but I shall nevertheless state the idea.

Imagine that we proceed in the following way. Take a circuit consisting of a capacitance and a self-inductance, the capacitor consisting, let us say, of two parallel plates. Imagine that on the capacitor there is an arbitrarily small charge, while at that moment there is no current. And at that moment move the plates apart. Since there is a charge, you will expend a certain amount of work. Now discharge takes place; all the potential energy is converted into kinetic energy, in the present case—magnetic energy. The work which we expended in moving the plates apart is likewise converted into magnetic energy. The current will therefore be somewhat stronger than it would have been had you not moved the plates apart. After a quarter of a period the charge is equal to 0, all the energy is in the magnetic field, and at that moment you can move the plates back together again, without doing any work in the process, and wait another quarter-period. Now all the energy has again passed into electrostatic energy, but now there is more of it than there was at first, namely by the energy which you created by moving the capacitor plates apart. The whole circuit, however, as regards capacitance and self-inductance, has returned to its former state. Thus, in one such cycle, you have obtained an increase of charge. Repeating the same procedure further and further, you will obtain a continuous growth of charges.

From everything that we have said here, it is clear how one must proceed: one must change the capacitance twice as fast as the natural period of my circuit. During one full period of the natural oscillations one must twice move the plates together and apart. This is the physical mechanism that underlies the mathematical treatment of the question. A more detailed analysis shows: 1) that there is no need to make the change of parameters by jumps; the same qualitative result is obtained also with a smooth periodic change of capacitance or self-inductance; 2) there is no need for the period of the change to be exactly half as small as the natural period; certain deviations are admissible here; finally, 3) quali-

... essentially the same phenomena are obtained when there is not too great a resistance in the circuit. As has already been noted above, in the model that was realized the self-induction was varied. In practice this leads to a more advantageous design. With a corresponding number of revolutions, the phenomenon discussed above occurs. The voltage and the current increase. This voltage, if it is not specially damped, reached in our experiments up to 15 thousand volts in a very short time. It was impossible to go further, because the insulation could not withstand it. If a resistance is introduced, the magnitude of which depends on the current, then a stationary state can be obtained. I shall note, incidentally, that a stationary state cannot, of course, be obtained in any way from a linear equation. Such a state can arise, and indeed does arise, only when nonlinear conductors are present in the circuit. Several hundred watts were obtained. The rotor was set in motion by a motor with a reduction gear, and the number of revolutions of the shaft reached 15 thousand per minute, while the number of periods was approximately 1700 per second. This corresponded to the excitation of oscillations at half frequency; hence the generator gave a frequency of 850 periods per second. The current was established, lamps burned, and so on. Whether this generator will have practical significance, and what kind, it is still premature to say anything about.

Allow me at this point to finish with the general questions relating to one degree of freedom. You see that this simplest case required a great deal of new mathematical apparatus. But we could not stop there. The practice of physics urgently demands the theoretical treatment of more complex systems, namely systems with many degrees of freedom. And here I can confine myself to a very brief remark, because I must frankly say that in this respect, from the theoretical point of view, we know little. Instead of the phase plane for one degree of freedom, here we must deal with a phase space of many dimensions. As a stationary, definitively established state with one degree of freedom, either positions of equilibrium or a periodic process are possible. Quasiperiodic phenomena, i.e. oscillations with two incommensurable periods, cannot arise there. In a more complex system, already in a system with two degrees of freedom, there are possible, first, states of equilibrium, then periodic processes, then quasiperiodic ones, and other still more complex ones.

Phenomena in systems with two or more degrees of freedom are of great interest. For these include beating phenomena, which have an extremely substantial signi...

can help here. We are unlikely to cope with these, apparently mathematically difficult, questions without the help of mathematicians.

