#### Goal of research

The main objective of the project is the development of the theory of dynamical systems and differential equations, the study of related problems of the theory of foliation and group theory, as well as numerical modeling and analytical study of systems with applications to physics, geophysics and engineering.

#### Methodology

In solving the problems of the project, modern methods of the theory of dynamical systems were used, including those created in the Nizhny Novgorod school of dynamical systems, founded by academician A. A. Andronov and for a long time led by L. P. Shilnikov. Topological and geometric methods of studying global properties, analytical methods of studying local properties of dynamical systems on manifolds and methods of the theory of global bifurcations were used to achieve the goals in the direction of qualitative research of dynamics. The approach used by Conley in proving the fundamental theorem of the theory of dynamical systems of choosing isolate dual pairs of attractors and repellers will be the basis of new methods of studying the global dynamics of cascades and constructing energy functions for them. Along with this, the Norton and Pugh method of characterization of the set of critical points of the smooth energy function was used. The project also created new mathematical methods for the study of multidimensional dynamical systems: the theory of pseudo-hyperbolic strange attractors of dissipative systems was developed, new global bifurcations of dissipative and Hamiltonian systems were studied, the concept of mixed dynamics for reversible systems was developed.

#### Empirical base of research

Modern methods of parallel programming for multicore and multiprocessor heterogeneous systems with the help of hardware and software architecture CUDA were applied for numerical study of dynamic systems. Python was used to visualize the calculations.

#### Results of research

All studies of the project are united by the general idea of developing a qualitative theory of dynamical systems. The dynamics is studied both in the classical sense, for one-parameter families of manifold transformations, and for algebraic-geometric structures and infinite-dimensional evolutionary equations. A special place in the project is occupied by applications of the theory of dynamical systems to the study of specific models of physics, biology, engineering.

Topological aspects of dynamics take a large part of the research. Understanding a dynamical system as a continuous or discrete family of maps of a topological manifold leads to a close connection of qualitative properties of the system with the topology of the carrier manifold and key invariant subsets. Thus, the effect of wild knotting of saddle separators of a three-dimensional cascade, discovered By D. Pickston in 1977, showed significant differences between cascades and flows. It became clear that even regular diffeomorphisms, starting from dimension 3, are much more complex than their continuous analogues. Thus, diffeomorphisms in general do not admit the Morse energy function and do not embed into the topological flow. The participants of this project, O. V. Pochinka and D. D. Shubin show that the suspension over a nontrivial Pixton diffeomorphism is a non-singular flow whose two-dimensional invariant manifold of the saddle periodic trajectory are not locally flat.

V. Z. Grines, O. V. Pochinka and E. Y. Gurevich obtained the solution of the J. Palis’s problem on the embedding of Morse-Smale diffeomorphisms in topological flows for a class of Morse-Smale diffeomorphisms without heteroclinic intersections given on a sphere of dimension four and greater. J. Palis obtained the necessary conditions of the embedding of Morse-Smale diffeomorphisms of a closed manifold M^n of dimension n in a topological flow and proved that these conditions are also sufficient for the case n=2. For the case n=3, the possibility of wild embedding of closures of separatrices of saddle periodic points is an additional obstacle for embedding of Morse-Smale cascades in the topological flow. During the work on the project, it was shown that for Morse-Smale diffeomorphisms without heteroclinic intersections given on the sphere S^n, n>3, such obstacles do not exist and the Palis conditions are again sufficient.

The project continues research of the possibility of the construction of energy functions for different systems. Thus, the project participants A. E. Kolobyanina, V. E. Kruglov and O. V. Pochinka constructed Morse energy functions for omega-stable flows without limit cycles on surfaces. It is also shown that for topological flows with a finite hyperbolic chain-recurrent set on the surface there exists a continuous Morse energy function. For chaotic three-dimensional diffeomorphisms with source-sink dynamics project participants M. K. Barinova, V. Z. Grines and O. V. Pochinka proved the existence of a smooth energy function.

