#### For citation:

Mukhin R. R. From the history of the theory of dynamical systems: Problem of classification. *Izvestiya VUZ. Applied Nonlinear Dynamics*, 2019, vol. 27, iss. 5, pp. 95-112. DOI: 10.18500/0869-6632-2019-27-5-95-112

# From the history of the theory of dynamical systems: Problem of classification

Aim. The aim of the work is to study the history of ideas about the classification of dynamical systems, which are the most important objects of modern mathematics and having a huge number of applications. Solving the problem of classification allows you to take the first steps in understanding the structure of the system as a whole. Method. The study is based on an analysis of original works involving some of the memories of participants in the described events. Results. The paper considers the development of ideas about the classification of dynamical systems, which allows to take the first steps in understanding their device as a whole. The statement of the problem goes back to A. Poincare, who divided differential equations into integrable and nonintegrable. In the language of dynamical systems, G. Birkhoff singled out non-ergodic and ergodic systems, taking the complexity of the nature of motion as the principle of classification. By the end of the 1950th a hierarchy of conservative dynamical systems has developed: integrable systems, ergodic systems, systems with mixing, K-systems, B-systems. In the dissipative case, analogues of integrable conservative systems and systems with complex, irregular motion were isolated. With the appearance in the 1960th of a hyperbolic theory, a hypothesis (S. Smale) was put forward about the existence of structurally stable systems in the multidimensional case. But it turned out that such systems (Morse–Smale systems) do not form a dense set; in the multidimensional case, systems with a homoclinic structure are typical. Then it turned out that real systems are heterogeneous, they have areas with regular and irregular motion with very complex topology: systems with divided phase space in the conservative case and quasi-attractors in the dissipative one. The forms of coexistence of order and chaos turned out to be very diverse. There are systems with «mixed» dynamics. In systems with homoclinic tangency, in general, even a complete qualitative analysis is impossible. Integrable systems, Morse–Smale systems themselves are complex sets, and their classification is a nontrivial task. The classification problem can be solved only for certain groups of dynamical systems. Discussion. Dynamic systems turned out to be an immense object both in their diversity and in the complexity of the device. An exhaustive classification of dynamic systems seems an insoluble task. This is also characteristic of other areas of mathematics, which is caused by the infinite variety of the external world.

