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.