PEOPLE OF SOVIET SCIENCE

YURII ALEKSEEVICH MITROPOLSKII

(On the occasion of his 50th birthday)

Yurii Alekseevich Mitropolskii was born on January 3, 1917, in the village of Shishaki, Poltava province. After completing a seven-year school, from 1932 to 1936 Yu. A. Mitropolskii worked at a factory; in 1938 he completed a ten-year school and entered the Faculty of Mechanics and Mathematics of Kiev State University. His studies at the university were interrupted by the war. In the spring of 1942 Yu. A. Mitropolskii graduated from the Faculty of Physics and Mathematics of Kirov Kazakh State University and was sent to an artillery school.

During the war years Yu. A. Mitropolskii fought in the ranks of the Soviet Army, serving as commander of a platoon of artillery reconnaissance. For his combat merits Yu. A. Mitropolskii was twice awarded the Order of the Red Star.

After the end of the war Yu. A. Mitropolskii worked at the Institute of Mechanics of the Academy of Sciences of the Ukrainian SSR as a junior research associate. There began his fruitful scientific activity under the guidance of the outstanding scientist Academician N. N. Bogolyubov.

In 1948 Yu. A. Mitropolskii successfully defended his Candidate’s dissertation, devoted to the study of resonance phenomena in nonlinear oscillatory systems; in 1951 he defended his Doctoral dissertation on topical problems of nonlinear mechanics and mathematical physics—the investigation of nonstationary processes in nonlinear oscillatory systems.

Since 1951 Yu. A. Mitropolskii has worked at the Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR, and since 1953 as head of the Department of Mathematical Physics.

In 1954 Yurii Alekseevich was awarded the title of professor.

In 1958 Yu. A. Mitropolskii was elected a corresponding member of the Academy of Sciences of the Ukrainian SSR, and in 1961 an academician of the Academy of Sciences of the Ukrainian SSR.

In 1958 Yu. A. Mitropolskii was elected director of the Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR, which he continues to head at the present time.

Yu. A. Mitropolskii has repeatedly been elected to the bureau of the Division of Physical and Mathematical Sciences of the Academy of Sciences of the Ukrainian SSR; from 1961 to 1963 he headed this bureau, and since 1963 he has been Academician-Secretary of the Division of Mathematics, Mechanics, and Cybernetics, and a member of the Presidium of the Academy of Sciences of the Ukrainian SSR.

Yu. A. Mitropolskii has published more than 100 scientific works, including

...including 6 monographs, two of which (one coauthored with N. N. Bogolyubov) have been translated into English, French, German, Chinese, and Japanese.

For outstanding achievements in the theory of nonlinear oscillations and nonlinear differential equations, Yurii Alekseevich was awarded the Lenin Prize for 1965.

The scientific results of Yurii Alekseevich Mitropol’skii belong to the following principal areas: the construction of asymptotic solutions and the detailed study of nonlinear differential equations with slowly varying coefficients; the development of a method for investigating single-frequency oscillatory processes in oscillatory systems with many degrees of freedom; the study of systems of nonlinear differential equations describing oscillatory processes in gyroscopic and strongly nonlinear systems; the development of the theory of integral manifolds as applied to problems of nonlinear mechanics and the consideration of the questions of stability of motion arising here; the development of the averaging method as applied to equations with slowly varying coefficients and to equations with nondifferentiable and discontinuous right-hand sides, and others.

With regard to the first area, it should be emphasized that problems of this type are of fundamental importance in a number of branches of modern technology and physics, and also are of great independent interest for the qualitative theory of differential equations. We mention, for example, the study of nonlinear oscillations in a nonstationary regime, in particular when passing through resonance, and the study of oscillations in systems with variable mass and stiffness. Questions of modulation of high-frequency oscillations by oscillations of lower frequencies, etc., also belong here.

From the mathematical point of view, problems of this kind are characterized by systems of differential equations close to linear ones, whose parameters (for example, effective natural and external frequencies) contain an independent variable (for example, time) in combination with a small parameter.

Despite the great fundamental significance of the indicated problems, before the investigations of Yu. A. Mitropol’skii there existed no sufficiently developed general methodology here, and only separate particular problems, mainly of a purely linear type, were considered. The methods of nonlinear mechanics applied to the investigation of systems close to linear ones had been developed exclusively for those cases in which the parameters of the linear system are constant.

