ALEXANDER ALEXANDROVICH
ANDRONOV

IN MEMORY OF A. A. ANDRONOV

G. S. Gorelik

Alexander Aleksandrovich Andronov has died—an outstanding scientist and a remarkable man, the creator of a new direction in the theory of oscillations and in the dynamics of machines, a talented figure in the Soviet higher school.

A. A. Andronov was born in 1901 in Moscow. Already in secondary school he dreamed of devoting himself to science. For a time he was most of all attracted by medicine; moreover, in order to seek new paths in it, he first wished to acquire the reliable scientific foundation provided by the physics-and-mathematics faculty of the University. In the upper grades he began reading books on higher mathematics.

A. A. completed secondary school in 1918, during the Civil War. He went to work at a factory, then in one of the military-provisioning detachments, with which he left for the Urals. Returning to Moscow in 1920, he was admitted to the Moscow Higher Technical School, to the electrical-engineering faculty. From 1921, while continuing his studies at MHTS, A. A. Andronov began attending certain lectures at the physics-and-mathematics faculty of Moscow University. He felt such a strong interest in physics that he transferred, in 1923, to the University and in 1925 graduated from the physics-and-mathematics faculty of Moscow University in the specialty “theoretical physics.”

Andronov’s student years coincided with the beginning of the flourishing of the Moscow mathematical school. At that time students of physics and mathematics at Moscow University attended the same mathematical lectures. A. A. Andronov devoted much effort to mathematics and acquired a mathematical culture considerably deeper and more versatile than that usually possessed by physicists, including theoreticians. During his university years A. A. also showed great interest in theoretical mechanics. S. A. Chaplygin made a strong impression on him. Studies in theoretical mechanics left a notable imprint on Andronov’s scientific works.

Even before graduating from the University, A. A. Andronov began teaching mechanics and theoretical physics at the Second Moscow State University (now the Lenin Moscow State Pedagogical Institute).

Of decisive importance for the formation of A. A. Andronov as a scientist was his postgraduate study at Moscow University (1925–1929). A. A.’s adviser in postgraduate study was L. I. Mandelstam. Under his direction A. A. carried out (jointly with M. A. Leontovich) his first work, concerning the theory of the scattering of light by the fluctuating surface of a liquid, and began that cycle of works on the theory of nonlinear oscillations which will be described in detail here. When A. A. Andronov became an independent scientist and his own scientific school grew up around him (chiefly in Gorky, where he moved in 1931), he continued to work in close contact with L. I. Mandelstam.

Although A. A. Andronov was by training a theoretical physicist, the main field of his activity proved to be rather far from what specialists in theoretical physics usually do. A. A. Andronov’s scientific development followed a distinctive path. He was not attracted by atomic physics—the new area of research into which most of the young theoreticians of the 1920s, his contemporaries, aspired to enter. During his postgraduate studies he worked on statistical physics and on certain questions of quantum physics. But toward the end of his postgraduate period A. A.’s creative energies became concentrated on questions of the generation of oscillations, which had been placed on the agenda by radio engineering in connection with the appearance of the electron tube. His concluding dissertation was devoted to these questions. Almost all of A. A. Andronov’s subsequent scientific investigations were a development of the ideas contained in it.

In order to understand the fundamental significance of A. A. Andronov’s first works on the theory of oscillations, it is necessary to bear in mind that almost the entire “oscillatory culture” possessed in the 1920s by physicists and engineers (including radio engineers) was linear—it was connected with problems solved by means of the superposition principle and linear differential equations. This includes, in particular, the ordinary theory of alternating currents, as well as the theory of coupled oscillations in circuits whose resistance obeys Ohm’s law. Meanwhile, processes of oscillation generation can be understood only with the aid of nonlinear differential equations. This is evident at least from the following observation: for a tube generator it is characteristic that undamped oscillations of a quite definite amplitude are established in it, independent of the initial conditions; whereas in a system described by a linear differential equation, either undamped oscillations cannot exist (without a variable external action), or undamped oscillations of arbitrary amplitude are possible, entirely dependent on the initial conditions.

In the 1920s most radio specialists had not yet truly grasped the fundamental difference between the processes of generation of oscillations and those processes that are described by linear differential equations. They often tried to apply to the processes occurring in a vacuum-tube generator concepts suitable only for linear systems, for example the principle of superposition. These concepts sometimes led to sharp contradiction with experiment. Some researchers well understood the fundamental difference between a vacuum-tube generator and linear systems and posed the problems associated with the vacuum-tube generator as nonlinear problems, reducing them to nonlinear differential equations. But it is not enough to write down a nonlinear differential equation; one must be able to investigate it. The solution of nonlinear equations is, generally speaking, a matter incomparably more difficult than the solution of linear equations. The researchers whom we have in mind here succeeded in solving a number of nonlinear problems and obtaining valuable results. But the methods by which these results were obtained were, if one may put it so, artisanal in character and did not possess sufficient generality. The results themselves were fragmentary.

A. A. Andronov was able to illuminate the questions of the generation of oscillations with the light of “big science,” pointing to the general mathematical apparatus adequate to these questions. The beginning was very modest. A. A. constructed the simplest theoretical models of a clock and of a vacuum-tube generator, idealized to the utmost. In the generator model the characteristic of the tube was made up of two horizontal half-lines (a “Z-characteristic”). The differential equations of these models—though nonlinear—were so simple that A. A. could integrate them without difficulty and construct the complete picture of the integral curves on the phase plane. This picture is as follows: the phase plane is filled with spirals nested one inside another, winding from inside and from outside onto a closed curve. The closed curve corresponds to undamped oscillations, the spirals to transient processes. Even earlier—Andronov knew this—an analogous picture had been obtained by Van der Pol, using the method of isoclines, for a vacuum-tube generator under the idealization of the tube characteristic by a cubic parabola.