- Liouville J. Remarques nouvelles sur l’equation de Riccati // J. Math. Pures et Appl. 1841. Vol. 6. P. 1–13, 36.
- Bour J. Sur l’integration des equations differentielles de la Mecanique Analytic // J. Math. Pureset Appl. 1855. Vol. 20. P. 185–200.
- Liouville J. Note a l’occasion du memoire precident de M. Edmond Bour // J. Math. Pures et ´ Appl. 1855. Vol. 20. P. 201–202.
- Poincare Н. Sur les proprietes des fonctions definies aux differences partielles. Ph. D. Thesis, Universite de Paris. Paris: Gautier-Villars, 1879. 93 p.
- Poincare Н. Memoire sur les courbes d ´ efinies par une ´ equations differentielle. ´ J. Math. Pures et Appl. Ser. 3 ´ , 1881, vol. 7, pр. 375–422; 1882, vol. 8, pр. 251–296; Ser. 4, 1885, vol. 1, ´ pр. 167–244; 1886, vol. 2, pр. 151–217.
- Poincare Н. Sur le probleme des trois corps et lesequations de la Dynamique // Acta Math. 1890. Vol. 13. P. 1–270.
- Poincare H. New Methods of Celestial Mechanics. N.Y.: Dover publication, 1957. Vol. I, II. ´ 413 p.; Vol. III. 401 p.
- Poincare Н. Sur unе theoreme de geometrie // Rendicont. Circolo mat. Palermo. 1912. Vol. 33.
- Aleksandrov P.S. Poincare and topology. ´ Russian Math. Surveys, 1972, vol. 27, issue 1, pp. 157–168.
- Birkhoff G.D. Dynamical Systems. Providence, Rhod Island: AMS, 1927. IX + 295 p.
- Vizgin V.P. The Development of the Relationship of the Principles of Invariance with Conservation Laws in Classical Physics. Moscow: Nauka, 1972. 242 p. (in Russian).
- Markov A.A. Sur une propriete generale des ensemble minimaux de Birkhoff // Comp. Ren. Acad. Sci. 1931. Vol. 193. P. 823–825.
- Arnold V.I., Avez A. Ergodic problems of classical mechanics. N.Y.: W.A. Benjamin, 1968. 286 p.
- Bolzmann L. Uber der Warmegleichgewicht zwischen meharatomigen Gasmolekulen // Sitzber. Akad. Wiss. Wien. 1871. B. 63. S. 397–418.
- Lo Bello A. On the Origin and History of Ergodic Theory // Bolletino di Storia delle Scienze Mathematiche. 1983. A. iii. No. 1. P. 37–75.
- Birkhoff G.D. Proof of recurrence theorem for strongly transitive systems and proof of the ergodic theorem // Proc. Nat. Acad. Sci. Amer. 1931. Vol. 17. P. 650–660.
- Hopf E. Эргодическая теория // УМН. 1949. Т. 4. В. 1. С. 113–182.
- Gibbs J.W. Elementary Principles of Statistical Mechanics. N.Y.: Charles Shribner’s Sons, L.; Edward Arnold, 1902.
- Zaslavsky G.M. Stochasticity of Dynamical Systems. Moscow: Nauka, 1984. 272 p. (in Russian).
- Kolmogorov A.N. The general theory of dynamical systems and classical mechanics. Proc. Intern. Congr. Math., 1954, Amsterdam, vol. 1, pр. 315–333.
- Ornstein D. Ergodic Theory, Randomness and Dynamical Systems. N.Y.: Yale Univ. Press, 1974. 141 p.
- Kolmogorov A.N. A new metric invariant of transitive dynamical systems and automorphisms of Lebesgue spaces. Rep. Acad. Sci. of USSR, 1958, vol. 119, no. 5, pp. 861–864 (in Russian).
- Kolmogorov A.N. On entropy in unit time as a metric invariant of automorphisms. Rep. of Acad. Sci. USSR, 1959, vol. 124, no. 4, pp. 754–755 (in Russian).
- Sinai Ja.G. Private communication from 26.03.2007.
- Sinai Ja.G. Ergodic theory. In: Kolmogorov A.N. Selected works. T. III. Information Theory and the Theory of Algorithms. Dordrecht: Springer Sci., 1993, pp. 247–250.
- Andronov A.A., Vitt A.A., Khaikin S.E. Theory of Oscillators. Oxford: Pergamon, AddisonWesley, 1966. 816 р.
- Nemytskii V.V., Stepanov V.V. Qualitative Theory of Differential Equations. Princeton, NJ: Princeton Univ. Press, 1960. 523 p.
- Peixoto M. Some recollections of the early work of Steve Smale // The Collected Papers of Stephen Smale. Vol. 1. Singapore: World Sci. 2000. P. 14–16.
- Smale S. On gradient dynamical systems // Ann. Math. 1961. Vol. 74. P. 199–206.
- Smale S. A structurally stable differential homomorphysm with an infinite number of periodic points // Тр. Межд. симпоз. по нелин. колебаниям. Киев 1961. Киев: АН УССР, 1963. С. 365–366.
- Smale S. Structurally stable systems are not dense. Am. J. Math., 1966, vol. 73, pр. 747–817.
- Andronov A.A., Pontryagin L.S. Coarse systems. Rep. of Acad. Sci. USSR, 1937, vol. 14, no. 5, pp. 247–252 (in Russian).
