Saturday, 09 April 2022 00:55

SAMOILENKO, Anatolii (1938 - 2020)

  • Author(s): I. O. Lukovs’kyi, V. L. Makarov, M. O. Perestyuk, O. A. Boichuk, I. O. Parasyuk, O. V. Antonyuk, V. I. Tkachenko, R. I. Petryshyn, M. V. Prats’ovytyi, O. M. Stanzhyts’kyi, M. I. Ronto

Anatolii Mykhailovych Samoilenko was an outstanding scientist-mathematician in the field of ordinary differential equations and the theory of nonlinear oscillations, founder of the famous scientific school in the theory of multifrequency oscillations and the theory of impulsive systems, doctor of physical and mathematical sciences, professor, Academician of the Ukrainian National Academy of Sciences, honored worker of science and engineering of Ukraine, winner of State Prizes of Ukraine and several name prizes of the Ukrainian National Academy of Sciences.

Samoilenko was born in the village of Potievka, Zhytomyr Region, to the family of Mykhailo Hryhorovych and Mariya Vasylivna. His father was a veterinarian and took part in the Second World War. His mother devoted her life to the family. After graduation from the secondary school in Malyn, Anatolii entered the Geological Department of the T. Shevchenko Kyiv State University. However, shortly thereafter, due to a great interest in mathematics, he cardinally changed his mind and decided to continue his studies at the Department of Mechanics and Mathematics. After this, all his life was devoted to the world of mathematics. For the last two yearsof studies, Samoilenko received the Lenin Scholarship. With deep gratitude, he remembered his teachers, professors B. V. Gnedenko, L. A. Kaluzhin, Yu. M. Berezanskii, H. M. Polozhii, Yu. A. Mitropol’skii, I. Z. Shtokalo, and B. Ya. Bukreev.

In 1960, Samoilenko graduated from the university with honors and, on the invitation of academician Mitropol’skii,entered the postgraduate courses at the Institute of Mathematics of the Academy of Sciences of Ukrainian RSR. The choice of the theme of his candidate-degree thesis “Application of Asymptotic Methods to the Investigation of Nonlinear Differential Equations with Irregular Right-Hand Side” was quite natural due to a rapid development of the world-known Kyiv school of nonlinear mechanics founded by Academicians N. M. Krylov and N. N. Bogolyubov. In 1967, Samoilenko defended his doctoral-degree thesis “Some Problems of the Theory of Periodic and Quasiperiodic Systems” and became the youngest doctor of sciences in Ukraine. In 1974–1987, he headed the Chair of Integral and Differential Equations at the T. Shevchenko Kyiv State University. Under his leadership, the research activities of the chair became much more intense. Numerous doctor- and candidate-degree theses were defended. The seminar of the chair in differential equations organized by Prof. Samoilenko became very popular not only in Ukraine but throughout the world. In 1978, Samoilenko was elected to become Corresponding Member of the Academy of Sciences of Ukrainian RSR.

In 1987, Prof. Samoilenko returned to the Institute of Mathematics of the Academy of Sciences of Ukrainian RSR and somewhat later was elected to become Director of the Institute of Mathematics. Since that time, for 32 years, he occupied this position. For this period, Anatolii Mykhailovych demonstrated that he is not only a prominent scientist but also a talented organizer of science. Thus, numerous prestigious international conferences were held on his initiative and with his direct participation (as Chairman of the Organizing Committee), including two Ukrainian Mathematical Congresses (in 2001 and 2009) with more than five hundred participants in each (including both Ukrainian and foreign mathematicians). Samoilenko was Editor-in-Chief of the “Ukrains’kyi Matematychnyi Zhurnal,” “Neliniini Kolyvannya,” “Ukrains’kyi Matematychnyi Visnyk,” and “Matematychnyi Visnyk Naukovoho Tovarystva im. Shevchenka” journals and member of the Editorial Boards of “Dopovidi Natsional’noi Akademii Nauk Ukrainy,” “Visnyk Natsional’noi Akademii Nauk Ukrainy,” “U Sviti Matematyky,” “Memoirs on Differential Equations and Mathematical Physics,” “Miskolc Mathematical Notes,” “International Journal of Dynamical Systems and Differential Equations,” and “Applied and Computational Mathematics” journals.