Here there occurred what determined A. A. Andronov’s entire subsequent scientific path: he perceived the identity of the closed curves on the phase plane, representing undamped oscillations of a clock and of a vacuum-tube generator, with limit cycles. He titled his graduate dissertation “Poincaré Limit Cycles and the Theory of Oscillations.”

A limit cycle is a closed integral curve of a nonlinear differential equation, to which neighboring integral curves approach asymptotically. Limit...

cycles were discovered and investigated by Poincaré, quite apart from any connection with physics, in his 1881 work “On curves defined by a differential equation.” This work marked the beginning of the qualitative (topological) theory of differential equations, whose aim was to clarify the general character of the behavior of integral curves. Before Andronov’s work, mathematicians engaged in the qualitative theory of differential equations did not suspect that limit cycles were related to physics and engineering; and physicists and engineers engaged in the study of processes connected with the generation of oscillations did not know that the mathematical apparatus needed for the creation of a general theory of these processes already existed.

A. A. Andronov also established the connection between the theory of the generation of oscillations and A. M. Lyapunov’s theory of stability, set forth in his famous work “The General Problem of the Stability of Motion” (1892).

What has been said about limit cycles requires clarification. Limit cycles may be stable or unstable: the representative point moves along neighboring integral curves toward the limit cycle or away from it. Properly speaking, the processes of generation of oscillations correspond to stable limit cycles. (Unstable limit cycles have another physical meaning, also explained by Andronov: they serve as a boundary between regions of initial conditions from which the system tends to various stable states.) A. A. Andronov showed that the motion represented by a stable limit cycle possesses the type of stability that has come to be called Lyapunov stability (the deviation of the representative point on the phase plane from a motion stable in the Lyapunov sense, if sufficiently small at the initial instant, remains—by definition—arbitrarily small for all time).

To designate undamped oscillations generated by systems possessing friction (resistance), like a clock or a tube generator, A. A. Andronov introduced a new term that has become firmly established in science—the term self-oscillations—and gave self-oscillations a precise mathematical definition*). According to Andronov, self-oscillations are motions represented on the phase plane (in the case of systems with one degree of freedom) by stable limit cycles.

Thus A. A. Andronov was the one who gave self-oscillations their name and mathematical definition, who connected their theory with the qualitative theory of differential equations, and therefore also with topology, as well as with the general theory of stability of motion.

*) Both he himself and L. I. Mandelstam attached essential importance to the latter circumstance.

His works contributed more than anyone else’s to the transformation of the theory of self-oscillations and phenomena related to them from a set of a few fragmentary results and computational recipes into a new, splendid chapter of the theory of oscillations.

This chapter lies outside not only classical linear oscillation theory, but also the theory of nonlinear conservative systems. Conservative systems have no limit cycles. One of the inalienable features of any generator of oscillations is that it contains friction (resistance).

The new chapter of oscillation theory, whose creation is associated with the name of A. A. Andronov, has not yet been developed as broadly as the classical chapters of oscillation theory—it was born only a quarter of a century ago. Nevertheless, even now it is not inferior to linear oscillation theory in the depth of its concepts, the clarity of its basic physical ideas, or the degree of correspondence of its mathematical images to real physical processes*).

After A. A. Andronov had clarified the significance, for the theory of oscillations, of the qualitative theory of differential equations and Lyapunov’s theory of stability, A. A. turned to applying them to the principal nonlinear problems of radiophysics**).

A. A. Andronov gave a theory of self-oscillations in a multivibrator, which have a sharply nonsinusoidal form. In this theory the self-oscillations of a multivibrator are considered as discontinuous (the currents change by a jump), and, in combination with the qualitative theory of differential equations, the “jump condition” stated by E. Friedländer, as well as by L. I. Mandelstam and N. D. Papaleksi (the postulate of continuity of energy), is applied to them.

On the other hand, A. A. Andronov showed that, for the quantitative calculation of self-oscillations close in their form to sinusoidal ones (oscillations in nonlinear nonconservative systems close to linear conservative ones), one can apply the method of expansion in a series in powers of a small parameter, developed

) A. A. Andronov was an opponent of the term “nonlinear mechanics,” by which the theory of self-oscillations and related phenomena is sometimes designated. He often emphasized that classical analytical mechanics of Lagrange–Hamilton is predominantly nonlinear mechanics. The solar system—one of the chief subjects of study of classical mechanics—is an essentially nonlinear system (the forces of gravitation depend nonlinearly on the distances). The distinguishing feature of the theory of self-oscillations, in comparison with classical analytical mechanics, is not nonlinearity, but something quite different: classical analytical mechanics deals predominantly with conservative systems (apart, for example, from the consideration of the solar system’s departure from primordial friction); the theory of self-oscillations deals with systems in which nonconservativity* plays a fundamental role.

**) A number of works in this cycle were carried out by A. A. Andronov jointly with A. A. Vitt.

Poincaré for the study of periodic solutions of three-body problems in celestial mechanics. With the help of this “small-parameter method” and Lyapunov’s theory of stability, A. A. Andronov gave a theory that fully clarified certain phenomena before which the linear “mode of thought” was powerless: capture (synchronization) of a tube generator by a periodic external force, as well as entrainment of the frequency and its jump-like changes under a smooth change in the tuning of a complex tube generator (with two oscillatory circuits).