In the traditional for the Nizhny Novgorod school of dynamical systems direction of topological classification, the participants of the project also obtained new results. After years of study V. Z. Grines and O. V. Pochinka, co-authored with the French mathematician C. Bonatti completed the topological classification of Morse-Smale diffeomorphisms on three-dimensional manifolds. V. E. Kruglov, O. V. Pochinka and D. D. Shubin showed the coincidence of topological conjugacy and equivalence classes for gradient-like flows without heteroclinic intersections on the n-dimensional sphere, and also modeled the dynamics of such systems by means of a root tree, thereby solving the realization problem, which is an integral part of the complete classification. A. I. Morozov and O. V. Pochinka made an important step to the combinatorial solution of the classification problem of surface Morse-Smale diffeomorphisms with orientable heteroclinics. Namely, the finiteness of the number of heteroclinic orbits for such systems is proved by the factorization method.

V. Z. Grines, E. D. Kurenkov classified orientation-preserving a-diffeomorphisms of orientable surfaces of the genus of a larger unit containing one-dimensional spaciously arranged perfect attractors. It is established that the question of topological classification of restrictions of diffeomorphisms on such basis sets is reduced to the problem of topological classification of pseudo-Anosov homeomorphisms with marked set of saddle singularities. In particular, the proof of announced by A. Yu. Zhirov and R. V. Plyikin topological classification of A-diffeomorphisms on the surfaces whose non-wandering set consists of a one-dimensional spaciously located attractor and zero-dimensional sources is given.

V. Z. Grines, E. Ya. Gurevich and E. D. Kurenkov determined a class of gradient-like flows with surface dynamics for which separatrices of saddle equilibrium states have a unambiguous asymptotic behavior. Systems with surface dynamics naturally arise as models of processes having at least one periodic variable. During the work on the project it is shown that if the separatrices of saddle equilibria of such systems have an additional property called unambiguous asymptotic behavior, then the classes of topological equivalence of such systems are described by model flows defined as skew products of flows on surface and on the circle. The topology of the carrier manifold of such systems is also studied.

Within the framework of the project, significant advances in the theory of bifurcations were obtained. It was found a partial solution to the 33rd Palis-Pugh problem about the existence of a stable path connecting two structurally stable systems. V. Z. Grines and O. V. Pochinka constructed a stable path in the space of diffeomorphisms of a three-dimensional torus connecting an arbitrary structurally stable diffeomorphism with a two-dimensional stretching attractor with a DA-diffeomorphism. E. V. Nozdrinova and O. V. Pochinka obtained a complete description of stable isotopic classes of gradient-like diffeomorphisms of a two-dimensional sphere.

N. I. Zhukova investigated completely geodesic foliations F of arbitrary dimension on n-dimensional pseudo-Riemannian manifolds, the metric on the layers of which does not degenerate, and the additional orthogonality distribution is the Ehresmann connectivity. The generally accepted graph G(F) of such a foliation is, in general, a non-Hausdorff manifold, so there was studied the graph GD(F) of a foliation F with Ehresmann connectivity D, introduced by N. I. Zhukova earlier, which is always Hausdorff manifold. It is proved that on the graph GD(F) a pseudo-Riemannian metric is defined, with respect to which the induced foliation and the simple foliations formed by the layers of canonical projections are quite geodesic. It is proved that the layers of induced foliation on the studied graph are reducible Riemannian manifolds and a description of their structure is given. An application to graphs of parallel foliations on nondegenerate reducible pseudo-Riemannian manifolds is considered. It is shown that any foliation obtained by the suspension over the homomorphism of the fundamental group of a pseudo-Riemannian manifold belongs to the studied class of foliations.

N. I. Zhukova together with A.V. Bagaev obtained the following results on the theory of orbifolds. According to Chern's hypothesis, the Euler characteristic of a closed affine manifold must vanish. The equivalence of this conjecture to the following conjecture for orbifolds is proved: the Euler -- Sataki characteristic of a compact affine orbifold is zero. The conditions under which the compact affine orbifold has zero Euler-Sataki characteristic are found. Examples are constructed.