Allow me to conclude this section with one more general observation. I have said, and I have often recalled, that we require a proof of the existence of periodic solutions, that this is a very important part of our theoretical investigations. One often hears the following considerations: indeed, it is very important to have a theory that allows us to calculate amplitude, to calculate periods, but why should you concern yourself with the question of the existence of periodic solutions? Is this not merely a mathematical amusement, since we are convinced by experience that such processes exist in the given system? When you write the differential equations for some physical problem, you always and inevitably simplify the given problem greatly. You always write the equation not for the given problem, but for an idealized, simplified one. How do you know that you have taken account of all the essential features of the given problem? And so, if the proof of existence leads to the fact that periodic solutions exist, and this is justified by experience, then this is already a certain argument that you have not missed essential features, because by essential features we mean those that determine the possibility of oscillations. But these are only indirect indications. But what will happen when, in the course of proving existence, you come upon the fact that our differential equations have no periodic solution, while the object for which you created it does have one? Then you can probably be convinced that you have failed to take account of the most essential features. Then you begin to look for what features you have missed. And practice knows examples where the search for existence proofs led you to considerations about where to look for these omissions, helped you find what was needed and thus put the whole problem on the right track. I shall give a completely elementary example. Of course, you all know how in many textbooks, especially older ones, the theory of a simple electric bell or, say, an electromagnetic interrupter is presented. A striker or an armature in the position of equilibrium closes the contact of an electric circuit into which

included is an electromagnet acting on the armature. When you switch on the battery, the electromagnet attracts the striker, the current is broken, the force of the magnets disappears, the spring drives the striker back, the contact closes again and, so to speak—I deliberately looked it up in one good textbook—“das Spiel geht weiter,” or “the game goes on.” And so, if you translate these considerations into the language of a differential equation, it is easy to prove that the existing differential equation does not allow the game to begin anew; it has no periodic solution. This means that something essential has been omitted here. And indeed, the theory of interrupters is not as simple as it seems. We know, for example, what an essential role self-induction plays in the question of the possibility of oscillations. Success in the problem of the interrupter has been achieved by the work of M. A. Leontovich. From it one can clearly see how self-induction influences the matter; it not only determines the possibility of the process itself, but also determines the period of the oscillations, which differs from the period of the tuning fork or of the striker itself. Thus the investigation of the existence or nonexistence of periodic solutions points to essential features, indicates how they must be taken into account. I could cite other examples as well, where it would be clear that the benefit of such investigations is quite real.

Allow me now to say a few more words briefly about the second main problem, the problem of reception. Since what is involved is reception by linear systems, I think no questions of principle arise. Here the matter stands approximately as follows. By studying the spectrum of the incoming signal and the spectrum of the disturbances, the problem is exhausted from the standpoint of principle. In concrete cases, sometimes more quickly one is led to the goal by methods different in form from the spectral approach, but this, of course, does not in principle violate the correctness of the proposition stated. The practically important and interesting questions here are determined, as is known, by the antagonism between two requirements—selectivity of reception and speed of receiving operation. You know what this antagonism consists in. If you wish, you can construct a receiver so that it practically reacts to one definite wavelength, one definite oscillation, one definite sinusoid. If one takes sufficiently small damping, then we shall achieve this with sufficient approximation. And today we have means of making the damping very small. Then this receiver will almost not react even to an oscillation close to its own. It protects itself very well against extraneous oscillations, including atmospheric disturbances. This insensitivity to an alien influence is at first glance very good. But, unfortunately,

such a receiving device does not solve the fundamental problem of communication. Such a receiver can report nothing other than that the transmitter is switched on.

The prerequisite for communication is the ability to receive signals. A signal consists in the fact that, at the transmitting station, in a definite way and at a definite rate, we change the form of the transmitter’s oscillation or, as we say, modulate it, producing dots and dashes with a key, or changing the amplitude in the rhythm of the acoustic oscillations of my speech.

But such a modulated oscillation already has a complex spectrum. This spectrum contains within it the characteristic, or form, of the signal. In order to receive the signal without distortion, one must receive its entire spectrum. A receiving device which protects itself from extraneous influences by receiving only a very narrow region of the spectrum is likewise incapable of receiving the spectrum of the signal itself.

