Development of methods for the qualitative theory of dynamical systems, including the construction of energy functions, topological classification, the inclusion of cascades in the flow, the search for stable arcs in the space of dynamical systems, the establishment of links between the dynamics of the system and the topology of the manifold, the algorithmization of distinguishing topological invariants. Development of new analytical and numerical methods for the study of pseudo-hyperbolic attractors, spiral quasi-attractors and mixed dynamics of multidimensional dissipative systems and their applications to specific models. Development of methods for the theory of deformations of modular Lie algebras and obtaining classification results for character algebras of characteristic 2. Development of methods for studying the qualitative behavior of foliations that are consistent with geometric structures. The study of the geometry of bundles, applications to dynamics. Development of the Chernov function calculus.
The development of topological methods in dynamics enriches both sciences - the qualitative theory of dynamical systems and topology. A remarkable example of such an interaction is the solution of one of the most important topological problems — the multidimensional Poincare hypothesis (in dimension greater than four). To solve this problem, S. Smale applied the theory of gradient dynamical systems induced by Morse functions. Namely, assuming that the manifold is a homotopy sphere, he proved that all critical points of a given Morse function can be successively removed and go to a function with exactly two critical points of index 0 and 1.
Using topological and geometric methods, the laboratory staff obtained deep results on the interrelation of the dynamic characteristics of flows and cascades with the topology of the manifold. The structure of manifolds admitting Morse-Smale diffeomorphisms was studied depending on the intersection structure of invariant manifolds of saddle periodic points. The interrelation of investments of invariant manifolds with the existence of Lyapunov energy functions, etc., was discovered. These results were published in a number of leading foreign and domestic journals, which indicates their compliance with the world level. The obtained reserve will allow us to further develop the relevant topics described above.
Another major achievement of the modern theory of dynamic systems, in which the laboratory staff are also recognized experts, was the discovery of mixed dynamics, which characterizes the fundamental impossibility to separate the attractors of a dynamic system from repellers. Despite the fact that such a phenomenon was discovered quite recently, we have already managed to detect a number of systems from various applications that demonstrate mixed dynamics. Thus, the development of methods for the study of mixed dynamics is an urgent task from both theoretical and practical points of view.
Thanks to the strong algebraic-geometric cell of the laboratory, it is planned to extend the concept of chaos in dynamical systems in the sense of Divani to arbitrary foliations and investigate the problem of the existence of chaos for Cartan foliations, as well as to make advancements in the classification of simple Lie algebras over fields of small characteristic.