Having overcome the great fundamental difficulties encountered here, Yu. A. Mitropol’skii succeeded, in a successive cycle of his works, in creating general effective methods for solving these problems and thus made an important contribution to the development of asymptotic methods of nonlinear mechanics.

In a number of works, convenient computational schemes for practical application were developed on the basis of the method created by Yu. A. Mitropol’skii.

Thus, in the work “Application of Symbolic Methods to the Investigation of Nonlinear Systems with Slowly Varying Parameters” (1949), an effective method was proposed for calculating nonlinear oscillatory systems containing one element with a nonlinear characteristic. The computational scheme makes it possible to investigate both stationary regimes and the processes by which oscillations are established.

Among the works in this area, which also have great physical significance, special mention should be made of those devoted to questions...

the passage of nonlinear oscillatory systems through resonance, and the passage through various demultiplicative resonances.

In the work “Forced Oscillations in Nonlinear Systems during Passage through Resonance” (1953), Yu. A. Mitropol’skii investigated in detail, both qualitatively and quantitatively, various complex phenomena that occur in nonlinear systems during passage through resonance.

The results obtained by Yu. A. Mitropol’skii in the first area of his work found wide application in a number of branches of modern physics and technology and played a major role in calculations and design in the most diverse fields of modern physics and engineering. Thus, for example, Yu. A. Mitropol’skii’s very first work, “Natural Oscillations in a Nonlinear System with Slowly Varying Parameters” (1948), immediately found application in the calculation of oscillations of a dredge bucket. Subsequently, the method developed by Yu. A. Mitropol’skii was applied in the calculation of modern accelerators, in the calculation of resonance phenomena in electrical systems, in the calculation of oscillations during passage through resonance in rotors of turbomachines, centrifuges, and other mechanical systems.

In the second area, Yu. A. Mitropol’skii developed a general method for studying single-frequency oscillatory processes in nonlinear systems with many and an infinite number of degrees of freedom, described by systems of differential equations containing a small parameter.

Here special mention should be made of Yu. A. Mitropol’skii’s work “Investigation of Oscillations in Nonlinear Systems with Many Degrees of Freedom and with Slowly Varying Parameters” (1949), in which the foundations were laid for constructing approximate solutions of the indicated systems of nonlinear differential equations in the case of a single-frequency regime.

A substantial contribution to the development of the methods he created is the work “Slow Processes in Nonlinear Oscillatory Systems with Many Degrees of Freedom” (1950), in which a rigorous mathematical justification of the method developed by Yu. A. Mitropol’skii was given. Of special note here are the error estimates and, in particular, a profound theorem establishing the stability properties of a family of solutions corresponding to the single-frequency processes under consideration.

For the study of single-frequency processes in systems with distributed parameters (with an infinite number of degrees of freedom), described by partial differential equations, Yu. A. Mitropol’skii developed a formalism (the energy method) that makes it possible to compose equations of the first and higher approximations, starting from expressions for the potential and kinetic energy.

The results of this area of Yu. A. Mitropol’skii’s work found extensive application in the calculation of oscillations in rotors of turbomachines (see V. A. Grobov’s monograph on this question), in the calculation of passage through resonance in the blades of gas turbines (see the works of G. S. Pisarenko), and also in control theory (see the monograph by E. P. Popov and I. P. Pal’tov).

The works of the third area include, first of all, the work “On Oscillations in Gyroscopic Systems during Passage through Resonance” (1953), where Yu. A. Mitropol’skii solves the problem of constructing asymptotic solutions in the presence, in the system of differential equations under study, of gyroscopic terms and multiple natural frequencies; the article “On Nonstationary Oscillations in Systems with

many degrees of freedom” (1954), in which B. V. Bulgakov’s method for reducing a system with gyroscopic terms to normal coordinates is generalized, as well as the extensive article “On Certain Equations Close to Exactly Integrable Ones” (1959), published in the collection Automatic Control and Computer Engineering, in which an original method is developed for constructing equations of the first approximation directly from the expression of the function appearing on the right-hand side of the equation.

The results of this direction are widely used in the calculation of complex phenomena in gyroscopic systems, as well as in theoretical investigations.

The fourth direction of Yu. A. Mitropol’skii’s work belongs to the field of integral manifolds.