And here there exists a remarkable dependence: the shorter the dot or dash, the higher the acoustic tone modulating the transmitter, the wider the range of frequencies in the signal spectrum, the less selective the receiver must be in order to be able to receive the spectrum of the signal intended for it. Rapid operation is incompatible with great selectivity. I note that the relation just discussed (it may be formulated schematically as follows: the shorter a sinusoidal oscillation lasts in time, the wider the spectral region it occupies; or, more generally: exact localization of a process in time is incompatible with a narrow spectrum) plays a large role in other areas of physics as well; for example, in the question of images of objects by means of optical apparatus—an unquestionably very important question connected with such problems as the resolving power of optical instruments, and so on. In these cases we encounter an analogous situation, but one pertaining to spatial rather than temporal relations.

But this antagonism between the exact localization of a process and the breadth of its spectrum acquires an entirely fundamental significance in modern wave mechanics, which, as I indicated at the beginning, is permeated by the spectral point of view. It is the basis of Heisenberg’s famous uncertainty principle, now regarded as the cornerstone of our physical worldview. Unfortunately, I cannot dwell on this in detail and must confine myself to these few remarks. I shall say only the following. Anyone who has grasped this antagonism in radio engineering will find it much easier than someone else to master this fundamental proposition of wave mechanics.

But all this applies to linear systems. Now, however, thanks to the introduction of nonlinear systems into receiving devices, we have other possibilities at our disposal.

I believe that, in principle, the antagonism between selectivity and speed of operation exists here as well. But here we have a whole series of new phenomena that gives hope for the possibility of achieving more advantageous operating conditions.

I shall not enumerate those very interesting phenomena which distinguish nonlinear systems in an essential way from linear ones. In general, some of them are commonly known. I shall point, for example, to the phenomenon of “capture,” specific to nonlinear systems, which also extends to acoustic phenomena (K. F. Teodorchik and S. E. Khaikin have recently been engaged with these questions).

But I should like to say a few words about one application of nonlinear systems to reception. Until now, even when nonlinear systems were used, one relied mainly on ordinary resonance—though, to be sure, proceeding here differently than in the linear case, but still not differing so very greatly, still having an analogous character, which also determines the characteristic shortcomings. A theoretical investigation of the behavior of nonlinear systems under the action of an external force—an investigation based, again, on Poincaré’s methods, developed by him for celestial mechanics—has shown that here one may expect, among other things, quite different “resonance” phenomena, namely the following. If one takes a nonlinear system in a definite regime predicted by the theory, then the following phenomenon is observed. So long as the signal frequency is substantially different from twice the frequency of the system, nothing special happens. The system behaves approximately as a linear one. But, under a corresponding regime of the receiver, there is a rather narrow band of frequencies in the neighborhood of twice the frequency of the system itself, possessing the property that if the signal frequency falls into this band, then the system becomes unstable and at once jumps to one-half the frequency of the signal. This phenomenon is essentially different from ordinary resonance, which, as is known, occurs when the frequencies are equal. And here one may speak of resonance curves—resonance curves of the second kind, essentially different in their whole character from ordinary resonance curves. This phenomenon may be made the basis of a receiving device.

In developing this phenomenon, N. D. Papaleksi and I were assisted, in the experimental part, by our co-workers: E. M. Rubchinskii, M. M. Vaissebein, and I. M. Borushko; and, in the theoretical part, we were assisted by co-workers

of our institute, A. A. Andronov and A. A. Witt. The experiments have passed beyond the stage of laboratory development. Under the direct direction of N. D. Papaleksi the corresponding devices were tested and tried in operation.

Whether these receiving devices will justify themselves, for example in the sense of a more effective freeing from interference—which is one of the basic tasks of radio engineering—the future will show. For the time being the results are quite good. In the sense of freedom from interference and of selectivity the new device apparently has advantages over others.

I have mentioned these experiments in order to conclude with the following. I am wholly convinced that the introduction of a nonlinear system into receiving devices, which has already yielded very much, still contains very great technical possibilities. The variety of physical phenomena here is much greater than in linear systems. It is quite possible that some of these phenomena, not yet utilized, can be used and valuable practical results obtained. Their physical interest for me, of course, has long been beyond doubt.