- Peixoto M. Structural stability on two-dimensional manifolds // Topology. 1962. Vol. 1, no. 2. P. 101–120.
- Smale S. On how I got started in dynamical systems 1959–1962 // The Chaos Avant-garde. Memoiries of the Early Days of Chaos Theory / Eds R. Abraham, Y. Ueda. Singapore: World Sci. 2000. P. 1–6.
- Andronov A.A. Mathematical problems in self-oscillation theory. Collected Works. Moscow: AN USSR, 1956, pp. 32–71 (in Russian).
- Anosov D.V. et al. Dynamic systems with hyperbolic behavior. Results of Science and Technology. Ser. Modern problems of math. Fundamental directions. Dynamical systems – 9. Moscow: VINITI, 1985. Vol. 66. 248 p. (in Russian).
- Anosov D.V., Sinai Ya.G. Certain smooth dynamical system. Russian Math. Surveys, vol. 22, no. 5, pp. 107–172 (in Russian).
- Hadamard J. Les surfaces a courbures opposees et leurs lignes geod esiques // J. Math. Pures et Appl. 1898. Vol. 4. Pр. 27–73.
- Hedlund G.A. The dynamic of geodesic flows // Bull. AMS. 1939. Vol. 45. Pр. 241–260.
- Anosov D.V. Coarseness of geodesic flows on compact Riemannian manifolds of negative curvature. Rep. of Acad. Sci. USSR, 1962, vol. 145, no. 4, pp. 707–709.
- Anosov D.V. Geodesic flows on compact Riemannian manifolds of negative curvature. Proceedings of MIAN. Moscow: Nauka, 1967, pp. 3–209 (in Russian).
- Grines et al. Classification of Morse–Smale systems and topological structure of the underlying manifolds. Russian Math. Surveys, 2019, vol. 74, issue 1, pp. 37–110.
- Kozlov V.V. Integrability and non-integrability in Hamiltonian mechanics. Russian Math. Surveys, 1983, vol. 38, issue 1, pp. 3–67 (in Russian).
- Zaslavsky G.M., Chirikov V.V. Stochastic instability of non-linear oscillations. Sov. Phys. Usp., 1972, vol. 14, pp. 549–568.
- Zaslavsky G.M. Physics of Chaos in Hamiltonian Systems. L.: Imper. College Press, 1998, 269 p.
- Zaslavsky G.M. Chaotic Dynamics and the Origin of Statistical Laws // Physics Today. 1999. Vol. 52. Pр. 39–45.
- Afraimovich V.A., Shilnikov L.P. On strange attractors and quasiattractors // Nonlinear dynamics and turbulence. Boston-London-Melbourn: Pitman, 1983. Pр. 1–34.
- Gonchenko S.V. et al. On Newhouse domains of two-dimensional diffeomorphisms that are close to diffeomorphism with a structurally unstable heteroclinic contour. Proceedings of MIAN. Moscow: Nauka, 1997, vol. 216, pp. 76–125 (in Russian).
- Gonchenko S.V. et al. Mathematical theory of dynamical chaos and its applications: Review. Izvestya VUZ. Applied Nonlinear Dynamics, 2017, vol. 25, issue 2, pp. 4–36 (in Russian).
- Shilnikov L.P. On a Poincare-Birkhoff problem. Mathematics of the USSR-Sbornik, 1967, vol. 174, no. 3, pp. 378–397.
- Shilnikov L.P. Homoclinic Trajectories from Poincare to the Present. Mathematical Events of the Twentieth Century. Berlin: Springer-Verlag, 2006, pp. 347–370.
- Newhouse S. Non-density of axiom A(a) on S 2 // Proc. AMS symp. pure math. 1970. Vol. 14. Pр. 191–202.
- Newhouse S. Diffeomorphisms with infinetly many sinks // Topology. 1974. Vol. 13. N 1. Pр. 9–18.
- Gavrilov N.K., Shilnikov L.P. On three-dimensional dynamical systems close to systems with structurally unstable homoclinic curve. Mathematics of the USSR-Sbornik, 1972, vol. 17, no. 4, pp. 467–485; vol 19, no. 1, pp. 139–156.
- Gonchenko S.V., Shilnikov L.P., Turaev D.V. Dynamical phenomena in systems with structurally unstable Poincare homoclinic orbits // Chaos. 1996. Vol. 6, no. 1. Pp. 15–31.
- Gonchenko S.V., Shilnikov L.P., Turaev D.V. Homoclinic tangencies of an arbitrary order in Newhouse domains. J. Math. Sci., 2001, vol. 105, issue 1, pp. 1738–1778.
- Markov A.A. The insolvability of the problem of homeomorphy. Rep. of Acad. Sci. USSR, 1958, vol. 121, no. 2, pp.218–220 (in Russian).
- Markov A.A. Unsolvability of certain problems of topology. Rep. of Acad. Sci. USSR, 1958, vol. 123, no. 6, pp. 978–980 (in Russian).
- Vavilov N.A. Simple Lie Algebras, Simple Algebraic Groups, and Simple Finite Groups. In: Mathematics of the XX Century. View from Petersburg, Moscow: MTSNMO, 2010, pp. 8–46 (in Russian).

- 1390 reads