E. I. Yakovlev together with T. A. Gonchar investigated the causal properties of stratified space-time manifolds. Any such manifold is the space of the principal bundle on which the Lorentz metric and temporal orientation invariant with respect to the action of the structural group are given. In the case of spatial similarity of layers, it induces the same constructions on the basis of the bundle. The following conditions of causality were studied: chronology, causality, stable and strong causality, global hyperbolicity. It is proved that if the base space-time satisfies one of these conditions, then this is also true for the stratified space-time. Examples showing that in general the converse statement is incorrect are constructed.

E. I. Yakovlev together with V. Y. Epifanov solved problems of computational topology on triangulated closed three-dimensional manifolds. Simplicial homology and cohomology groups modulo 2 were used. Two effective algorithms for calculation of indices of intersection of cycles of dimensions 1 and 2 are developed and strictly justified. With the help of these algorithms, it is also possible to construct bases of cohomology groups by given cycles that generate a basis of the homology group of additional dimension.

A.V. Vedenin constructed historically the first example of rapidly converging Chernov’s approximations to the solution of the evolutionary equation in the model case of the equation of thermal conductivity on the line. V. D. Galkin and E. Y. Karatetskaya have written a computer program that allows numerically investigate the convergence rate of Chernov’s approximation. I. D. Remizov introduced the concept of approximation subspaces arising in connection with Chernov's theorem; the simplest properties of these subspaces are studied; on the example of the semigroup of shifts, it is shown that the convergence in Chernov's theorem can be arbitrarily fast or arbitrarily slow. In addition, I. D. Remizov constructed (and interpreted as Feynman formulas with generalized functions under the sign of the integral) Chernov’s approximations for solving a parabolic partial differential equation with variable coefficients with respect to a function defined on a multidimensional real linear space. T-his is the first step to the study of diffusion on non-compact manifolds.

A significant part of the results of the project is related to the development of the theory of pseudo-hyperbolic attractors, quasi-attractors, mixed dynamics and the application of the results for the study of applied problems. Pseudo-hyperbolic attractors are understood as a wide class of attractors whose chaotic dynamics is preserved during perturbations of the system (changing of parameters). The following results are obtained in this direction: the bifurcation boundaries of the existence of a discrete pseudo-hyperbolic Lorentz attractor in a nonholonomic model of Celtic stone are studied; numerical methods to verify of pseudohyperbolicity of strange attractors of multidimensional flows and diffeomorphisms are developed, new types of pseudohyperbolic attractors are discovered; in the three-dimensional flows possible classes of pseudohyperbolic homoclinic attractors are determined, and proved that this class contains only attractors of Lorenz-type and saddle-attractors of Shil'nikov. Unlike the pseudo-hyperbolic attractors, quasiattractors either contain stable periodic trajectories (with small basins of attraction), or such trajectories occur at arbitrarily small perturbations. The following results are obtained in this direction: the classification of homoclinic quasiattractors of three-dimensional flows by the type of equilibrium state belonging to the attractor is constructed; the systems demonstrating new types of quasiattractors of three-dimensional flows (Shilnikov saddle attractor, Rovella attractor) are proposed); scenarios of the emergence of hyper-chaotic quasi-attractor in multidimensional flows are described; and the chaotic dynamics of the system describing the interaction of two gas bubbles in a liquid and having a great practical importance in medical applications (ultrasound studies, noninvasive treatment, smart drug delivery) is investigated; new mechanism is proposed for the sudden extinction of neural activity associated with the emergence in neural ensembles of homoclinic quasi-attractor of Shilnikov’s type; the influence of electrical and memristor connections on the dynamics of an ensemble of two identical Fitzhugh-Nagumo elements is studied. In the direction of the study of mixed dynamics - a new third type of chaos is characterized by the inseparability of dissipative elements of dynamics from conservative ones, the following results were obtained: a new phenomenon of fusion of a strange attractor and a repeller leading to the formation of mixed dynamics was discovered; new mechanisms of transition from conservative and dissipative dynamics to mixed ones were discovered in the nonholonomic model of Celtic stone.