Now, if you—quite naturally—wish to make a quantitative calculation of certain questions, say, to calculate what advantages should be expected here in the sense of interference, then although qualitatively you will obtain a known answer, in a quantitative calculation you encounter fundamental difficulties. And one of such difficulties, which arises not only here but also earlier, as soon as you turn to nonlinear reception, consists in the following. Suppose you know how the signal of the station which you are receiving separately affects reception; you know how the interference acts separately. But what interests you is not this, but what will happen when both the station and the interference act simultaneously. Thus, with linear systems, knowing how one station or one interference acts, you knew how they act jointly. Now, however, knowing how each part acts, you still cannot say what will happen when both act. The principle of superposition is inapplicable here.

And then you see that this alone is already sufficient in order to be convinced that a vast number, I think, of most valuable practical and physical experimental possibilities is purchased at no very cheap price; it is purchased by the fact that the harmonious and integral theoretical conception which we had up to now is disturbed.

Here, indeed, the situation is as follows. In the early days of radio telegraphy, its physical and technical content was comparatively poor. It was built in its main part on linear systems. But the mathematical side of the phenomena, and especially the spectral approach, which played the decisive role, were transparent, clear, and integral. The introduction of nonlinear

transmitters has made the picture more complicated. But in evaluating the transmitted signal and in reception we still for the time being think mainly by the spectral method, i.e., we stand at a point of view which is essentially adequate to linear systems. Now, with the violation of linearity, this approach loses its ground also in reception, and we have, at any rate in part, to abandon this point of view in reception as well. The spectral point of view is gradually beginning to outlive itself. And along with this there arises such a question: is a harmonic oscillation, generally speaking, an “elementary” oscillation? To what extent is it expedient to require of a transmitter the absence of overtones and, in general, periodicity, etc.? And of course this is unpleasant. The integrity of the theoretical picture is always desirable. We have already for quite a long time been in a position where, with the introduction of nonlinear systems, strongly differing from linear ones in transmitters and receivers, we must renounce the majority of the theoretical concepts that have guided us. How should one regard this—is it regrettable or not regrettable? On the one hand, yes; but, on the other hand, the invasion of the new must never be obstructed. And my point of view is as follows. It is possible—and I myself am sometimes instinctively inclined—to reason approximately thus: as long as a thing is theoretically unclear to me, I shall try to avoid it and shall try to build my systems, my devices, in such a way that those things which are unclear to me do not get mixed in there. Then the whole picture is clear, but I deprive myself of very valuable possibilities. No, it seems to me that the right path is this: to seek new theories, new points of view, without being distressed that at first we lose coherence. And we are now compelled to take this path.

I consider that in questions of oscillations, in the theory of oscillations, the present state of affairs, in the theoretical sense, is rather acute. We are in fact gradually losing part of those guiding principles by which we have been guided until now. But the way out is not to strive to narrow the experiment, but to broaden the theory. I remember how in former times people engaged in radiotelegraphy knew: one must avoid iron. This was a slogan, because very little was known about the behavior of iron at high frequencies; iron was considered a contaminant, and they acted correctly, since they did not know its action. At one time it interfered, at another time it helped; it would have been better to get rid of it. Now we know better how iron behaves, and already many circuits operate with the use of iron. Did not the same thing repeat itself with gas tubes, with gas kenotrons: we knew discharges in gases poorly and therefore turned to vacuum tubes. Remember, there was a slogan: gas always does harm. I think that now hardly anyone ...

it asserting this. As we study phenomena more and more, we gain mastery over them and no longer fear them; on the contrary, we consider them increasingly useful.

Therefore, I believe that our task now, alongside the full development of experimental work, is to try to find and develop an adequate theoretical method. I think that we cannot manage without the help of mathematicians, and I think that our conference will provide an impetus and will enable us to move forward in this respect as well. Allow me to conclude here.