The theory of differential equations originated at the end of the seventeenth century in the works of I. Newton, G. W. Leibniz and others. During the first century of its existence, this theory consisted only of isolated methods of solving certain types of differential equations; but the problem of the existence of a solution and its representability in quadratures was posed already in the second. As a result of numerous investigations it became clear that integrability in quadratures is an extremely rare phe nomenon and that the solution of many differential equations arising in applications cannot be expressed in quadratures. Also the methods of numerical integration of equations did not open the road to the general theory since these methods yield only one particular solution and this solution is obtained on a finite interval. Applications - especially the problems of celestial mechanics - required the clarification of at least the nature of the behavior of integral curves in the entire domain of their existence without integration of the equation. In this connection, at the end of the last century there arose the qualitative theory of differential equations, the creators of which one must by all rights consider to be H. Poincare and A. M. Lyapunov.

Inhalt
I. General properties of dynamical systems.- § 1. General definition of a dynamical system.- § 2. Simplest properties of dynamical systems.- § 3. The classification of motions and trajectories.- § 4. Invariant sets.- § 5. Theorems on rest points.- § 6. Dynamical systems on the real line. The isomorphism of dynamical systems.- II. Limiting properties of dynamical systems.- § 7. Dynamical limit points. Properties of limit sets.- § 8. Lagrange stability.- § 9. The classification of motions according to the properties of dynamical limit sets.- § 10. Examples of Poisson stable motions on the torus.- § 11. Properties of Poisson stable points and motions.- III. Nonwandering points. Central motions.- § 12. Wandering and nonwandering points.- § 13. Properties of the set of nonwandering points.- § 14. The set of central motions.- § 15. Minimal center of attraction.- IV. Minimal sets and recurrent motions.- § 16. Minimal sets.- § 17. Almost recurrent motions and recurrent motions.- § 18. Interrelationships among minimal sets, almost recurrent motions; and recurrent motions.- § 19. The Shcherbakov classification of Poisson stable motions. Pseudorecurrent motions.- § 20. The Bebutov dynamical system.- V. Almost periodic motions. Lyapunov stability.- § 21. Uniformly Poisson stable motions and almost periodic motions.- § 22. Lyapunov stability.- § 23. Lyapunov stable motions on the real line.- § 24. Interrelationship between periodicity and Lyapunov stability.- § 25. Motions in dynamical limit sets.- § 26. Stability of rest points.- VI. Generalized theory of dynamical systems.- § 27. General dynamical systems. Topological transformation groups.- § 28. Discrete dynamical systems.- § 29. Partially ordered dynamical systems.- § 30. Dispersive dynamicalsystems.- Literature.