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Systems and foliations with a complex structure of limit sets

Priority areas of development: mathematics
Department: Laboratory of Theory and Practice of Decision-Making Support Systems (Nizhny Novgorod)
The project has been carried out as part of the HSE Program of Fundamental Studies.

Purpose: To develop methods of qualitative theory of dynamical systems, foliations and associated algebraic structures in order to obtain new fundamental results. Apply these results to problems of mechanics, neural networks, meteorology, etc.

Methods: methods of qualitative theory of dynamical systems, bifurcation theory, group theory and topological methods.

Results: Therewere obtained results in the following scientific directions:

1)   topological aspects of dynamics and their applications;

2)   classification of dynamical systems and construction of energy functions

3)   bifurcation theory;

4)   dynamics of foliations with additional structures and associated algebraic structures;

5)   dynamical chaos theory and exploration of chaotic dynamics in particular dynamical systems.

Direction 1) topological aspects of dynamics and their applications

1.1 The problem of interrelation between existence of separators in magnetic field in the solar corona and number and type of saddle singularities and charges was solved. Notion of equivalent magnetic fields was introduced. Topological classification of such fields was obtained.

1.2 An example of diffeomorphism of 3-sphere with positive topological entropy was constructed. This diffeomorphisms has a solenoidal basic set with two-dimensional unstable and one-dimensional stable invariant manifolds at every point (consequently, the basic set is neither attractor nor repeller). This diffeomorphism forms the basis for construction of a fast non-dissipative kinematic dynamo with one-dimensional invariant solenoidal set.

1.3 A review of Anosov-Weil theory was written. It summarizes the results obtained during many years (including the results of laboratory staff members V.Z. Grines and E.V. Zhuzhoma) concerning asymptotic properties of preimages of flow trajectories, fiber of foliations, stable and unstable manifolds of points from basic sets of diffeomorphisms, orientable surfaces whose universal cover is either Euclidean plane or Lobachevsky plane was written.

1.4 The restriction of endomorphisms on codimansion one basic set in case when such basic sets are topological submanifolds of an ambient manifold was studied. A criterion for basic set of endomorphism satisfying axiom A to be an attractor was obtained.

1.5 It was numerically shown the existence of an endomorphism of two-torus satisfying axiom A whose nonwandering set has one-dimensional invariant contracting repeller locally homeomorphic to the product of the Cantor set with an interval.

1.6 It was shown that if smooth orientable manifold of topological dimension greater or equal three admits Morse-Smale system without heteroclinic intersections (for Morse-Smale flows absence of periodic trajectories is required), then such manifold is homeomophic to the connected sum of manifolds whose structure is connected with number and type of non-wandering points.

 1.7 Euler characteristic and criterion of orientability of surface obtained from a polygon with an even number of edges were obtained by means of colored graph.1.8 A review of application of dynamical systems to investigation of topology of magnetic fields in conducting medium was written.

1.9 A review of dynamics and classification of structurally stable diffeomorphisms with codimension one basic sets was written.

1.10 A class of homeomorphisms of topological manifolds with regular dynamics was introduced. These homeomorphisms were called Morse-Smale homeomorphisms. For this class an analogue of Smale’s theorem (that was earlier proved by Smale for smooth systems) on interrelation of dynamics and topology of ambient manifold was proved.

1.11 For a class of diffeomorphisms of 3-manifolds with surface dynamics a precise lower bound on a number of non-compact heteroclinic curves was given. It was shown that an ambient manifold of such diffeomorphisms is a fiber bundle over the circle (mapping torus). Moreover, it was shown that any mapping torus admits surface dynamics.

1.12 A class of Morse-Smale flows with non-wandering set consisting of exactly four points (two nodes and two saddles) on an n-sphere was considered. It was shown that for any flow from this class an intersection of invariant manifolds of saddle points is not empty and consists of a finite number of connected components.

1.13 It was shown that two periodic translations on n-torus of the same period are topologically conjugate by means of a countable family of toral automorphisms. Moreover, it was shown that for two fixed topologically conjugated translations every homotopy class of the set of conjugating homeomorphisms consists of at least continuum number of conjugating homeomorphisms.

1.14 It was shown that for any orientation-preserving gradient-like diffeomorphism of a smooth orientable closed surface there exists a dual pair attractor-repeller of topological dimension not greater than one with a characteristic space homeomorphic to 2-torus. As an immediate consequence of this result one has a fact that all separatrices of such diffeomorphisms have the same period.

1.15 It was shown that polar diffeomorphisms, diffeomorphisms with exactly one saddle orbit exist on surfaces of any genus. In this case saddle orbit always has a negative orientation type. Moreover, all possible types of periodic data for such diffeomorphisms were found.

1.16 It was shown that any orientable surface admits an orientation-preserving Morse-Smale diffeomorphism with one saddle orbit. It was also shown that such diffeomorphisms have exactly three node orbits. Moreover, all possible types of periodic data for such diffeomorphisms were found.

