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Kruglov V. E., Pochinka O. V. Classification of the Morse – Smale flows on surfaces with a finite moduli of stability number in sense of topological conjugacy. Izvestiya VUZ. Applied Nonlinear Dynamics, 2021, vol. 29, iss. 6, pp. 835-850. DOI: 10.18500/0869-6632-2021-29-6-835-850

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Classification of the Morse – Smale flows on surfaces with a finite moduli of stability number in sense of topological conjugacy

Kruglov Vladislav Evgenievich, National Research University "Higher School of Economics"
Pochinka Olga Vitalievna, National Research University "Higher School of Economics"

Purpose. The purpose of this study is to consider the class of Morse – Smale flows on surfaces, to characterize its subclass consisting of flows with a finite number of moduli of stability, and to obtain a topological classification of such flows up to topological conjugacy, that is, to find an invariant that shows that there exists a homeomorphism that transfers the trajectories of one flow to the trajectories of another while preserving the direction of movement and the time of movement along the trajectories; for the obtained invariant, to construct a polynomial algorithm for recognizing its isomorphism and to construct the realisation of the invariant by a standard flow on the surface. Methods. Methods for finding moduli of topological conjugacy go back to the classical works of J. Palis, W. di Melo and use smooth flow lianerization in a neighborhood of equilibrium states and limit cycles. For the classification of flows, the traditional methods of dividing the phase surface into regions with the same behavior of trajectories are used, which are a modification of the methods of A. A. Andronov, E. A. Leontovich, and A. G. Mayer. Results. It is shown that a Morse – Smale flow on a surface has a finite number of moduli if and only if it does not have a trajectory going from one limit cycle to another. For a subclass of Morse – Smale flows with a finite number of moduli, a classification is done up to topological conjugacy by means of an equipped graph. Conclusion. The criterion for the finiteness of the number of moduli of Morse – Smale flows on surfaces is obtained. A topological invariant is constructed that describes the topological conjugacy class of a Morse – Smale flow on a surface with a finite number of modules, that is, without trajectories going from one limit cycle to another. 

This work was supported by the Russian Science Foundation (project 17-11-01041) except section 3 which was funded by RFBR according to the research project No. 20-31-90067. Subsection 4.4 was carried out with the support of the HSE Dynamic Systems and Applications Laboratory, a grant from the Ministry of Science and Higher Education of the Russian Federation, agreement № 075-15-2019-1931.
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