Even earlier, applying the qualitative theory of differential equations to the approximate nonlinear equations obtained by Van der Pol in his theory of capture, A. A. Andronov clarified the question, which had remained controversial, of whether a “threshold of capture” exists. He showed that capture is possible for an arbitrarily small amplitude of the external force.

Let us add that, with the aid of the small-parameter method introduced into radiophysics by A. A. Andronov, L. I. Mandelstam and N. D. Papaleksi developed the theory of the phenomenon, discovered by them, of resonance of an unexcited tube generator under the action of an external emf of frequency that is a multiple of its own frequency (resonance of the \(n\)-th order).

Alongside the small-parameter method there exist other methods for the approximate quantitative solution of problems on oscillations, close to sinusoidal ones, in nonlinear systems. One of these methods is known as the quasilinear method. It operates with concepts borrowed from linear theory (and therefore familiar to the radio engineer), modifying them in accordance with the special features of nonlinear problems. In his last years A. A. Andronov was somewhat disturbed by the fact that, on the question of the relation between the small-parameter method and the quasilinear method, certain misunderstandings had, in his opinion, become rooted among physicists and engineers. A. A. Andronov believed that it was necessary to dispel these misunderstandings. It is appropriate here to set forth his point of view in a few words.

Using the small-parameter method, we represent the sought periodic motion in the form of a series in powers of a certain quantity \(\mu\) (the “small parameter”). The first term of this series (the “zero approximation”), to which one usually confines oneself, coincides with the approximate solution obtained by means of the quasilinear method. It is sometimes asserted that the small-parameter method has here the advantage that it is rigorous. This is incorrect. The small-parameter method is “rigorous” only in the sense that, for sufficiently small values of \(\mu\), the series with which it operates certainly converge. But nowhere has it been proved that these series converge for those values of \(\mu\) which characterize the real system whose oscillations we are calculating.

A. A. Andronov saw the chief significance of the small-parameter method in the fact that it naturally connects the problem of approximate-

calculation of periodic solutions with the qualitative theory of differential equations and with the problem of the birth of limit cycles (the meaning of this term will become clearer from what follows).

It is necessary to note, for a correct understanding of what was new that A. A. Andronov introduced into the theory of oscillatory phenomena, that (as he himself believed) the small-parameter method occupies in his works—even if one speaks only of their mathematical aspect—a completely secondary place. What is principal in them is the application to the study of nonlinear oscillations of the qualitative theory of differential equations and the topological methods connected with it. Let us take up the book in which some of A. A. Andronov’s basic ideas are set forth.*) Among the pages written by A. A. Andronov, the most characteristic are those in which a topological study of integral curves is carried out and a classification is given of the phase trajectories of rough systems.**) Let us note in passing that one of the characteristic features of A. A. Andronov’s scientific creativity was the striving to create, in every question he developed, a coherent logical system with an exhaustive classification of all possible cases by families, types, and subtypes. In The Theory of Oscillations there are entire pages of drawings showing various qualitative types of division into trajectories (Andronov’s term) of the phase plane of a rough system, various types of “coexistence” (also Andronov’s term) of singular points, limit cycles, and certain other characteristic curves (separatrices) describing the behavior of the system on the phase plane.

A. A. Andronov did not confine himself to applying to the physics of oscillations mathematical results that were already available. In connection with problems in the theory of oscillations he undertook the further development of the qualitative theory of differential equations. He introduced into it certain new ideas and obtained a number of essential mathematical results.

What is meant here above all is A. A. Andronov’s fruitful physical and mathematical idea of rough systems, developed by him with the participation of L. S. Pontryagin. A rough system is one whose qualitative character of motion does not change under a sufficiently small change of the parameters. Conservative systems are not rough: the oscillations of an ideal pendulum without friction are periodic (not damped); but there is no periodicity in the presence of even arbitrarily small friction. Every generator of undamped oscillations possesses characteristic properties that are not preserved under conservative idealization but are correctly reflected by the notion of a “rough system.” For example, a tube generator generates

*) A. A. Andronov and S. E. Khaikin, Theory of Oscillations, Moscow–Leningrad, 1937.
**) This term is explained below.

of oscillations (possesses periodic motions) in some region of values of any parameter: the steepness of the tube, the resistance of the circuit, the supply voltage. Any motor possesses an analogous property. The periodic motion of a motor changes its period (a quantitative change), but remains periodic (does not change qualitatively) under a not too large change of load. Hence it is clear that the concept of a rough system has for “terrestrial” mechanics—and for all technology—no less important a significance than the concept of a conservative system has for celestial mechanics. The last example is important for understanding one of the main lines of development of A. A. Andronov’s research—from the theory of oscillations in the proper sense to the general dynamics of machines.

Another fruitful idea with which A. A. Andronov enriched the qualitative theory of differential equations consists, roughly speaking, in the following: alongside the “systematics” of differential equations, another approach to them is possible—they can be investigated “historically” or, if you like, “embryologically.” One may be interested in the change, the development, of the qualitative picture of the integral curves of a differential equation when a parameter entering into the equation is varied (I do not know whether this idea was suggested by physics; physically, such an approach corresponds, say, to the study of changes in the operating regime of a machine under a smooth change in the position of the control member). Under a continuous change of the parameter, the change in the picture of the integral curves does not always occur continuously: for certain values of the parameter, qualitative changes take place; for example, limit cycles appear or disappear, singular points (equilibrium states) merge, and so on. One such case is the birth of a limit cycle for \(\mu > 0\), which was discussed in connection with the method of the small parameter.