Direction 2) classification of dynamical systems and construction of energy functions

2.1 Orientation-preserving Morse-Smale diffeomorphism was constructed according to its abstract scheme for every class of topological conjugacy.

2.2 A complete topological classification of Ω-stable flows on surfaces by means of directed bipartite graphs equipped with four-colored multigraph was obtained. For such four-colored and equipped graphs a polynomial recognition algorithm was constructed. A standard Ω-stable flow on a surface was constructed for every isomorphism class of such graphs.

2.3 A review of results on existence of energy function for discrete dynamical systems and construction of such energy functions for some classes of Ω-stable and structurally stable diffemorphisms of 2 and 3-manifolds was written.

2.4 It is shown that, up to topological conjugation,} the equivalence class of a Morse-Smale diffeomorphism without heteroclinic curves on a 3-manifold is completely defined by an embedding of two-dimensional stable and unstable heteroclinic laminations to a characteristic space.

Direction 3) bifurcation theory

3.1 One of possible scenarios of appearing heteroclinic separators in solar corona was described and constructed. The suggested joining scenario connects a magnetic field with two null-points of different sign whose fan surfaces do not intersect and a magnetic field with two null-points and two heteroclinic curves connecting these null-points.

3.2 A bifurcation between different types of solenoidal basic sets was constructed.

3.3 Some obstructions for existence a simple arc connecting multidimensional Morse-Smale diffeomorphisms of non-simply connected manifolds were found.

3.4 For an arbitrary number of charges (irrespective of their location) and without assuming the potential of the field B, estimates are given that relate the number of charges of a certain type to the number of zero points. For boundary estimates, the topological structure of the magnetic field is described. A bifurcation of the production of a large number of separators is given.

3.4 A smooth arc without bifurcations connecting source-sink diffeomorphisms on two-sphere was constructed.

Direction 4) dynamics of foliations with additional structures and associated algebraic structures

4.1 A review paper on geometry of foliations with transversal linear connection was prepared. The review consists of three parts. The first one is dedicated to automorphism groups of foliations with transversal linear connection in category of foliations. The second one describes results on the equivalence of different approaches to the notion of completeness for a considered class of foliations. The third part contains theorems concerning pseudo-Riemannian foliations that constitute an important class of foliations with transversal linear connection.

4.2 An investigation of pseudo-Riemannian foliations and their graphs was conducted. A criterion for a smooth foliation of an arbitrary dimension on n-dimensional pseudo-Riemannian manifold to be pseudo-Riemannian was proved. Graphs structures of pseudo-Riemannian foliations were described.

4.3 Necessary and sufficient conditions for a Lorentz foliation admitting Eresman connection to be Riemannian foliation were found. A description of a structure of non- Riemannian transversally analytic Lorentz foliations of codimension two with Eresman connection was given.

4.4 By means of Lie algebra deformation theory for Lie algebra of type G2  isomorphisms between known simple 14-dimensional Lie algebras over fields of even characteristic and Lie algebras of Cantor type S or H were construncted.

Direction 5) dynamical chaos theory and exploration of chaotic dynamics in particular dynamical systems

5.1 A review paper on theory of pseudo-hyperbolic strange attractors was prepared. Although such attractors are not structurally stable, since they admit homoclinic tangencies, they as well as hyperbolic attractors belong to the class of real attractors, since homoclinic tangencies do not lead to appearance of stable orbits. It was shown that such attractors can appear in 3-dimensional maps. Examples of pseudo-hyperbolic attractors appearing in generalized Henon maps were given.

5.2 An investigation of spiral chaos in Lotka–Volterra and Rosenzweig-MacArthur systems was conducted. It was shown that strange attractors in such systems appear according to Shilnikov scenario based on homoclinic trajectory to saddle-focus equilibrium with two-dimensional unstable manifold. By means of numerical experiment bifurcation curves corresponding to appearance of homoclinic trajectory to saddle-focus equilibriums in such systems were constructed. New computational methods for exploring dynamical systems with spiral attractors were developed.

5.3 An investigation of appearance of chaotic dynamics in system describing a movement of two vortices undergoing wave disturbance and shift stream was conducted. It was shown that chaos in such system is associated with strange attractors as well as with strongly dissipative mixed dynamics that appears as a result of join of homoclinic attractor with homoclinic repeller. Local and global bifurcations that lead to appearance of mixed dynamics were studied.

5.4 An investigation of scenarios of appearance of mixed dynamics in the oscillator Pikovsky- Topaj model describing interconnection of four connected rotators was conducted. It was shown that in this system mixed dynamics appears in a “soft way” as a result of sequence of local and global symmetry-breaking bifurcations as well as in a “hard way” as a result of heteroclinic contours appearing after the crisis of simple (regular) attractors and repellers.


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