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Dynamics in Siberia - 2019
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Topological conjugacy of gradient-like flows on В. Е. Круглов |
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Аннотация: Gradient-like flows are continuous dynamical systems whose non-wandering set consists of a finite number of hyperbolic fixed points. Their invariant manifolds cross each other transversally. Depending on research goals there are two important things: qualitative behaviour of the system (i.e. partition of a manifold into trajectories) and moving along the trajectories by the time. In dynamical systems theory topological equivalence is an existence of a homeomorphism sending trajectories of one flows into trajectories of another one preserving direction of moving; if such a homeomorphism preserves time of moving along the trajectories, then it is called topological conjugacy of flows. Searching for invariant determining topological equivalence class for a system is topological classification. The non-wandering set is a finite. Hence, the problem of topological classification may be reduced to a combinatorial one. First time it was done by E. Leontovich and A. Mayer in [2], [3] for classification of flows with finite number of singular trajectories on 2-dimensional sphere. These results were developed in researches by M.Peixoto [5], A.Oshemkov, V.Sharko [4], S.Pilyugin [6], A.Prishlyak [7], where similar problem was solved for Morse–Smale flows on closed manifolds of dimensions 2,3 and higher. These works were dedicated to topological equivalency. In [1] there is proved that topological equivalent flows on surfaces are also conjugate, hence, all equivalence results are also true for conjugacy. In our work we obtained similar result for class Acknowledgements. The work was done in collaboration with O.Pochinka and D.Malyshev with support of Russian Science Foundation, project No. 17-11-01041. References [1] Kruglov V. Topological conjugacy of gradient-like flows on surfaces // Dinamicheskie sistemy. 2018. V. 8. no. 36. 15–21. [2] Leontovich E.A., Mayer A.G. On trajectories determining qualitative structure of sphere partition into trajectories // Doklady Akademii nauk SSSR. 1937. V. 14. no. 5. 251–257. [3] Leontovich E.A., Mayer A.G. On scheme О схеме, determining topological structure of partition into trajectories // Doklady Akademii nauk SSSR. 1955. V. 103. no. 4. 557–560. [4] Oshemkov A.A., Sharko V.V. On classification of Morse–Smale flows on 2-dimensional manifolds // Matematicheskiy sbornik. 1998. V. 189. no. 8. 93–140. [5] Peixoto M. On the classification of flows on two manifolds // Dynamical systems Proc. 1971. [6] Pilyugin S.Yu. Phase diagrams determining Morse–Smale systems without periodic trajectories on spheres // Differencial’nye uravneniya. 1978. V. 14. no. 2. 245–254. [7] Prishlyak A.O. Morse–Smale vector fields without closed trajectories on three-dimensional manifolds //Matematicheskie zametki. 2002. V. 71. no. 2. 254–260. Язык доклада: английский |