The new approach often makes it possible to draw far-reaching conclusions about a given differential equation whose integral curves do not lend themselves to direct investigation: sometimes one can construct a simpler equation (A), from which the given equation (B) is obtained by changing a parameter; knowing the “laws of development” of the picture of integral curves and the picture of integral curves of equation (A), one can draw a number of conclusions about the picture of integral curves of equation (B). As a result of the new approach to differential equations developed by A. A. Andronov, he found (in collaboration with E. A. Leontovich) the laws of the appearance, disappearance, mutual transformation of singular points, limit cycles, and separatrices under variation of parameters. These laws have a direct physical meaning. They enabled A. A. Andronov to give (as one of the simplest examples) the picture of two types of evolution of the phase plane of a tube generator under variation of the parameter characterizing the feedback—the “soft” and “hard” onset of oscillations.

Here the word “picture” is not accidental. Andronov’s theory of generators begs to be put on film. A multiplied film on the birth of limit cycles could be a remarkable teaching aid.

The mathematical investigations of A. A. Andronov and his school, devoted to the qualitative theory of differential equations, developed continuously. Substantial results in this field were obtained by A. G. Maier. In his last years A. A. Andronov and his closest colleagues among the mathematicians worked a great deal on the creation of a monograph setting forth their work on qualitative theory. (This monograph remained unfinished.) In one of the last lectures delivered by A. A. Andronov at Gorky University, he spoke with enthusiasm about topology and expressed the conviction that in the future it would become an obligatory part of the mathematical equipment of physicists.

Alongside radio, in the prewar years A. A. Andronov became increasingly interested in other branches of technology. Subsequently this line of development led to fundamental achievements in the theory of automatic control. But before turning to them, let us briefly illuminate another line of development, close to the circle of interests of many physicists.

Physicists and radio engineers now know well what fluctuations are. They constantly encounter the fluctuation threshold of sensitivity of measuring and radio-receiving apparatus. Far less widespread is knowledge of how fluctuations manifest themselves in self-oscillatory systems and systems related to them. To this day one encounters in the literature the notion that, in contrast to optical radiation, which is in principle nonmonochromatic, the radio emission of a tube generator is, in principle, ideally monochromatic. As L. I. Mandelstam long ago explained, such a juxtaposition is incorrect: fluctuations inevitably lead to a blurring of the frequency of a self-oscillatory system.

At the suggestion and under the guidance of A. A. Andronov, I. L. Bershtein carried out a theoretical investigation of the action of fluctuations on a self-oscillatory system. This investigation made it possible, in particular, to give a quantitative estimate of the relative line width of a tube generator caused by fluctuations; this is a quantity of order \(10^{-13}\).

Still earlier, A. A. Andronov, jointly with L. S. Pontryagin, had carried out a theoretical investigation of the influence of fluctuations on nonconservative systems possessing several stable equilibrium states. In particular, the mathematical expectation of the time of a spontaneous transition of the system, in the presence of fluctuations, from one stable state to another was calculated.

The works in the field of fluctuations begun under the guidance of A. A. Andronov exerted a great influence on the development of experi-

mental investigations at the Physics and Technology Institute of Gorky University. The experimental verification of the theoretical estimate of the line width of a tube generator required—because of the smallness of this quantity—the creation of a new interference method (I. L. Bershtein), which opened up a number of unexpected possibilities in radiophysics and in optics.

Let us return, however, to what stood at the center of A. A. Andronov’s scientific interests.

He early appreciated the enormous role of automatic-control devices in modern technology and the importance of the development of automation for the national economy of the USSR. Even before the war he began to work on the theory of automatic control. These works underwent great development in the last years of the Great Patriotic War and in the postwar years.

The transition from self-oscillations in radiophysics to automatic control was quite natural for Andronov. A system with automatic control (for example, an airplane equipped with an autopilot) has a characteristic tendency toward self-oscillations (usually undesirable). The analogy between the self-oscillations of systems with automatic control and self-oscillations in radiophysics is obvious. But A. A. Andronov discerned between the theory of self-oscillations and the theory of systems with automatic control a far deeper kinship. Systems with automatic control are the most important (and ever more widespread) class of machines in the broadest understanding of this term, including, in particular, electronic devices that play an ever increasing role in modern technology. As has already been said, a self-oscillatory system and an engine possess common physical properties, reflected in the concept of a “rough system.” These properties are characteristic of machines in general (or at least of an extensive class of machines). Connected with them is the fact that not only self-oscillations, but also the periodic motion of an engine, is represented in phase space by a closed curve, to which neighboring phase trajectories approach asymptotically. In considering questions connected with automatic control, A. A. Andronov became ever more convinced that the theory of self-oscillations should be regarded as one of the parts of the general dynamics of machines.

The general dynamics of machines in Andronov’s understanding is, above all (translated into mathematical language), the study of their phase space, the determination of the regions of parameter values corresponding to various types of its subdivision into trajectories, and the classification of machines according to the character of this subdivision. At the same time, of course, the structure of the phase space, reflecting the dynamic behavior of machines, depends essentially on the automatic-control systems included in the machine.

A number of fundamental problems in the theory of automatic control are essentially nonlinear problems, and, moreover, considerably more difficult than the nonlinear problems that are of primary interest in radio engineering. This is connected with two circumstances. First, with the fact that, in contrast to the simplest vacuum-tube generators, even the simplest practically interesting systems of automatic control must be considered as systems having more than one degree of freedom; consequently, their theoretical investigation leads to the consideration not of the phase plane, but of a phase space of three, four, etc. dimensions. As the number of dimensions of the phase space increases, the difficulties grow to a significant degree. Second, with the fact that the theory of automatic-control devices more often than radio engineering deals with self-oscillations that are very far from sinusoidal. Such oscillations cannot be investigated by means of approximate methods that operate with sinusoidal oscillations as the zeroth approximation.

