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Dynamical systems and their applications (DSA)

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.

Goal of research : Development of methods of the qualitative theory of dynamical systems on manifolds for obtaining  new fundamental results and their application to solving  problems in mechanics, astrophysics, neural networks, meteorology, and others.

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.



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