In the works of Yu. A. Mitropol’skii, the method of integral manifolds, whose fundamental theorems had been formulated and proved by N. N. Bogoliubov as early as 1945, underwent substantial development.

Thus, in the work “On Certain Differential Equations Occurring in the Theory of Relaxation Oscillations” (1957), Yu. A. Mitropol’skii, applying this method to the study of one relaxation system, reduced the original system of equations on a manifold to a single equation and, as a result of a detailed analysis of the solutions of this equation, derived criteria for the existence of zones of parametric resonance for the relaxation system under consideration, discovered the phenomenon of quasi-synchronization, refined in the second approximation the frequency of asynchronous oscillations, and so on.

In the work “On the Stability of a One-Parameter Family of Solutions of a System of Equations with Variable Coefficients” (1958), with the aid of the method of integral manifolds developed by Yu. A. Mitropol’skii as applied to the study of equations with variable coefficients, the one-frequency method was substantiated for nonlinear oscillatory systems described by nonlinear differential equations with slowly varying parameters.

In recent years he has also obtained a number of theorems on the existence and properties of integral manifolds for nonlinear differential equations with variable coefficients, both in the finite-dimensional case and in the infinite-dimensional case. In the latter case a theorem was obtained that justifies the application of the one-frequency method to the study of systems with distributed parameters.

The methods developed by Yu. A. Mitropol’skii for proving theorems on the existence of integral manifolds, as well as effective methods for their calculation, constitute a major contribution to the theory of ordinary differential equations.

Significant results were obtained by Yu. A. Mitropol’skii in the study of the behavior of trajectories on toroidal manifolds. In the works “On the Question of the Structure of Trajectories on Toroidal Manifolds” (1964), “On the Construction of Solutions of Linear Differential Equations with Quasiperiodic Coefficients by Means of the Accelerated-Convergence Method” (1965) (jointly with A. M. Samoilenko), Yu. A. Mitropol’skii investigated the structure of differentiable trajectories on a torus of arbitrary dimension, and also considered the question of the reducibility of equations with quasiperiodic coefficients.

An important class of equations in problems of nonlinear mechanics is formed by equations of the form

\[ \frac{dh}{dt}=Hh+\varepsilon F(h,\varphi,\varepsilon), \qquad \frac{d\varphi}{dt}=\nu+\varepsilon \Phi(h,\varphi,\varepsilon) \tag{a} \]

\((h = h_1, \ldots, h_{n_0};\quad \varphi = \varphi_1, \ldots, \varphi_{n_1};\quad n_0 = n_1 = n)\). For systems of equations of this form in the case when the matrix \(H\) degenerates into a vector \(\boldsymbol{\beta} = (\beta_1, \ldots, \beta_n)\) (with all \(\beta_i\) having negative real parts), Yu. A. Mitropol’skii, in the paper “On the construction of the general solution of nonlinear differential equations by means of a method ensuring ‘accelerated convergence’” (1964), relying on the works of N. N. Bogoliubov, as well as of A. N. Kolmogorov and V. I. Arnold, developed a method for constructing the general solution of the problem under consideration, based on the introduction of successive changes of variables of the type

\[ h_{i-1}=h_i+u_i(h_i,\varphi_i),\quad \varphi_{i-1}=\varphi_i+v_i(h_i,\varphi_i)\quad (i=1,2,3,\ldots), \]

which give accelerated convergence, and proved a theorem on the tendency of any solution of system \((\alpha)\) to a quasiperiodic solution.

A number of important results were obtained by Yu. A. Mitropol’skii in the area of the further development of the averaging method, and also of its application to systems of nonlinear differential equations with variable coefficients, with nondifferentiable and discontinuous right-hand sides, etc. These questions are the subject of the monograph published in 1966, Lectures on the Averaging Method in Nonlinear Mechanics.

Yu. A. Mitropol’skii is coauthor of the fundamental monograph on asymptotic methods (together with N. N. Bogoliubov), Asymptotic Methods in the Theory of Nonlinear Oscillations, which appeared in three editions in the Soviet Union (1955; 1958; 1963) and in six editions abroad.