Undertaking new investigations, A. A. Andronov always carried out a powerful “mobilization of information.” In this he was helped by his rare bibliographical memory. A. A. Andronov very quickly became the greatest—probably not only in the USSR, but in the whole world—specialist in the literature and history of the theory of automatic control. Of great importance for the development of Andronov’s work on the theory of automatic control was his acquaintance and collaboration with I. N. Voznesensky, an outstanding figure in the technology of automatic control. An exhaustive knowledge of the literature helped A. A. Andronov to concentrate his efforts on questions having fundamental significance for the theory of automatic control.

Andronov’s principal work in this field (jointly with A. G. Maier) is a direct continuation and completion of Vyshnegradsky’s classical work (1876), “On Direct-Action Regulators.”

The question concerns a steam engine with a centrifugal governor. Despite the apparent simplicity of the device, the creation of its dynamic theory proved to be a very complicated problem. In order to convey the essential features of the behavior of a machine equipped with a centrifugal governor, it is necessary to investigate a system of three first-order differential equations, i.e., not a phase plane, as in the case of the simplest vacuum-tube generator, but a three-dimensional phase space. Because of dry friction in the governor coupling the equations are nonlinear. The problem of the stability of operation of a machine equipped with a governor was solved by Vyshnegradsky under the assumption that dry friction in the governor coupling is absent. In this approximation the equations of motion of the system become linear. The three-dimensional nonlinear problem that arises when dry friction in the coupling is taken into account remained unsolved because of its great

of mathematical difficulty, despite the fact that it had been studied by N. E. Zhukovskii, Stodola, Grdina, Mises, and many other investigators. This problem was solved by Andronov and Maier.

Success was ensured by the fact that they managed to develop a powerful mathematical method for solving a vast class of nonlinear problems in three-dimensional and four-dimensional phase space (problems with piecewise-linear characteristics): the method of transforming a surface into a surface (for three-dimensional phase space) and a space into a space (for four-dimensional phase space). This method is a generalization of the method of transforming a line into a line,* which Andronov applied in his early work that led him to the discovery of the connection between self-oscillations and limit cycles.

With the aid of the method of transforming a surface into a surface, Andronov and Maier were able, for a machine with a regulator possessing dry friction, to carry out an investigation of the behavior of integral curves throughout the entire three-dimensional phase space for all parameter values.

This first major success, achieved with the aid of the method of transforming a surface into a surface, was followed by a number of others. In particular, A. A. Andronov and his pupil N. N. Bautin solved the three-dimensional nonlinear problem of the motion of an airplane equipped with an autopilot. Later A. A. Andronov and his collaborators solved a further series of three-dimensional nonlinear problems in control theory. The solution of all the problems just mentioned was carried through to numerical calculations of the boundaries of stability regions for various values of the parameters—precisely what is of direct interest to engineers designing automatic-control devices.

The fundamental significance of that cycle of works by Andronov and his collaborators, of which we have just spoken, lies in the fact that in them, for the first time, without linear idealization, an exhaustive picture was given of the dynamic behavior of a number of systems with automatic regulation—for all possible positions of the control organs and all possible initial conditions.

A special place in the general dynamics of a machine should, in the opinion of A. A. Andronov, be occupied by the theory of clocks. A. A. Andronov always showed the liveliest interest in clocks. When he was beginning his research on self-oscillations, a model of a clock stood on his desk. Subsequently Andronov devoted much time to studying the history of clocks—from pendulum-free clocks to modern electromagnetic clocks. In doing so it became clear that “although clocks have served as the subject of numerous theoretical investigations, one cannot regard as existing a theory of clocks and of such devices equivalent to them

* Theory of Oscillations, p. 169.

in a dynamical sense, such devices as anchor-type escapement regulators is satisfactory. Only a model of a clock having one degree of freedom has been fully studied. However, such a model, leaving out of consideration the process of interaction between the balance and the escape wheel, cannot serve for the investigation of a number of fundamental questions in the theory of clocks. As for those theoretical works in which two degrees of freedom were taken into account, they [...] were confined to an approximate analysis of separate mechanical questions posed by designers, and did not consider the clock as a closed dynamical system with two degrees of freedom.”

A. A. Andronov began the investigation of clocks “as a closed dynamical system” with the model of the simplest Galilean clocks*). Soon, at A. A. Andronov’s suggestion, N. N. Bautin took up the study of two-degree-of-freedom models of Galileo–Huygens clocks (clocks with a pendulum or balance subjected to the action of a restoring force). He gave, in particular, the solution of a problem posed in 1944 by L. I. Mandelstam during discussion with A. A. Andronov of the work on the theory of clocks begun in Gorky: it was clarified precisely which dynamical features of Galileo–Huygens clocks ensure the stability of the period of self-oscillations. In recent years N. N. Bautin has succeeded in obtaining new results which, as A. A. Andronov believed, are of very substantial importance for the theory of clocks. In them, for the first time, the general dynamics of machines in Andronov’s sense comes into contact with the engineering problems of clock technology.

Naturally, given his striving for an all-round coverage of every subject under investigation, A. A. Andronov was also interested, in connection with the general dynamics of machines, in the theory of electrical machines. Knowledge of their phase space is necessary for a complete understanding of their behavior in automatically regulated systems. He studied a very large number of works, in particular the investigations of Maxwell and Poincaré devoted to elementary models of collector machines, and became an expert not only in the history but also in the prehistory of electrical machines. Andronov undertook a careful analysis of the initial premises of existing theories of electrical machines. Here he discovered great deficiencies. There do not exist sufficiently general and correct methods for composing the equations of motion of unipolar and collector machines. The composition of the equations of motion by the Lagrange–Maxwell method here runs into difficulties. For unipolar machines they are connected with the presence of a contact sliding over the surface of a moving bulky conductor

*) The characterization given above of the state of the theory of clocks is taken from the article by A. A. Andronov and Yu. I. Neimark devoted to this model, DAN 51, 17 (1946).

