Abstract
In this paper, we consider regular topological flows on closed \(n\)-manifolds. Such flows have a hyperbolic (in the topological sense) chain recurrent set consisting of a finite number of fixed points and periodic orbits. The class of such flows includes, for example, Morse – Smale flows, which are closely related to the topology of the supporting manifold. This connection is provided by the existence of the Morse – Bott energy function for the Morse – Smale flows. It is well known that, starting from dimension 4, there exist nonsmoothing topological manifolds, on which dynamical systems can be considered only in a continuous category. The existence of continuous analogs of regular flows on any topological manifolds is an open question, as is the existence of energy functions for such flows. In this paper, we study the dynamics of regular topological flows, investigate the topology of the embedding and the asymptotic behavior of invariant manifolds of fixed points and periodic orbits. The main result is the construction of the Morse – Bott energy function for such flows, which ensures their close connection with the topology of the ambient manifold.
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Funding
The work on Section 3 was partially supported by the Laboratory of Dynamical Systems and Applications NRU HSE, by the Ministry of Science and Higher Education of the Russian Federation (ag. 075-15-2019-1931) and by the Foundation for the Advancement of Theoretical Physics and Mathematics BASIS (project 19-7-1-15-1); the work on Section 4 was funded by RFBR, project number 20-31-90069.
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MSC2010
37D05, 37B20, 37B35
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Pochinka, O.V., Zinina, S.K. Construction of the Morse – Bott Energy Function for Regular Topological Flows. Regul. Chaot. Dyn. 26, 350–369 (2021). https://doi.org/10.1134/S1560354721040031
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DOI: https://doi.org/10.1134/S1560354721040031