The principal achievements of Yurii Alekseevich Mitropol’skii relating to the investigation of nonstationary oscillatory processes in nonlinear systems both with one degree of freedom and with many degrees of freedom, as well as with an infinite number of them, are set forth in the monograph Problems of the Asymptotic Theory of Nonstationary Oscillations, published in 1964; it also includes the contents of Yu. A. Mitropol’skii’s earlier monograph Nonstationary Processes in Nonlinear Oscillatory Systems (1955), published in the USA, Japan, and China.

These monographs are at present handbooks for a wide circle of specialists engaged in the investigation of nonstationary processes described by nonlinear differential equations with variable coefficients.

It should be noted that a characteristic feature of the works of Yurii Alekseevich Mitropol’skii is the comprehensive study of a problem—the finding of a convenient algorithm for constructing approximate solutions, and then a deep theoretical justification that makes it possible to obtain estimates of the \(m\)-th approximations in general cases, as well as a profound qualitative investigation.

Alongside the high theoretical level, a distinctive feature of the works of Yu. A. Mitropol’skii is also constant attention to the physical essence of the phenomena under investigation.

Yurii Alekseevich Mitropol’skii brilliantly combines his scientific activity with scientific-organizational, pedagogical, and public activity.

For many years Yurii Alekseevich has lectured at Kiev State University.

Yurii Alekseevich has trained a large number of scientific workers—under his supervision more than 20 Candidate’s dissertations and 3 doctoral dissertations have been defended.

Much attention is devoted by Yurii Alekseevich to the mathematical education of young people. On his initiative and with his participation, beginning in 1963 summer mathematical schools were organized for the first time in the Soviet Union, where young scientists from Kiev and other cities of the Soviet Union were given lectures by leading specialists in various fields of mathematics.

Yurii Alekseevich Mitropol’skii is an outstanding scientist, an excellent organizer of science, and a talented educator of young people. His name is widely known and enjoys well-deserved authority among the scientific community both in the Soviet Union and abroad.

In January 1967 it was 50 years since the birth of Yurii Alekseevich; we wish him good health and new creative successes in his many-sided scientific activity.

N. P. ERUGIN, Yu. D. SOKOLOV, S. F. FESHCHENKO, O. B. LYKOVA

LIST OF THE PRINCIPAL WORKS OF Yu. A. MITROPOL’SKII

A. Articles

1. Natural oscillations of a nonlinear system with slowly varying parameters. Collection of Works of the ISM, No. 11, 107–114, 1948.

2. Investigation of natural oscillations of a nonlinear system close to exactly integrable, in the presence of slowly varying parameters. Collection of Works of the ISM, No. 11, 99–106, 1948.

3. Investigation of oscillations in nonlinear systems with many degrees of freedom and slowly varying parameters. Ukr. Mat. Zhurnal, vol. 1, No. 2, 85–98, 1949.

4. Application of symbolic methods to the investigation of nonlinear systems with slowly varying parameters. Collection of Works of the ISM, No. 13, 99–111, 1949.

5. On stationary oscillations in nonlinear systems with many degrees of freedom. Collection of Works of the ISM, No. 12, 228–233, 1950.

6. Investigation of oscillations of a nonlinear system with slowly varying parameters. Collection of Works of the ISM, No. 14, 134–144, 1950.

7. Slow processes in nonlinear oscillatory systems with many degrees of freedom. Applied Mathematics and Mechanics, vol. XIV, No. 2, 139–170, 1950.

8. On passage through resonance in a nonlinear oscillatory system with many degrees of freedom. Collection of Works of the ISM, No. 17, 47–56, 1952.

9. Forced oscillations in nonlinear systems during passage through resonance. Engineering Collection, vol. XV, 89–98, 1953.

10. On oscillations in gyroscopic systems during passage through resonance. Ukr. Mat. Zhurnal, vol. V, No. 3, 333–349, 1953.

11. On the influence of elastic elements with a nonlinear characteristic on small oscillations in certain gyroscopic systems. Scientific Notes of the Kiev State University named after T. G. Shevchenko. Mathematical Collection, No. 5, 107–114, 1954.

12. On nonstationary oscillations in systems with many degrees of freedom. Ukr. Mat. Zhurnal, vol. VI, No. 2, 176–189, 1954.

13. On the action of a sinusoidal force with modulated frequency on a nonlinear vibrator. Ukr. Mat. Zhurnal, vol. VI, No. 4, 442–447, 1954.