The mathematical gift and organizational activities of Prof. Samoilenko gave him honored authority and respect of the scientific community. He was elected to become Full Member of the National Academy of Sciences of Ukraine (1995) and European Academy of Sciences (2002), Corresponding Member of Academia Peloritana dei Pericilanti (Messina, Sicily, 2006), and Foreign Member of the Academy of Sciences of Tadzhikistan Republic (2011). Since 2006, Samoilenko occupied the position of Academician-Secretary of the Department of Mathematics of the National Academy of Sciences of Ukraine.

His scientific results are widely known for experts in the fields of differential equations, mathematical physics, and theory of nonlinear oscillations. Academician Samoilenko is generally recognized as a founder of several important directions of investigations in these fields. Thus, in 1965, he proposed and substantiated a new efficient method for finding periodic solutions of essentially nonlinear differential equations, which isnow known as the “Samoilenko numerical-analytic method.” This method was thoroughly developed and used for the solution of nonlinear boundary-value problems in numerous works both by Samoilenko himself and by his disciples. The corresponding results were published in a series of monographs.

In 1964–1968, Samoilenko published a cycle of works (both himself and together with Mitropol’skii) in which, he solved a series of new important problems of the theory of multifrequency oscillations by the method of successive changes of variables with accelerated convergence. A typical object of analysis in this theory is, e.g., a system of (weakly) coupled oscillators both harmonic with rationally independent frequencies and anharmonic whose frequencies are functions of the amplitudes of oscillations.

In a more general case, the researchers study the n-dimensional autonomous system of ordinary differential equations whose m-dimensional invariant torus is filled with quasiperiodic trajectories. It is assumed that this system is unperturbed and, in addition, that the so-called Floquet coordinates can be introduced in a neighborhood of the invariant torus. These are the angular coordinates on the invariant torus in which the trajectories of quasiperiodic motions of the system are straight lines and the coordinates normal to the torus in which a system linearized with respect to the torus has a constant matrix. For the solution of posed problems, it is necessary, first, to construct the general solution of the indicated unperturbed system by reducing it to the simplest possibleform and, second, to develop an informal perturbation theory and prove a similar result on reduction for theweakly perturbed system. Prof. Samoilenko managed to solve both problems. Moreover, they were solved in a much more general form, i.e., for systems of finite classes of smoothness.

It is worth noting that the results of A. N. Kolmogorov, V. I. Arnold, and N. N. Bogolyubov were obtained at that time only for analytic systems. In two J. Moser's works published in the early 1960s, the number of required derivatives of the right-hand sides in the analyzed systems is far from being optimal. In one of the first Arnold's works devoted to the KAM-theory, he wrote that the possibility to omit the requirement of analyticity was a very unexpected and important advance in the solution of the problems with small denominators. Note that Arnold was one of the opponents of the doctoral-degree thesis of A. M. Samoilenko. In the course of discussion of the results, he proposed Anatolii to solve an important problem of the theory of singularities of maps: on the equivalence of a finitely differentiable function of several variables to its Taylor polynomial. At that time, the corresponding result was known in the analytic case. In 1967, Samoilenko solved the posed problem in the pure analytic form by combining the method of accelerated convergence with the technique of smoothening.

He successfully applied this approach to the investigation of the well-known problem of rectification of a vector field on the torus. The classical A. Poincaré and A. Denjoy results in this field were obtained solely for the vector fields on two-dimensional tori. At that time, Samoilenko's results were the best from the viewpoint of minimization of the class-of-smoothness index for almost parallel vector fields on a torus of any dimension.

By the method of successive changes of variables characterized by accelerated convergence and the technique of smoothing, Prof. Samoilenko established a series of important results for finitely smooth nonconcervative systems of nonlinear mechanics and, in particular, proved the theorems on the existence of a linearizing diffeomorphism in the vicinity of a toroidal manifold swept by a quasiperiodic trajectory, on the reducibility of linear quasiperiodic systems with almost constant coefficients, and on the measure of reducible systems fromthis class. These and other results were presented in the monograph by Bogolyubov, Mitropol’skii, and Samoilenko “Method of Accelerated Convergence in Nonlinear Mechanics” published in 1969. This monograph was later translated into English [“Methods of Accelerated Convergence in Nonlinear Mechanics,” Springer, Berlin–Heidelberg–New York (1976)]. It is worth noting that important results on the spectral characteristics of the one-dimensional Schrödinger equation with quasiperiodic potential were obtained by using the method of accelerated convergence in the works by E. I. Dinaburg, Ya. G. Sinai, and E. D. Belokolos, which appeared soon after publishing this monograph.