(questions of the electrodynamics of moving bodies arise), and, for collector machines, because, owing to the switching of circuits, the number of degrees of freedom is variable. The question of the equations of motion of unipolar and collector machines was studied, under the direction of A. A. Andronov, by his graduate student A. V. Gaponov. It proved possible to show that, under certain reasonable simplifications, collector and unipolar machines belong, from the general dynamical point of view, to the class of nonholonomic systems of the type of S. A. Chaplygin. Subsequently, general equations of motion were obtained that are valid for any combination of collector, brushless, and unipolar machines.

A. A. Andronov was concerned with self-oscillations not only in connection with technology. He had long been interested in astrophysics and, while still a graduate student, put forward the supposition that Cepheids (stars with periodically varying brightness) are self-oscillatory systems. In 1941 he suggested to one of his graduate students, S. A. Zhevakin, that he undertake a theoretical investigation of the mechanism of Cepheid oscillations. This work was interrupted by the war and resumed in 1946. Recently S. A. Zhevakin succeeded in constructing a theory explaining the self-excitation of oscillations in Cepheids and a number of characteristic features of these oscillations. A. A. Andronov’s interest in astrophysical questions became still more active in connection with the development of radio astronomy. In the last months of his life he was especially strongly preoccupied with the mystery of the “radio stars.”

In the very recent period, in work carried out under A. A. Andronov’s direction, one more line of research began to take shape—the study of certain types of radio circuits used in electronic automation. One of the works in this direction (N. A. Zheleztsov and L. V. Rodygin) is a very substantial step forward in the theory of the multivibrator and related devices. Under an idealization that discards “parasitic capacitances,” the self-oscillations of a multivibrator are discontinuous; “jumps” of currents occur. Instead of postulating, as had been done earlier, definite jump conditions (continuity of energy), N. A. Zheleztsov and L. V. Rodygin derived the jump conditions from the differential equations written with the parasitic capacitances taken into account. This made it possible to indicate what happens in cases when continuity of energy does not uniquely determine the course of the “jump.” In addition, this compelled a correction of the former theory at certain essential points.

In developing the theory of nonlinear oscillations, A. A. Andronov was far from underestimating the significance of the linear theory of oscillations, both for radiophysics and radio engineering and for the theory of automatic control. A number of investigations on the linear theory of oscillations and related questions were carried out by A. A. Andronov’s school. Thus, for example, in 1934, in the work of A. G. Mayer and E. A. Leontovich, the least possible value of the product of the diffuseness of a signal (in time) and its nonmonochromaticity was established.

Thus an exact formulation was given of the classical analogue of the uncertainty relation, which has fundamental significance for all communications technology. When A. A. Andronov’s studies in the theory of automatic regulation developed widely, he set before his graduate student Yu. I. Neimark the task of analyzing, from a mathematical point of view, the stability criterion for linearized systems proposed by Nyquist without rigorous justification. This criterion is widely used in the calculation of amplifiers and automatic-control systems. Under the influence of the “embryological” approach to equations characteristic of Andronov’s school, Yu. I. Neimark approached the question of stability in a new way. He made the coefficients of the characteristic equation vary and began to follow how, in doing so, its roots move in the complex plane. As a result of this approach a new, practically important stability criterion was obtained, which has already entered a number of textbooks.

With this we shall conclude our far from complete survey of the scientific investigations of A. A. Andronov and his pupils. It is generally recognized that in the field of the theory of nonlinear oscillations our country long ago took first place in the world. In this connection it should be noted that the first Soviet works on the theory of nonlinear oscillations were the works of A. A. Andronov in 1928–1930.

The chief place in the life of A. A. Andronov, alongside scientific research (and at times, perhaps, even without this reservation), was occupied by a deeply patriotic concern for the growth of Soviet science.

Already in his young years A. A. Andronov began to regard the creation of genuine centers of science in the provinces as a most important task of the state. On his own initiative he moved in 1931 from Moscow to Gorky in order to work at the Gorky Physico-Technical Institute then being organized, and later also at Gorky University, which opened on November 1, 1931, and of which he remained a professor to the end of his life. It is hard to convey how much enthusiasm, how much spiritual strength A. A. Andronov gave to Gorky University, to its Physico-Technical Institute, and to the university library. He created at Gorky University a course in the theory of oscillations, lectured on courses in electrodynamics and the theory of relativity, and organized the teaching of theoretical physics. The vivid, deeply thought-out lectures of A. A. Andronov invariably aroused enormous interest among the students. A. A. Andronov trained a large number of young scholars at Gorky State University. He tirelessly concerned himself with attracting new scientific forces, often entering into all the details of the everyday arrangements of the people invited. He tirelessly fought for improving the quality of teaching, for high standards in the defense of dissertations and in nominations for academic ranks. Nothing in university life left him indifferent. He watched—even then, when he was already seriously ill—over

with the growth of young physicists, keenly shared in their successes and failures in scientific work and teaching.

A. A. Andronov also took a large part in the scientific work of the Institute of Automatics and Telemechanics of the Academy of Sciences of the USSR. Here, in particular, work was carried out on the analysis of the classical heritage in the theory of automatic control. At the Institute of Automatics A. A. Andronov trained a group of scholars who are now successfully conducting independent research (M. A. Aizerman, M. V. Meerov, V. V. Petrov, and others).

