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DEVELOPMENT OF THE QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS:

– Development of the theory of multivariate dynamic systems with homoclinic and heteroclinic Poincare trajectories;
– Development of methods for researching multivariate nonlinear dynamic systems with complex dynamics;
– Development of the theory of strange attractors in multivariate dissipative systems, studying the structure of wild hyperbolic sets;

– Studying the structure of multivariate integrated Hamiltonian systems, development of the theory of global bifurcations in Hamiltonian systems, research into the structure of the systems close to integrable Hamiltonian ones;

– Development of the theory of non-autonomous systems with homoclinic sequences, the theory of smooth cascades on surfaces, the theory of hyperbolic attractors of codimension one on closed manifolds, the theory three-dimensional Morse-Smale diffeomorphisms;

– Research into localized solutions of the equations with partial derivatives.

Main results:

Fundamental results in the field of the qualitative theory, the theory of bifurcations and strange attractors of multivariate nonlinear dynamic systems have been obtained, the groundwork has been laid for the mathematical theory of dynamic chaos . These results are unparalleled in the world and testify to the priority positions in the theory of non-local bifurcations and strange attractors. They are universally recognized by experts in the field of the theory of differential equations, the theory of dynamic systems and nonlinear dynamic analysis.
- The theory of bifurcation of multivariate dynamic systems with homoclini с and hetroclinic tangencies has been developed;
- Dynamic properties of systems from multidimensional Newhouse regions have been established. It has been proved that small analytical perturbations of systems with square-law homoclinic tangency can lead to homoclinic tangencies of an arbitrarily high order. Bifurcations of two-dimentional mappings retaining their area with homoclinic tangencies have been studied. The existence of Newhouse regions with mixed dynamics near three-dimensional mappings with non-coarse heteroclinic contours has been proved;
- For the case of two-dimentional mappings with homoclini с tangencies, bifurcation boundaries of hyperbolicity intervals have been described. Bifurcations of three-dimensional mappings with homoclini с tangencies have been described for the cases when the pseudohyperbolicity condition is violated. A deion is given of bifurcation phenomena during he transition from spiral quasiattractors to wild spiral attractors;
- For three-dimensional Henon mappings of various types, bifurcation diagrams of fixed points and period two points have been constructed, hyperbolic dynamics with the proof of the existence of three-dimensional Smale horseshoes of various differentiated types has been investigated, chaotic dynamics, including the proof of the existence of wild strange attractors of the Lorenz type, has been studied. Existence of Newhouse regions with a countable set wild Lorenz attractors near three-dimensional mappings with non-coarse heteroclinic contours has been proved;
- The advanced theory and methods of research of multivariate systems with complex dynamics have been applied to the research of concrete basic models of natural sciences – when studying package activity in a number of models of neurodynamics, when investigating the modes and stability of synchronization in the networks of periodically and chaotically connected dynamic systems, when studying the resonant phenomena and irregular dynamics in pendular systems, when establishing the source of travelling fronts, pulses and droplets in the generalized Swift-Hohenberg equation, when solving some specific problems in radiophysics, mechanics, atmosphere dynamics, photo- and electrochemistry, etc.

The methods of the theory of nonlinear dynamic systems and bifurcations theory that have been developed serve as the main tool for studying dynamic models from various areas of natural sciences and engineering (in particular, for the analysis of processes of instability development in equilibrium and periodic modes, alteration of behaviour with the changes in parameters, for explaining complex chaotic behaviour). Application of these methods allows an adequate model to be produced for describing nonlinear processes developing in complex objects and systems, including the systems with chaotic dynamics.

Leading experts:

- Leonid Pavlovich Shilnikov, С and.Sc. (Physics and Mathematics), Professor;
- Lev Mikhailovich Lerman, D.Sc. (Physics and Mathematics), Professor;
- Sergey Vladimirovich Gonchenko, D.Sc. (Physics and Mathematics).

Main partners:

- Mathematical institute of the Russian Academy of Sciences,
- Institute of applied mathematics of the Russian Academy of Sciences,
- Institute of mathematics of the Siberian Branch of the Russian Academy of Sciences,
- Institute of radio engineering and electronics of the Russian Academy of Sciences,
- State Universities of Moscow, St.-Petersburg, Kazan, Saratov, Yaroslavl.

Key projects (sources of financing):

- The grant of the President of the Russian Federation for the State support of leading scientific schools of the Russian Federation , NSh-9686.2006.1.
- Projects of the Russian Foundation for Basic Research 01-01-00905а, 02-01-00273а, 05-01-00558а, 06-01-72023 MNTIа, 06-01-03010а, 08-01-00083а.
- Projects of the Program « Universities of Russia », UR.03.01.033, UR.03.01.015, UR.03.01.180.
- Projects of ESPRIT (EC), INTAS (EC), CRDF (USA).
- The analytical departmental target program of the Federal Agency for Education «Development of scientific potential of higher education institutions», Research Project 1.51.06.

Main publications:

- Shilnikov L.P., Shilnikov A.L., Turaev D.V., Chua L. Methods of the qualitative theory in nonlinear dynamics. Part 1. – Moscow-Izhevsk: Institute of Computer Studies. 2004. 416 pp.
- Gonchenko S.V., Shilnikov L.P., Turaev D.V. On dynamical properties of multidimensional diffeomorphisms from Newhouse regions // Nonlinearity, 2008. 21 (5). Pp. 923-972.
- Turaev D.V., Shilnikov L.P. Pseudo-hyperbolicity and the problem of periodic disturbance of Lorentz-type attractors // Reports of the Academy of Sciences . 2008. 418 (1). Pp. 23-27.
- Gonchenko S.V., Li M.C., Malkin M.I. Generalized Henon maps and Smale horseshoes of new types // International Journal of Bifurcation and Chaos, 2008. 18 (10). Pp. 1-24.
- Lerman L.M., Markova A.P., On stability at the Hamiltonian Hopf Bifurcation // Regular and Chaotic Dynamics. 2008. 17 (6).

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