#### For citation:

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

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

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.

- Andronov AA, Pontryagin LS. Rough systems. Proceedings of the USSR Academy of Sciences. 1937;14(5):247–250 (in Russian).
- Leontovich EA, Mayer AG. About trajectories determining qualitative structure of sphere partition into trajectories. Proceedings of the USSR Academy of Sciences. 1937;14(5):251–257 (in Russian).
- Leontovich EA, Mayer AG. On a scheme determining the topological structure of the separation of trajectories. Proceedings of the USSR Academy of Sciences. 1955;103(4):557–560 (in Russian).
- Peixoto MM. On the classification of flows on 2-manifolds. In: Dynamical Systems. Proceedings of a Symposium Held at the University of Bahia. 26 July–14 August 1971, Salvador, Brasil. Cambridge, Massachusetts: Academic Press; 1973. P. 389–419. DOI: 10.1016/B978-0-12-550350-1.50033-3.
- Oshemkov AA, Sharko VV. Classification of Morse–Smale flows on two-dimensional manifolds. Sbornik: Mathematics. 1998;189(8):1205–1250. DOI: 10.1070/SM1998v189n08ABEH000341.
- Palis J. A differentiable invariant of topological conjugacies and moduli of stability. Asterisque. 1978;51:335–346.
- Takens F. Heteroclinic attractors: Time averages and moduli of topological conjugacy. Bol. Soc. Bras. Mat. 1994;25(1):107–120. DOI: 10.1007/BF01232938.
- Kruglov V. Topological conjugacy of gradient-like flows on surfaces. Dynamical Systems. 2018;8(1):15–21.
- Kruglov V, Pochinka O, Talanova G. On functional moduli of surface flows. Proceedings of the International Geometry Center. 2020;13(1):49–60. DOI: 10.15673/tmgc.v13i1.1714.
- Kruglov V, Malyshev D, Pochinka O. Topological classification of Ω-stable flows on surfaces by means of effectively distinguishable multigraphs. Discrete & Continuous Dynamical Systems. 2018;38(9):4305–4327. DOI: 10.3934/dcds.2018188.
- Palis J, de Melo W. Geometric Theory of Dynamical Systems: An Introduction. New York: Springer-Verlag; 1982. 198 p. DOI: 10.1007/978-1-4612-5703-5.
- Robinson C. Dynamical Systems: Stability, Symbolic Dynamics, and Chaos. Boca Raton: CRC Press; 1995. 468 p.
- Irwin MC. A classification of elementary cycles. Topology. 1970;9:35–47.
- Smale S. Differentiable dynamical systems. Bull. Amer. Math. Soc. 1967;73(6):747–817. DOI: 10.1090/S0002-9904-1967-11798-1.
- Kruglov VE, Malyshev DS, Pochinka OV. A multicolour graph as a complete topological invariant for Ω-stable flows without periodic trajectories on surfaces. Sbornik: Mathematics. 2018;209(1):96–121. DOI: 10.1070/SM8797.

- 351 reads