14. On the question of passage through resonance of the second kind. Ukr. Mat. Zhurnal, vol. VII, No. 1, 121–123, 1955.

15. The asymptotic methods of N. M. Krylov and N. N. Bogolyubov and their further development. Revue de mathématiques Pures et Appliquées (Académie de la RPR), vol. I, No. 3, 15–26, 1956.

16. Asymptotic methods in the theory of nonlinear oscillations. Proceedings of the Third All-Union Mathematical Congress, vol. 2, 90–91, 1956.

17. Nonstationary processes in nonlinear oscillatory systems. Proceedings of the Third All-Union Mathematical Congress, vol. I, 224, 1956.

18. The action on nonlinear oscillatory systems of external forces with variable frequencies. Buletinul Institutului Politehnic din Iași. Serie nouă, vol. III (VII), 1–2, 15–24, 1957.

19. On the question of internal resonance in nonlinear oscillatory systems. Scientific Notes of the Kiev State University named after T. G. Shevchenko, vol. XVI, issue II, Mathematical Collection No. 9, 53–61, 1957.

20. On certain differential equations encountered in the theory of relaxation oscillations. Ukr. Mat. Zhurnal, vol. XI, No. 3, 296–309, 1957.

List of Principal Works of Yu. A. Mitropol’skii

1. On nonstationary processes in certain relaxation oscillatory systems. Scientific Notes of Kyiv State University named after T. G. Shevchenko, vol. XVI, issue XVI. Mathematical collection No. 10, 93–101, 1957.

2. On the asymptotic representation of solutions of a system of nonlinear equations with variable coefficients. Scientific Yearbook of KSU for 1956, 504–506, 1957.

3. On the investigation of an integral manifold for a system of nonlinear equations with variable coefficients. Ukrainian Mathematical Journal, vol. X, No. 3, 270–279, 1958.

4. On the stability of a one-parameter family of solutions of a system of equations with variable coefficients. Ukrainian Mathematical Journal, vol. X, No. 4, 1958, 389–393.

5. On the question of equations close to exactly integrable ones. Bulletin of Kyiv University, No. 1, series of astronomy, mathematics, and mechanics, issue 1, 97–100, 1958.

6. On the question of the order of error in the asymptotic integration of equations close to exactly integrable ones. Bulletin of Kyiv University, No. 1, series of astronomy, mathematics, and mechanics, issue 2, 3–6, 1958.

7. Asymptotic methods in the theory of nonlinear oscillations. Proceedings of the Third All-Union Mathematical Congress, vol. III, 531–542, 1958.

8. Quelques questions de l’intégration asymptotique des équations différentielles non linéaires. Publishing House of the Academy of Sciences of the Ukrainian SSR, 1–17, 1958. (Report at the International Mathematical Congress in Edinburgh in August 1958.)

9. Investigation of nonstationary oscillatory regimes in systems with distributed parameters. Bulletin of Kyiv University, No. 2, series of astronomy, mathematics, and mechanics, issue 1, 3–17, 1959 (jointly with B. I. Moseenkov).

10. On certain equations close to exactly integrable ones. Collection “Automatic Control and Computational Techniques,” issue 2, 221–248, 1959.

11. On periodic solutions of a system of nonlinear differential equations with nondifferentiable right-hand sides. Reports of the Academy of Sciences of the USSR, 128, No. 6, 1118–1121, 1959.

12. On periodic solutions of systems of nonlinear differential equations close to autonomous ones. Reports of the Academy of Sciences of the Ukrainian SSR, No. 11, 1175–1178, 1959 (jointly with O. B. Lykova).

13. On periodic solutions of systems of nonlinear differential equations whose right-hand sides are nondifferentiable. Ukrainian Mathematical Journal, vol. XI, No. 4, 366–379, 1959.

14. On the question of nonlinear equations with periodic coefficients. Bulletin of Kyiv University, No. 2, series of astronomy, mathematics, and mechanics, issue 2, 3–12, 1959 (jointly with O. B. Lykova).

15. On the question of periodic solutions of nonautonomous systems in the case of an isolated generating solution. Reports of the Academy of Sciences of the Ukrainian SSR, No. 1, 3–6, 1960 (jointly with O. B. Lykova).

16. Sur les multiplicités intégrales des systèmes d’équations différentielles non linéaires ayant un petit paramètre. Annali di Matematica pura ed applicata. Serie IV, vol. XLIX, Bologna, 181–192, 1960.