A special place in Samoilenko's scientific interests was occupied by the theory of invariant toroidal manifolds of nonlinear dynamical systems. He developed the efficient approach to the investigation of the problem of preservation of invariant tori under perturbations. This approach is based on his notion of Green function of the linear extension of a dynamical system on the torus (in the contemporary mathematical literature, this notion is known as the Green–Samoilenko function). This notion turned out to be very useful in the analysis of numerous problems of the theory of dynamical (discrete, impulsive, and countable) systems, systems inBanach spaces, differential equations with delay, and problems of mathematical physics.

Note that the proposed approach has the following advantages: unlike the traditional methods of perturbation theory, the Green–Samoilenko function used for the solution of the problem of preservation of invariant manifold enables one not to introduce coordinates in its neighborhood reducing the unperturbed system to a system with constant coefficients and not only to prove the theorems on existence of stable and hyperbolic invariant tori but also to study their smoothness. By using the apparatus of Green functions, Samoilenko not only proved theorems of existence but also developed and substantiated an approximate projection-iterative method for finding invariant tori in the form of convergent sequences of trigonometric polynomials.

As a specific feature of Samoilenko's scientific work, we can mention a harmonic combination of deep theoretical investigations aimed, e.g., at the proof of the theorems of existence, with the development of efficient constructive methods. This feature is brightly illustrated by his monograph “Elements of the Mathematical Theory of Multifrequency Oscillations. Invariant Tori,” Nauka, Moscow (1987), which was later translated into English [“Elements of the Mathematical Theory of Multi-Frequency Oscillations,” Kluwer Academic Publishers, Dordrecht (1991)].

In his investigations of the behavior of trajectories of a dynamical system in a neighborhood of its toroidal manifold, Samoilenko established fairly general conditions for the existence of a diffeomorphism that connects this system with the corresponding canonical form. It turns out that the indicated form has the structure of linear (homogeneous) extension of the dynamical system induced by the system on its invariant manifold. In connection with the investigation of the behavior of trajectories, it is necessary to mention the following fine ergodic result: In the case of quasiperiodic winding, the integral mean of an arbitrary continuous function along the trajectories of a nonlinear extension of the system on the torus approaches the integral with respect to the ergodic measure on the torus. A. M. Samoilenko, together with his colleagues, developed a profound theory to explainthe relationships between various important characteristics of extensions, such as the property of exponential dichotomy and splitting, the presence of an (alternating) Lyapunov function, and the existence of the Green–Samoilenko function. The results obtained in this field were summarized in the monograph written together with Yu. A. Mitropol’skii and V. L. Kulyk, “Investigation of the Dichotomy of Linear Systems of Differential Equations with the Help of Lyapunov Functions,” Naukova Dumka, Kyiv (1990). Later, this monograph was translated into English as “Dichotomies and Stability in Non-Autonomous Linear Systems,” Taylor & Francis, London–New York (2003).

In the 1970–1980s, Samoilenko, together with his colleagues, created and developed the theory of differential systems with impulsive actions. Numerous publications of the authors from Samoilenko's scientific school laid the foundations for the systematic investigation of various problems of the qualitative and analytic theory of impulsive systems. The results of investigations in this field were presented in the monograph by A. M.Samoilenko and M. O. Perestyuk, “Impulsive Differential Equations,” Vyshcha Shkola, Kiev (1987). This monograph was the first book in the world literature containing a systematic presentation of the main results of the theory of differential equations with impulsive action. In 1995, this monograph was supplemented with new results andpublished in English as “Impulsive Differential Equations,” World Scientific, Singapore (1995).

