The main task of the project: the creation of new methods for the study of multidimensional systems with non-trivial dynamics. The chaotic behavior of smooth differential equations was discovered at the end of the 19th century; the study of this phenomenon was devoted to the methods of outstanding mathematicians and physicists of the past. Today, the theory of dynamical systems, due to its interdisciplinary nature (from a mathematical point of view, it is both an area of analysis, and a part of geometry, and a section of group theory and, at the same time, probability theory, applications of number theory, etc.) is most intensively in the field of mathematics. The number of applied problems in which dynamic chaos is observed is enormous, and the question of the statistical properties of multidimensional systems with a chaotic problem belongs to the fundamental problems of physics. Nevertheless, the current state of the theory does not provide almost any mathematically rigorously substantiated information about the structure of chaotic dynamics in almost any randomly chosen physical system. The reason is researches focused on devices that have some kind of "convenient" mathematical structure (this or that kind of hyperbolicity, symmetry, etc.), which leads to a description of only specially developed examples or only unstable dynamic modes. On the contrary, the present project is aimed at the study of mathematical structures that most adequately formalize the stable properties of the dynamics of systems of natural origin, and at the creation of a strictly justified corresponding study of the dynamic chaos observed in applied problems.

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Implemented results of research:**** **

As part of the work on the project, a software package which implements high-performance methods for the numerical analysis of multidimensional dynamical systems that arise, in particular, in various applications has been developed. With the help of the developed complex, the following results of the study of applied models were obtained: in the direction associated with the study of models of the dynamics of encapsulated gas bubbles in a liquid, regions with hyperchaotic oscillations of bubbles were identified, the possibility of transition from synchronous oscillations to asynchronous oscillations was established, it was shown that this process is carried out according to the scenario of a bubble transition; in the direction devoted to the study of dynamic systems that simulate the functioning of gene networks, the mechanisms of transition from regular periodic regimes corresponding to in-phase and antiphase modes of activity to chaotic ones are described; In the direction of studying ensembles of interacting neural-like elements with a time-varying connection topology, a new effect has been discovered and described, when the periodic closure and opening of the connection between the elements of the ensemble leads to an exponential increase in energy.

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Education and career development:**** **

The following lecture courses were developed and implemented: "Dynamics of endomorphisms", "Modern theory of dynamic chaos", "Theory of bifurcations of multidimensional systems", "Quantum mechanics for mathematicians".

International schools-conferences were held:

- International student school-conference "Mathematical spring 2020" February 17 - 21, 2020

- International student school-conference "Mathematical spring 2021" March 30 - April 1, 2021

The preliminary defense of the candidate's thesis by E.V. Nozdrinova has passed. on the topic “On classes of stable isotopic connection of gradient-like diffeomorphisms of surfaces”.

Internship at the Saratov National Research State University named after N. G. Chernyshevsky, school "Non-linear days".

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Organizational and structural changes:**** **

Participation in the mathematical workshop with the project "Investigation of periodic data of surface homeomorphisms"

Scientific and educational group "Topological and computational methods in dynamics"

Scientific and educational group "Evolutionary semigroups and their applications".

Cooperation with the laboratory "Laboratory of Theoretical Nonlinear Dynamics" in Saratov on the project "Model and Radiophysical Dynamical Systems: Theory and Experiment".

A collective use center equipped with a heterogeneous computing facility has been created. On this computing facility, a developed software package is deployed, access to which is carried out over a local network. The users of the complex have the opportunity to explore model problems that allow them to improve their understanding of various dynamic phenomena, and to create and study their own problems described by finite-dimensional dynamical systems.

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Collaborations:**** **

Institut de Mathématiques de Bourgogne (France), Moscow State University (Russia), University of Hradec Kralove (Czech Republic), YarGRU (Russia), University of Warwick (Great Britain), SSU (Russia), Imperial College (UK), Ogarev Mordovia State University (Russia), University of Potsdam (Germany), Georgia State University (USA), Tongji University (China), Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation named after V.I. N.V. Pushkov of the Russian Academy of Sciences (Russia): joint research. As a result of the collaborations, more than twenty articles have been published in top-rated journals included, in particular, in the WOS list with quartiles Q1-Q2.

Student exchanges, joint scientific events: within the laboratory, there is a regular student exchange program with the German University of Passau (Germany).

Regular internships for trainees-researchers of the laboratory at the Mathematical Center in Novosibirsk (Russia), at the Saratov State University (Russia) have been established.