**Scientific results:**

The main goal of the project is to create new methods of research of systems with a nontrivial dynamics. The chaotic behavior of solutions of smooth differential equations was discovered already in the late 19^{th} century; efforts of many outstanding mathematicians and physicists of the past were devoted to the research of this phenomenon. As of today, the theory of dynamic systems, thanks to its interdisciplinary nature (from the mathematical standpoint it is at the same time a domain of analysis, a part of geometry and a branch of the group theory and, simultaneously, probability theory, with applications to number theory etc.) is one of the most steadily developing domains of mathematics. The number of applied problems in which dynamic chaos can be observed is vast and the question of statistical properties of multidimensional systems with chaotic dynamics is one of the fundamental problems of physics. Nevertheless, the modern state of the theory does not give almost any mathematically strictly substantiated information on the structure of the chaotic dynamic in practically any randomly selected physical system. The cause is the focus of the majority of researchers on systems that possess some «convenient» mathematical structure (one or another variety of hyperbolicity, symmetry etc.), which leads to description of only specifically prepared examples or only unstable dynamic modes. This project, to the contrary, is aimed at researching mathematical structures that formalize the main persistent properties of the dynamics of systems of natural origin in the most adequate way, as well as at creating and strictly substantiating corresponding methods for studying dynamic chaos found in applied problems.

In the theory of systems on manifolds and the theory of strange attractors, the following results have been achieved:

- we researched flows that satisfy Smale's axiom A on an arbitrarily oriented 3-manifold. It has been demonstrated that two-dimensional basic sets of such flows are either hyperbolic attractors or hyperbolic repellers;
- a complete description has been provided for the structure of the basins of attraction of such attractors
- a complete description has been provided for the topological structure of orientable 2-manifolds that allow А-diffeomorphisms with a specified number of one-dimensional basic sets;
- we have provided an exhaustive topological classification of non-singular Morse–Smale flows with two periodical trajectories of manifolds of arbitrary dimensionality;
- it has been demonstrated that for manifolds that allow such flows there exist from one to three classes of topological equivalence of such flows, depending on the type of the manifold;
- we obtained a classification of Morse–Smale diffeomorphisms of 2-manifold with respect to stable isotopism in the Newhouse–Palis–Takens sense.

In the theory of bifurcations, holomorphic dynamics and the theory of reversible systems:

- our researchers have proposed a scenario that allows to arbitrarily change the homotopy type of a closure of an unstable manifold of a saddle periodic point for polar diffeomorphisms of a two-dimensional torus;
- we have studied the structure of the closure of the set of critical points of multipliers of a family of quadratic maps of the complex plane;
- our researchers have studied reversible non-conservative perturbations of cubic Hénon map. In particular, we researched rearrangements related to 1:3 resonance. It was demonstrated that the destruction of this resonance leads to bifurcations of symmetry destruction and to the birth of a new mixed dynamic (intersection of the attractor and the repeller) due to the formation of nontrivial heteroclinic cycles. The results were generalized to the case of resonances of arbitrary odd order.

In the domain of applications to problems of mathematical physics:

- our researchers have obtained universal topological ratio between the number of null points and magnetic field points for branching configurations of separators connecting sources and null pints that are responsible for the structure of anemone-like Solar flare;
- we have studied rearrangements of sine waves in modified Korteweg–De Vries equations with third- and fifth-order nonlinearities. In the limit of null dispersion we provided analytical expressions for the time and level of wave breaking. We provided a description of the main laws of inelastic interaction of soliton-like solutions. The dynamics of a gas of interacting breathers has been studied, including irregular waves and turbulence, as well as the statistical properties of obtained wave solutions. We found waves of extreme amplitude;
- our researchers have proposed an efficient procedure for reconstructing the phase of a oscillator system by iterative application of Hilbert transform. We demonstrated that this method significantly improves the quality of reconstruction for high-frequency oscillations.

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Implemented results of research:**** **

As part of the project, we have developed a software complex that implements high-performance methods of numerical research of multidimensional dynamic systems. Using the developed complex, we have obtained the following results of our research of applied models: in the field of models of the dynamics of encapsulated gas bubbles in fluid we found regions with hyperchaotic oscillations of bubbles, established the possibility of transfer from synchronous oscillations to asynchronous ones, it was shown that this process is performed according to the bubble transition scenario; in the area of dynamic systems modeling the functioning of gene networks we have described the mechanisms of transition from regular periodic modes corresponding to in-phase and anti-phase modes of activity, to chaotic ones; in the domain of ensembles of interacting neuron-like elements with topology of connection changing over time we have found and described a new effect, when periodic opening and closing of the link between the elements of the ensemble leads to an exponential growth of energy; in hydrodynamic models describing convection in a layer of liquid we determined the existence of pseudo-hyperbolic attractors, as well as quasi-attractors existing on sets of values of parameters that have a positive Lebesgue measure.

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Education and career development:**** **

Two Candidate of Sciences dissertations have been prepared and defended, as well as one Doctor of Sciences dissertation.

We have developed and implemented the lecture courses «Dynamics of endomorphisms», «Modern theory of dynamic chaos», «Theory of bifurcations of multidimensional diffeomorphisms», «Quantum mechanics for mathematicians», «Applied theory of dynamic systems», «Introduction to one-parameter semigroups of operators and linear evolution equations», «Dynamic systems and applications», «Additional chapters of the qualitative theory of dynamic systems, geometry and real analysis», «Foundations of the theory of oscillations».

In 2021–2022, under the supervision of the leading scientist six employees of the Laboratory completed internships at a laboratory at Imperial College London (United Kingdom) on the topic of the project.

We have conducted international schools and conferences:

- International student school and conference «Mathematical spring 2020» 17–21 February 2020
- International student school and conference «Mathematical spring 2021» 30 March–1 April 2021

Yelena V. Nozdrikova has passed the preliminary defense of her Candidate of Sciences dissertation «On classes of stable isotopic connectivity of surfaced gradient-like diffeomorphisms».

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Organizational and structural changes:**** **

- Participation in a mathematics workshop with the project «Research of periodic data of surface homomorphisms».
- Research and education group «Topological and computational methods in dynamics».
- Research and education group «Evolutionary semigroups and their applications».
- Collaboration with the Laboratory of Theoretical Nonlinear Dynamics in Saratov on the project «Model and radiophysical dynamic systems: theory and experiment».
- We have created a center for the collective use of scientific equipment that features a heterogeneous computing device. On this facility we deployed the developed software complex, which can be accessed via the local network. Users of the complex have the ability to research model problems, allowing to enhance the understanding of various dynamic phenomena and to create and research their own problems described with finite-dimensional dynamic systems.

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Collaborations:**** **

Institut de Mathématiques de Bourgogne (France), Moscow State University (Russia), University of Hradec Kralove (Czech Republic), Yaroslavl State University (Russia), University of Warwick (United Kingdom), Saratov State University (Russia), Imperial College London (United Kingdom), Ogarev Mordovia Swtate University (Russia), University of Potsdam (German), Georgia State University (USA), Tongji University (China PR), Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation of the Russian Academy of Sciences (Russia), South Ural State University (Russia), Space Research Institute of the Russian Academy of Sconces (Russia), Jawcobs University Bremen (Germany), University of Barcelona (Spain): joint research. As a result of the collaboration, we have published more than twenty articles in highly-ranked Q1 and Q2 a journals according to the Web of Science database.

The Laboratory has a student exchange program with the University of Passau (Germany).

We have established regular internships for our researchers at Mathematical Center in Akademgorodok and Saratov State University.