17. Najnowsze osiągnięcia w dziedzinie mechaniki nielineowej. Rozprawy Inżynierskie, CXLVI, vol. VIII, issue 2, 125–135, 1960.

18. On the question of periodic solutions of nonlinear equations with a small parameter. Ukrainian Mathematical Journal, vol. XII, No. 4, 391–401, 1960 (jointly with O. B. Lykova).

19. On nonlinear differential equations with periodic coefficients and slowly varying parameters. Bulletin of Kyiv University, No. 3, series of mathematics and mechanics, issue 2, 3–10, 1960 (jointly with O. B. Lykova).

20. On the question of periodic solutions of a system of nonlinear differential equations with nondifferentiable right-hand sides. Buletinul Institutului Politehnic din Iași. Serie nouă, 7–12, 1960 (jointly with O. B. Lykova).

21. Construction of an asymptotic solution of an autonomous system with strong nonlinearity. Reports of the Academy of Sciences of the Ukrainian SSR, No. 7, 839–843, 1961 (jointly with P. M. Sennik).

22. Analytic methods in the theory of nonlinear oscillations. Proceedings of the All-Union Congress on Theoretical and Applied Mechanics, Publishing House of the Academy of Sciences of the USSR, 25–35, 1962 (jointly with N. N. Bogolyubov).

23. Méthodes Analytiques de la Théorie des Oscillations Nonlinéaires. Proceedings of the Tenth International Congress of Applied Mechanics, Stresa, 1960. Amsterdam–New York, 9–25, 1962 (jointly with N. N. Bogolyubov).

24. The Method of Integral Manifolds in the Theory of Nonlinear Differential Equations. (Paper presented to the International Mathematical Congress in Stockholm.) Kiev, Institute of Mathematics, Academy of Sciences of the Ukrainian SSR, 1962.

25. The Method of Integral Manifolds in the Theory of Nonlinear Oscillations. International Symposium on Nonlinear Differential Equations and Nonlinear Mechanics. Academic Press, New York and London, pp. 1–15, 1963.

26. The method of integral manifolds in nonlinear mechanics. Proceedings of the International Symposium on Nonlinear Oscillations. Publishing House of the Academy of Sciences of the Ukrainian SSR, 1, 93–154, 1963 (jointly with N. N. Bogolyubov).

27. Translation. The Method of Integral Manifolds in Nonlinear Mechanics. Contributions to Differential Equations, vol. II, pp. 123–196, 1963 (jointly with N. N. Bogolyubov).

LIST OF THE PRINCIPAL WORKS OF Yu. A. MITROPOL'SKII

1. Investigation of nonstationary oscillations in nonlinear systems. Proceedings of the International Symposium on Nonlinear Oscillations. Publishing House of the Academy of Sciences of the Ukrainian SSR, vol. III, 241—274, 1963.

2. The method of integral manifolds in the theory of differential equations. Proceedings of the Fourth All-Union Mathematical Congress, vol. II, 432—437, 1964 (jointly with N. N. Bogoliubov).

3. Some questions in the theory of nonlinear oscillations. Nonlinear Vibration Problems, 5, 27—36, Warszawa, 1964.

4. On an integral manifold of nonlinear differential equations containing slow and fast motions. Ukrainian Mathematical Journal, vol. XVI, No. 2, 157—163, 1964 (jointly with O. B. Lykova).

5. On the investigation of an integral manifold for a system of nonlinear equations close to equations with variable coefficients in a Hilbert space. Ukrainian Mathematical Journal, vol. XVI, No. 3, 334—337, 1964.

6. On the construction of a general solution of nonlinear differential equations by means of a method ensuring “accelerated” convergence. Ukrainian Mathematical Journal, vol. XVI, No. 4, 475—501, 1964.

7. On the question of the structure of trajectories on toroidal manifolds. Reports of the Academy of Sciences of the Ukrainian SSR, No. 8, 984—986, 1964 (jointly with A. M. Samoilenko).

8. Investigation of the behavior of solutions of nonlinear equations in a neighborhood of an equilibrium position. Institute of Mathematics of the Academy of Sciences of the Ukrainian SSR, Institute of Technical Information. Kiev, 3—44, 1964 (jointly with O. B. Lykova).