A great talent and experience of Anatolii Mykhailovych as a researcher, organizer of scientific investigations, and leader of the Kyiv mathematical school, made it possible for him to guide the research work of his colleagues in several different directions simultaneously. Thus, the theory of alternating Lyapunov functionswas developed for the analysis of dichotomies, globally bounded solutions, and invariant manifolds of linear extensions of dynamical systems on the torus. At the same time, the researchers developed the theory of Fredholm boundary-value problems for systems with delay, impulsive equations, and singularly perturbed systems. Later, the authors of these theories,proposed their efficient application to the investigation of the solutions of nonautonomous systems bounded on the entire axis, which have the property of exponential dichotomy on the semiaxes.

One more direction of Samoilenko’s research interests was connected with the study of resonance phenomena in multifrequency systems, including the systems with slowly varying parameters. His fine estimates of theoscillating integrals appearing in the analysis of transitions of trajectories through the resonance subsets of the phase space laid the foundations for getting new profound results in the substantiation of the averaging method for oscillating systems in the case where the number of frequencies is greater than two. These and related results can be found in Samoilenko's monographs “Numerical-Analytic Methods for the Investigation of Periodic Solutions,” Vyshcha Shkola, Kyiv (1976) (together with M. I. Ronto), which was translated into English as “Numerical-Analytic Methods of Investigating Periodic Solutions,” Mir, Moscow (1979), “Numerical-Analytic Methods for the Investigation of Boundary-Value Problems,” Naukova Dumka, Kyiv (1985) (together with M. I. Ronto), which was translated into English as “Numerical-Analytic Methods in the Theory of Boundary-Value Problems,” World Scientific, Singapore (2000), “Multi-Frequency Oscillations of Nonlinear Systems,” Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv (1998) (together with R. I. Petryshyn); Englishtranslation: “Multi-Frequency Oscillations of Nonlinear Systems,” Kluwer, Dordrecht–Boston–London (2004),“Mathematical Aspects of the Theory of Nonlinear Oscillations,” Naukova Dumka, Kyiv (2004) (together with R. I. Petryshyn), “Generalized Inverse Operators and Fredholm Boundary-Value Problems,” Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv (1995) (together with O. A. Boichuk and V. P. Zhuravl’ov), “Generalized Inverse Operators and Fredholm Boundary-Value Problems,” VSP, Brill, Utrecht (2004) (together with O. A. Boichuk; this monograph survived the second edition by the De Gruyter Publishing House, Berlin (2016)), and “Normally Solvable Boundary-Value Problems,” Naukova Dumka, Kyiv (2019) (together with O. A. Boichuk and V. P. Zhuravl’ov).

Acad. Samoilenko is the author of more than 600 scientific works, including 30 monographs and more than 20 textbooks. Among his disciples, there are 36 doctors and 89 candidates of sciences. We also especially mention great pedagogic activities of Prof. Samoilenko at the T. Shevchenko Kyiv National University, I. SikorskyKyiv Polytechnic Institute, National Technical University of Ukraine, and other institutions of higher education. Due to his unique talent as a lecturer, he always produced unforgettable impression on the audience by his ability to give clear and emotional presentation of the material of his original lecture courses.

For the long-term scientific, pedagogic, and public activity, Samoilenko was awarded the Order of Friendship of Peoples (1984), Order “For Services” of Degree III (2003), and Orders of Prince Yaroslav Mudryi of Degrees V (2008), IV (2013), and III (2018), and the Honorary Diploma of Presidium of the Supreme Soviet of Ukraine (1987). He was also the winner of State Prizes of Ukraine in the Field of Science and Engineering (1985 and 1996), State Prize of Ukraine in the Field of Education (2012), Republican Ostrovskii Prize (1968), Krylov Prize (1981), Bogolyubov Prize (1998), Lavrent’ev Prize (2000), Ostrogradskii Prize (2004), Mitropol’skii Prize (2010), and Krein Prize (2020) of the National Academy of Sciences of Ukraine. He wasalso given the titles of “Honored Worker of Science and Engineering of Ukraine” (1998), “Soros Professor” (1996), and honorary doctor of the T. Shevchenko Kyiv National University. He was awarded a medal of the Junior Academy of Sciences of Ukraine “200th Birthday of T. H. Shevchenko” (2014) and a medal of the Drahomanov National Pedagogic University “For Scientific Achievements” (2015).

Anatolii Mykhailovych Samoilenko will always remain in the history of world mathematics and in the memory of everybody who knew him as a prominent scientist, great teacher, talented organizer of science, and wonderful and delicate man.