In his last years A. A. Andronov devoted a considerable part of his time to historical investigations.

A. A. Andronov (jointly with I. N. Voznesenskii) was responsible for the highly substantive study “On the Works of J. C. Maxwell, I. A. Vyshnegradskii and A. Stodola in the Field of the Theory of Machine Regulation.”*) On the basis of a careful analysis the authors came to the following conclusion: “Only by comparing the works of Maxwell and Vyshnegradskii can one truly understand what was done by Vyshnegradskii, and that it is precisely with Vyshnegradskii that the engineering theory of machine regulation begins.”

At the initiative and under the direction of A. A. Andronov, a major effort was undertaken in the Gorky Regional Archive to determine where Lobachevskii was born: the information available earlier had been contradictory. Andronov was interested in Lobachevskii not only as a brilliant mathematician, but also as a university figure, organizer, and rector of Kazan University. The examination of a large number of documents proved beyond doubt that Lobachevskii was born in Nizhnii Novgorod, now the city of Gorky. It was established where the house in which Lobachevskii was born had stood, and much interesting information about his family was obtained.

The scientific works and public activity of A. A. Andronov were highly appreciated by the Communist Party, the Soviet Government, and our scientific community.

For work carried out during the years of the Great Patriotic War, he was awarded the Order of the Red Star in 1944. In 1946 he was elected a full member of the Academy of Sciences of the USSR in the Division of Technical Sciences. In 1947 A. A. Andronov was elected a deputy of the Supreme Soviet of the RSFSR from the Sverdlovsk electoral district of the city of Gorky—the district where Gorky University is located. He was also elected a member of the Presidium of the Supreme Soviet of the RSFSR. In 1950 A. A. Andronov was elected a deputy of the Supreme Soviet of the USSR. A. A. Andronov understood the full responsibility,

) It is included in the collected volume published under the editorship of A. A. Andronov and I. N. Voznesenskii, “Classics of Science” series: J. C. Maxwell, I. A. Vyshnegradskii, A. Stodola, Theory of Automatic Control*, Publishing House of the Academy of Sciences of the USSR, 1949.

which the voters’ trust imposed upon him. He devoted much time to his duties as a deputy, leaving without attention not a single letter, not a single appeal.

The impression of rare unity and consistency of development created by those works of A. A. Andronov and his school that we have been able to touch upon here is preserved also upon a fuller acquaintance with his scientific legacy. One of A. A. Andronov’s characteristic traits as a researcher was his purposefulness. Another characteristic trait of his was a passionate need for the fullest, absolute logical clarity. Connected with it was his striving, in working out any scientific question, for an exhaustive knowledge of its history and of all its connections with other questions, for a systematic classification of all possible cases, and for the application of the most general mathematical methods possible. Andronov’s need for logical clarity was deeply in harmony with his adherence to principle, as well as with his extraordinary exactingness regarding the quality of the presentation of scientific results. A. A. Andronov could not read without irritation works in which it was unclear what was being postulated, what was being proved, and under what assumptions. “Let us first clarify the logical structure”—this was his typical introduction to the discussion of a scientific report or manuscript. It must be added that Andronov’s logic was not cold and abstract. The attainment of logical clarity was somehow very easily, of itself, combined in him with the emergence of fruitful concepts, vivid images, and expressive terms.

A. A. Andronov was an integral and life-loving man, who knew very much and was eagerly interested in everything. For all his purposefulness in scientific research he was the complete opposite of what is meant when people say “narrow specialist.” He possessed a broad mind and a rich, many-sided culture. The circle of his immediate scientific interests included all of physics, mathematics, engineering, and astronomy. He was keenly interested in all of natural science, medicine, history, literature, and painting. He was a connoisseur of the history of Russian culture. A. A. Andronov’s speech was powerful, witty, and irresistible. His directness often reached the point of sharpness. At the same time he was simple in his dealings with people, responsive, and pure-hearted. There was in him no egoism or insecure petty vanity.

The last years of A. A. Andronov’s life were darkened by a tormenting illness (a severe form of hypertension). On October 31, 1952, he passed away.

The duty of A. A. Andronov’s colleagues and numerous students is to continue his investigations in the theory of oscillations, the dynamics of machines, and the qualitative theory of differential equations, to develop further his remarkable scientific ideas, and to make them the property of a broad circle of Soviet physicists, mathematicians, and engineers.

G. S. GORELIK

LIST OF WORKS BY A. A. ANDRONOV *)

1926

1. Zur Theorie der molekularen Lichtzerstreuung an Flüssigkeitsoberflächen (jointly with M. A. Leontovich), Zeits. f. Phys., 38, 485 (1926).

1927

1. On oscillations of systems with periodically varying parameters (jointly with M. A. Leontovich), Journal of the Russian Physico-Chemical Society, Physical Part, 59, 429 (1927).

1928

1. Poincaré limit cycles and the theory of oscillations. In the book VI Congress of Russian Physicists. Moscow, Nizhny Novgorod, Kazan, Saratov (August 5–16, 1928). List of papers presented at the congress with brief summaries. M. — L., Gos. izd-vo, 1928, pp. 23–24.

2. On the theory of adiabatic invariants (jointly with L. I. Mandelstam and M. A. Leontovich), Journal of the Russian Physico-Chemical Society, Physical Part, 60, 413 (1928).

