Master
2020/2021

## Research seminar "Modern theory of dynamical systems"

Type:
Compulsory course (Mathematics)

Area of studies:
Mathematics

Delivered by:
Department of Fundamental Mathematics

When:
1 year, 1-4 module

Mode of studies:
offline

Open to:
students of one campus

Instructors:
Vyacheslav Grines

Master’s programme:
Mathematics

Language:
English

ECTS credits:
5

### Course Syllabus

#### Abstract

The course is devoted to the description of the current state of the theory of dynamic systems from the point of view of topological classification of important classes of systems with regular and chaotic dynamics arising in the modeling of various processes of natural science.

#### Learning Objectives

- To show the current state of research in the field of dynamic systems, the beginning of which dates back to the classical works of A. Poincare, A. M. Lyapunov, A. A. Andronov and his school of nonlinear oscillations, which determined many achievements of specialists in the direction of studying the stability and theory of bifurcations of dynamic systems for many years to come.
- To explain the phenomenon of "hyperbolic revolution", begun in the works of D. V. Anosov and S. Smale, and the associated cascade of works in the field of topological classification of dynamic systems on manifolds with complex dynamics.
- To give an idea of the methods of the qualitative theory of dynamic systems, closely related to the study of the topology of the phase spaces on which they are defined.

#### Expected Learning Outcomes

- understanding of dynamics on the circle
- Know calssification theorem for cascades on surfaces
- Understanding of the interrelation between dynamics and topology of ambient manifold. T
- Understanding resuluts on topological structure of multidimensional manifolds (dimensions greater than three) that admit Morse-Smale flows and cascades without heteroclinic intersections of saddle periodic points with Morse index 1 or n-1.

#### Course Contents

- Theme 1. Dynamics of one-dimensional mappings.Topological classification of homeomorphisms of a circle with a finite nonwandering set. Periodic circle mappings and their classification. Poincaré rotation number and classification of transitive homeomorphisms. The construction of circle homeomorphisms with a nowhere dense nonwandering set and their topological classification. Denjoy theorem. Endomorphisms of the circle, which are local homeomorphisms. Classification of expanding maps. The concept of chaotic dynamics. Three attributes of chaotic dynamics.
- Theme 2. Classification of flows, foliations and cascades on surfaces with regular and chaotic dynamics.Topological classification of two-dimensional manifolds. Vector fields on surfaces, equilibrium states and their index. The Euler-Poincaré-Hopf formula on the relationship between the indices of equilibrium states of a vector field and the Euler characteristic. Flows on a torus without equilibrium states. Asymptotic behavior of the preimages of the trajectories on the universal covering. Poincaré rotation number in the interpretation of A. Weil. Topological classification of flows without equilibrium states on a torus. Flows on surfaces of negative Euler characteristic. The role of the Lobachevsky plane in the description of flows and foliations on surfaces of negative curvature. Homotopy class of rotation (Aranson-Grines) and topological classification of transitive flows on surfaces. The concept of geodesic laminations and the representation of minimal sets of flows by geodesic laminations. Two-webs on surfaces and their topological classification. Morse-Smale flows on surfaces. Topological classification of structurally stable flows without closed trajectories by means of Peixoto graphs. Cascades of Morse-Smale. Topological classification of gradient-like cascades by means of flows without closed trajectories and periodic mappings. Diffeomorphisms of surfaces with a countable set of periodic points. The concept of a homoclinic trajectory. Classification of Anosov diffeomorphisms. Axiom A. S. Smale, and the classification of basiс sets.
- Theme 3. The topology of three-dimensional manifolds and a description of structurally stable flows and cascades, the dynamic properties of which are most closely related to the topological characteristics of the carrier phase space.Representation of three-dimensional manifolds in the form of a connected sum of simple manifolds (Milnor-Kneser theorem). Heegaard splitting of a three-dimensional manifold. The concept of the genus of Heegaard. The concept of heteroclinic trajectories of Morse-Smale flows on 3-manifolds. Representation of a manifold admitting a Morse-Smale flow without closed trajectories in the form of a connected sum of finitely many copies of the product of the two-dimensional sphere and the circle. A sufficient condition for the existence of heteroclinic flow trajectories, the concept of a separator in the theory of magnetic hydrodynamics. Representation of a manifold admitting a Morse-Smale flow without closed trajectories in the form of a Heegaard splitting with a Heegaard surface whose genus is determined by the number and type of equilibrium states of a given flow. A sufficient condition for the existence of a closed trajectory of an arbitrary Morse-Smale flow expressed in terms of the Heegaard genus of the supporting manifold. Morse-Smale diffeomorphisms of three-dimensional manifolds with invariant manifolds of saddle periodic points whose closures are wildly embedded in the ambient manifold. Topological invariants describing wild embeddings of separatrix closures. Knot invariant of Bonatti-Grines. Topological classification of diffeomorphisms from the Pexton class. Classification of three-dimensional manifolds which admit structurally stable diffeomorphisms whose nonwandering set has dimension three or consists of basic sets of dimension two.
- Theme 4. Topological structure of multidimensional manifolds (dimensions greater than three) that admit Morse-Smale flows and cascades without heteroclinic intersections of saddle periodic points with Morse index 1 or n-1.The topological structure of manifolds admitting rough flows with a nonwandering set consisting of three points. The concept of projective-like manifolds. The concept of a generalized Heegaard splitting of a manifold of dimension n> 3. The structure of the Heegaard hypersurface of a manifold admitting a Morse-Smale flow without closed trajectories, all saddle equilibrium states have a Morse index of 1 or n-1. The concept of embedding Morse-Smale diffeomorphism in a topological flow. Necessary Palis conditions and finding of sufficient conditions for embedding. Topological classification of Morse-Smale cascades on manifolds of dimension n> 3 under the assumption that the Morse index of periodic points belongs to the set {0,1, n, n-1} and the invariant manifolds of saddle periodic points do not intersect. A topological classification of structurally stable diffeomorphisms of the n-dimensional torus (n> 3), nonwandering sets of which contain basic sets of dimensions n and n-1.