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Uspekhi Mat. Nauk, 2019, Volume 74, Issue 1(445), Pages 41–116 (Mi umn9855)  

Classification of Morse–Smale systems and topological structure of the underlying manifolds

V. Z. Grines, E. Ya. Gurevich, E. V. Zhuzhoma, O. V. Pochinka

National Research University "Higher School of Economics", Nizhnii Novgorod

Abstract: Morse–Smale systems arise naturally in applications for mathematical modelling of processes with regular dynamics (for example, in chains of coupled maps describing diffusion reactions, or in the study of the topology of magnetic fields in a conducting medium, in particular, in the study of the question of existence of separators in magnetic fields of highly conducting media). Since mathematical models in the form of Morse–Smale systems appear in the description of processes of various nature, the first step in the study of such models is to distinguish properties independent of the physical context but determining a partition of the phase space into trajectories. The relation preserving the partition into trajectories up to a homeomorphism is called topological equivalence, and the relation preserving also the time of motion along trajectories (continuous in the case of flows, and discrete in the case of cascades) is called topological conjugacy. The problem of topological classification of dynamical systems consists in finding invariants that uniquely determine the equivalence class or the conjugacy class for a given system. The present survey is devoted to a description of results on topological classification of Morse–Smale systems on closed manifolds, including results recently obtained by the authors. Also presented are recent results of the authors concerning the interconnections between the global dynamics of such systems and the topological structure of the underlying manifolds.
Bibliography: 112 titles.

Keywords: Morse–Smale systems, topological classification, topology of the underlying manifold.

Funding Agency Grant Number
Russian Science Foundation 17-11-01041
National Research University Higher School of Economics
This research was supported by a grant from the Russian Science Foundation (project no. 17-11-01041), except for § 5, written with the support of a grant from the Russian Science Foundation (project no. 14-41-00044), and § 4, written in 2018 in the framework of a programme of the Centre for Fundamental Research of the National Research University “Higher School of Economics”.


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English version:
Russian Mathematical Surveys, 2019, 74:1, 37–110

Document Type: Article
UDC: 517.9+513.8
MSC: Primary 37--02; Secondary 37B30, 37B35, 37C15, 37C27, 37C29, 37C70, 37D15
Received: 16.09.2018

Citation: V. Z. Grines, E. Ya. Gurevich, E. V. Zhuzhoma, O. V. Pochinka, “Classification of Morse–Smale systems and topological structure of the underlying manifolds”, Uspekhi Mat. Nauk, 74:1(445) (2019), 41–116; Russian Math. Surveys, 74:1 (2019), 37–110

Citation in format AMSBIB
\by V.~Z.~Grines, E.~Ya.~Gurevich, E.~V.~Zhuzhoma, O.~V.~Pochinka
\paper Classification of Morse--Smale systems and topological structure of the underlying manifolds
\jour Uspekhi Mat. Nauk
\yr 2019
\vol 74
\issue 1(445)
\pages 41--116
\jour Russian Math. Surveys
\yr 2019
\vol 74
\issue 1
\pages 37--110

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