Dynamical systems and differential equations are a domain of mathematics which studies models of structures that vary in time. This domain appeared as a part of the classical mathematical analysis but at present, this is a part of mathematics which intensively uses methods of algebra, geometry, topology, and probability.

#### Local and global qualitative theory

Qualitative theory of dynamical systems and differential equations studies the structure of trajectories both in neighborhoods of invariant sets (for example, such as fixed points and periodic trajectories) and in the whole phase space. An important part of the qualitative theory is the classical stability theory. It is of real interest to study invariant sets with complicated structure (chaotic invariant sets).

Among the people working in this field are S.G. Kryzhevich, S.Yu. Pilyugin and S.B. Tikhomirov.

#### Theory of perturbations

Theory of perturbations studies the evolution of the qualitative structure of trajectories of a dynamical system when the equations which determine the dynamics are changed. An important problem is the problem of structural stability (i.e., of preservation of the topological structure of trajectories in the phase space under small perturbations of the system). This theory also studies small discontinuous perturbations which correspond, for example, to the replacement of the exact system by its approximation in computer modeling.

Among the people working in this field are S.G. Kryzhevich, S.Yu. Pilyugin and S.B. Tikhomirov.

#### Ergodic theory

Ergodic theory studies the behavior of invariant measures of a dynamical system. One of the main objects of study is the entropy of the system which characterizes the complexity of dynamics. This direction is closely related to methods of probability theory.

Among the people working in this field are S.G. Kryzhevich and S.B. Tikhomirov.