9. On the investigation of quasiperiodic regimes in nonlinear oscillatory systems. Les Vibrations Forcées dans les Systèmes Non-Linéaires, Paris, 181—192, 1965 (jointly with N. N. Bogoliubov).

10. Translation. Regimes Quasi-Périodiques dans les Systèmes Oscillants Non-Linéaires. Les Vibrations Forcées dans les Systèmes Non-Linéaires. Paris, 193—204, 1965 (jointly with N. N. Bogoliubov).

11. On the existence of quasiperiodic solutions of a perturbed canonical system. Les Vibrations Forcées dans les Systèmes Non-Linéaires. Paris, 407—414, 1965 (jointly with O. B. Lykova).

12. Translation. Sur l’existence de solutions quasi-périodiques d’un systèmes canonique troublé. Les Vibrations Forcées dans les Systèmes Non-Linéaires. Paris, 415—423, 1965 (jointly with O. B. Lykova).

13. Application of asymptotic methods of nonlinear mechanics to the investigation of nonlinear oscillatory systems with distributed parameters. III Konferenz über Nichtlineare Schwingungen (Berlin vom 25 bis 30 Mai 1964), 1 Akademie-Verlag. Berlin, 10—20, 1965.

14. Investigation of the behavior of solutions of nonlinear equations in a neighborhood of an equilibrium position. Republican interdepartmental collection Mathematical Physics. Kiev, 74—96, 1965 (jointly with O. B. Lykova).

15. On an integral manifold of a nonlinear system in a Hilbert space. Ukrainian Mathematical Journal, 17, No. 5, 43—53, 1965 (jointly with O. B. Lykova).

16. On the construction of solutions of linear differential equations with quasiperiodic coefficients by means of the accelerated-convergence method. Ukrainian Mathematical Journal, 17, No. 6, 42—59, 1965 (jointly with A. M. Samoilenko).

17. Asymptotic methods of nonlinear mechanics as applied to nonlinear equations with a delayed argument. Ukrainian Mathematical Journal, 18, No. 3, 65—84, 1966 (jointly with V. I. Fodchuk).

18. The accelerated-convergence method in problems of nonlinear mechanics. Journal Funkcialaj Ekvacioj. Japan, vol. II, 1—27, 1967.

B. Monographs

1. Nonstationary Processes in Nonlinear Oscillatory Systems. Publishing House of the Academy of Sciences of the Ukrainian SSR, 1955.

2. Nonstationary Processes in Nonlinear Oscillatory Systems (in Chinese). Publishing House “Nauka,” Peking, 1958.

3. Nonstationary Process in Non-Linear Oscillatory Systems. Air Technical Intelligence Translation ATIC-270579, F-TS-9085/V, 1961.

4. Nonstationary Processes in Nonlinear Oscillatory Systems (in Japanese). Tokyo, 1962.

5. Asymptotic Methods in the Theory of Nonlinear Oscillations. Gostekhizdat, 1955 (jointly with N. N. Bogoliubov).

6. Second edition, revised and enlarged. Fizmatgiz, 1958.

7. Third edition, revised and enlarged. Fizmatgiz, 1963.

8. Asymptotic Methods in the Theory of Nonlinear Oscillations (in Japanese). Tokyo.

9. Asimptotic Methods in the Theory of Non-Linear Oscillations. Cordon and Breach Science Publishers. New York, Hindustan Publishing, Corpn., Delhi, 6, 1961.

1. Asymptotic Methods in the Theory of Nonlinear Oscillations (in Chinese). Beijing, 1962.

2. Les Méthodes Asymptotiques en Théorie des Oscillation Non-Linéaires, Gauthier-Villars. Paris, 1962.

3. Translation. Asymptotische Methoden in der Theorie der Nichtlinearen Schwingungen. Academie-Verlag. Berlin, 1965.

4. Investigation of Oscillations in Systems with Distributed Parameters (Asymptotic Methods). Publishing House of Kiev State University, 1961 (jointly with B. I. Moseenkov).

5. Problems of the Asymptotic Theory of Nonstationary Oscillations. Publishing House “Nauka,” Moscow, 1964.

6. Lectures on the Averaging Method in Nonlinear Mechanics. Publishing House of the Academy of Sciences of the Ukrainian SSR, Kiev, 1966.