1929

1. Les cycles limites de Poincaré et la théorie des oscillations auto-entretenues, Comptes Rendus, 189, 559 (1929).

1930

1. Sur la théorie mathématique des auto-oscillations, Comptes Rendus, 190, 256 (1930).

2. Sur les mouvements quasi-périodiques, Journal of Applied Physics, 7, 119 (1930).

3. Zur Theorie des Mitnehmens von Van der Pol, Archiv für Elektrotechnik Berlin, 24, 99 (1930).

4. Unstetige periodische Bewegungen und die Theorie des Multivibrators von Abraham und Bloch, DAN SSSR, issue 8, 189 (1930).

5. On the mathematical theory of entrainment, Journal of Applied Physics, 7, 1 (1930).

1933

1. On Lyapunov stability, Journal of Experimental and Theoretical Physics, 3, 373 (1933).

2. Mathematical problems of the theory of self-oscillations. In the book First All-Union Conference on Oscillations, Collection 1. M. — L., GTTI, 1933. Reports, resolutions, and materials of the conference.

3. On the statistical consideration of dynamical systems (jointly with L. Pontryagin), Journal of Experimental and Theoretical Physics, 3, 165 (1933).

4. Zur Stabilität nach Liapunow, Phys. Zeits. d. Sowjetunion, 4, 606 (1933).

1934

1. On the mathematical theory of self-oscillatory systems with two degrees of freedom, Journal of Technical Physics, 4, 122 (1934).

*) The list of A. A. Andronov’s works does not include: articles in abstract journals, popular articles, biographies, translations, etc.

1935

1. Application of Poincaré’s theory of “bifurcation points” and “change of stability” to the simplest self-oscillatory systems, Journal of Experimental and Theoretical Physics, 5, 296, 1935 (with A. G. Lyubina).

2. Exposé des recherches récentes sur les oscillations non-linéaires (with L. I. Mandelstam, N. D. Papaleksi and S. E. Khaikin), Techn. Phys. of the USSR, 2, 81 (1935).

1936

1. New research in the field of nonlinear oscillations. Moscow, State Publishing House for Questions of Radio, 1936, 96 pp. (with L. I. Mandelstam, N. D. Papaleksi, S. E. Khaikin and G. S. Gorelik).

1937

1. Rough systems, DAN SSSR, 14, 247 (1937) (with L. Pontryagin).

2. Theory of oscillations, Part I. Moscow–Leningrad, ONTI, 1937, XII, 518 pp. (with S. E. Khaikin).

1938

1. On the theory of changes in the qualitative structure of the partition of the plane into trajectories (with E. Leontovich), DAN SSSR, 21, 427 (1938).

1939

1. Certain cases of dependence of limit cycles on a parameter, Scientific Notes of Gorky State University, issue 6, 3 (1939) (with E. A. Leontovich).

1944

1. Mises’ problem in the theory of direct control and the theory of point transformations of surfaces, DAN SSSR, 43, 58 (1944) (with A. G. Maier).

2. Motion of a neutral airplane equipped with an automatic pilot, and the theory of point transformations of surfaces (with N. N. Bautin), DAN SSSR, 43, 197 (1944).

1945

1. L. I. Mandelstam and the theory of nonlinear oscillations. Izv. AN SSSR, physical series, 9, 30 (1945).

2. On resonance phenomena in the motion of a relativistic particle in a cyclotron (with G. S. Gorelik), DAN SSSR, 11, 664 (1945).

3. Stabilization of the course of a neutral airplane by an automatic pilot with constant servomotor speed and a zone of insensitivity (with N. N. Bautin), DAN SSSR, 46, 158 (1945).

4. On one degenerate case of the general problem of direct control (with N. N. Bautin), DAN SSSR, 46, 304 (1945).

5. Self-oscillations of the simplest circuit containing an automatic variable-pitch propeller (with N. N. Bautin and G. S. Gorelik), DAN SSSR, 47, 265 (1945).

6. On the Vyshnegradsky problem in the theory of direct control (with A. G. Maier), DAN SSSR, 47, 345 (1945).

1946

1. Simplest linear systems with delay (jointly with A. G. Mayer), Avtomatika i telemekhanika, 7, 95 (1946).

2. On the motions of an ideal model of a clock having two degrees of freedom. I. A model of pre-Galilean clocks (jointly with Yu. I. Neimark), Dokl. Akad. Nauk SSSR, 51, 17 (1946).

3. The theory of indirect control with allowance for Coulomb friction in a sensitive element (jointly with N. N. Bautin and G. S. Gorelik), Avtomatika i telemekhanika, 7, 15 (1946).

1947

1. The Vyshnegradsky problem in the theory of direct control. Communication 1. Theory of a direct-action regulator in the presence of Coulomb and viscous friction (jointly with A. G. Mayer), Avtomatika i telemekhanika, 8, 314 (1947).

2. Some investigations in the field of the theory of nonlinear oscillations carried out in the USSR beginning in 1935 (jointly with N. D. Papaleksi, G. S. Gorelik, and S. M. Rytov), UFN, 33, 335 (1947).

1949

1. I. A. Vyshnegradsky and his role in the creation of the theory of automatic control. Izv. Akad. Nauk SSSR, OTN, No. 6, p. 805 (1949). The same, in the book Questions of the History of Russian Science, General Meeting of the Academy of Sciences of the USSR devoted to the history of Russian science, January 5–11, 1949, Moscow–Leningrad, Academy of Sciences of the USSR Press, pp. 500–517 (1949).

2. On the works of D. K. Maxwell, I. A. Vyshnegradsky, and A. Stodola in the field of the theory of control of machines (jointly with I. N. Voznesensky). In the book D. K. Maxwell, I. A. Vyshnegradsky, and A. Stodola. Theory of Automatic Control (Linearized Problems). Moscow, Academy of Sciences of the USSR Press, pp. 253–301 (1949).