Methodology: methods of the qualitative theory of dynamical systems and topology.
Results of research: during the work on the project results in several scientific fields were obtained. Namely,
In direction 1. Regular dynamics and its application to the study of magnetic fields of conductive flows
- It was found that the structure of the domains in the solar corona is describes by three-colored graph and proved the effectiveness of the distinguishing algorithm of two such graphs.
- It was described a class of gradient-like dynamical systems on surfaces whose topological classification is reduced to the classification of structurally stable systems on the circle, received by A.G. Maier.
- It was proved that Morse-Smale diffeomorphism f on the sphere of dimension greater that 3 whose wandering set NW (f) satisfies the following conditions: i) NW (f) consists of fixed points with a positive type of orientation; ii) stable and unstable manifolds of saddle points of the NW (f) do not intersect, embeds in topological flow.
- There were obtained the topological classification of manifolds admitting a Morse-Smale flows whose non-wandering set consists exactly on three equilibrium states. Namely, it was proved that all the flows from this class defined on two-dimensional and four-dimensional manifolds are topologically equivalent. On manifolds of dimension 8 and 16 the complete topological invariant is defined by embedding of the unstable manifold of the saddle separatrices. On manifolds of other dimensions the flows of described class do not exist.
- There was povided a definition of the coherent equivalence of Morse-Bott energy functions for Morse-Smale flows on surfaces and proved that the coherent equivalence of the energy functions is necessary and sufficient condition for topological equivalence of such flows. The proposed result eliminates the inaccuracy in the proof of the analogous fact by K. Meyer, which was noticed by A.A. Oshemkov and V.V. Charco.
- It was proved that the phase space of omega-stable flows without periodic trajectories on surfaces is cutted by saddle separatrices and t-curves for similar polygonal region. Then each of these dynamical system corresponds to colored graph whose class of isomorphism is complete topological invariant
- It was shown that for 3-diffeomorphisms with a finite hyperbolic non-wandering set and a finite number of modules of stability the complete topological invariant is a scheme that contains information about the geometry of the intersection of separatrices of saddles and about the eigenvalues of saddle points, whose two-dimensional separatrices tangents.
In direction 2. Building energy functions and topological classification of systems with chaotic behavior
- It was proved that any rough 3-diffeomorphism with non-trivial basic sets of dimension 2 is topologically conjugated to locally direct product of Anosov diffeomorphisms and a rough map on the circle.
- It was proved that the class of topological conjugacy of semi-direct product of DA-diffeomorphism on the two-dimensional torus and rough transformation of the circle is completely defined by combinatorial invariants, namely by hyperbolic automorphism of the torus and some a subset of its periodic orbits and by the number of periodic orbits and serial number of the map on the circle.
- There was defined a class of pseudo-coherent diffeomorphisms on three-dimensional manifolds, and was proved than any diffeomorphism of this class is topologically conjugated by locally direct product of pseudo-anosov homeomorphism of the surface and rough map on the circle.
- There was built an energy function (smooth Lyapunov function, the set of critical points coincides with the set of non-wandering points of the system) for omega-stable cascades on surfaces with expanding attractors of codimension 1 and for three-dimensional cascades with two-dimensional non-trivial basic sets;
- There was built an one-parameter family of pairwise topologically non-conjugate endomorphisms that are skew product of expanding endomorphism on the circle and source-sink cascade on the circle.
In direction 3. Theory of bifurcations in systems with regular and chaotic dynamics
- In the discrete Rulkov model of three elements joined by inhibitory connections, there were studied sequent switching of activity. It is shown that in this model there are various types of sequent activity, as well as other modes for the neuron-specific ensembles with a similar topology of connections.
- There were obtained classification of reversing movements observed in the Celtic stones. It was shown that the chaotic behavior of Celtic stone is one of the causes of the multiple reversing movement.
- In the system of four coupled rotators researched by D.Topazh and A.Pikovskii it was founded out that the transition from coservative to dissipative dynamics is associated with the occurrence of bifurcations of loosing of symmetry.
- In the system describing the motion of two point vortices under the influence of acoustic wave there were detected the bifurcations of loosing of symmetry. It is shown that the chaos in the system is associated with no strange attractors, but with mixed dynamics.
- In the system describing the motion of Chaplygin top on a plane without slippage, it was discovered a new chaotic attractor called ring heteroclinic strange attractor.
- For the non-autonomous two-dimensional system of Duffing type slowely periodically depending on time it was shown that in a certain region the dynamics of the system is chaotic. In addition, in this region exists chaotic behavior as well as stability islands (elliptical orbit) of high periods.
In direction 4. The geometry and dynamics of foliations agreed with additional structures
- for Cartan foliations with Ehresmann connectivity was introduced algebraic invariant - Structural Lie algebra -- and was proved that the vanishing of this invariant is a sufficient condition for the existence and uniqueness of the structure of a Lie group of automorphisms in the group of basic Cartan foliation. There were obtained exact estimates of the dimension of this Lie group an built examples;
- it was proved that for any proper Cartan foliation with Ehresmann connectivity there exists an open dense subset of the ambient manifold, all of whose fibers are diffeomorphic to each other, have trivial holonomy group and locally stable in the Reeb sense;
- there were found necessary and sufficient conditions for a smooth foliation on the pseudo-rimanian manifold to be pseudo-rimanian. The structure of graphs of pseudo-rimanian bundles were researched.
- there were proved the equivalence of different approaches to the notion of completeness of foliations with transverse linear connectivity and, in particular, for a transversely affine bundles.