Criterion for the Topological Conjugacy of Multi-Dimensional Gradient-Like Flows with No Heteroclinic Intersections on a
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Journal of Mathematical Sciences, Vol. 250, No. 1, October, 2020
CRITERION FOR THE TOPOLOGICAL CONJUGACY OF MULTI- DIMENSIONAL GRADIENT-LIKE FLOWS WITH NO HETEROCLINIC INTERSECTIONS ON A SPHERE V. E. Kruglov National Research University Higher School of Economics 25/12, Bol’shaya Pechorskaya St., Nizhny Novgorod 603155, Russia [email protected]
O. V. Pochinka ∗ National Research University Higher School of Economics 25/12, Bol’shaya Pechorskaya St., Nizhny Novgorod 603155, Russia [email protected]
UDC 517.9
We study gradient-like flows with no heteroclinic intersections on an n-dimensional (n 3) sphere from the point of view of topological conjugacy. We prove that the topological conjugacy class of such a flow is completely determined by the bicolor tree corresponding to the frame of separatrices of codimension 1. We show that for such flows the notions of topological equivalence and topological conjugacy coincide (which is not the case if there are limit cycles and connections. Bibliography: 18 titles. Illustrations: 3 figures.
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Introduction
As known, it is possible to introduce Morse functions and, consequently, gradient flows generated by the Morse functions on any manifold. In general, gradient flows are not structurally stable, but rough flows are dense [1] in the set of gradient flows. The nonwandering set of structurally stable gradient flows consists of finitely many hyperbolic fixed points whose invariant manifolds intersect transversally. Such flows are distinguished as a separate class of the so-called gradientlike flows. Gradient flows are used to describe regular processes in various applications (cf., for example, [2]). In particular, they simulate the processes of reconnecting magnetic lines in the solar corona (cf., for example, [3]). Therefore, it is important to have the possibility to compare the dynamics of such models independently of their origin. Furthermore, it is important to study the qualitative behavior (a partition into trajectories), as well as the time motion along ∗
To whom the correspondence should be addressed.
Translated from Problemy Matematicheskogo Analiza 104, 2020, pp. 21-27. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2501-0022
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trajectories. According to the theory of dynamical systems, we say that flows f t , f t : S n → S n are topological equivalent if there exists a homeomorphism sending trajectories of one flow to trajectories of the other and preserving the motion direction and topologically conjugate if there exists a homeomorphism h : S n → S n such that h ◦ f t = f t ◦ h for t ∈ R. The topological classification of flows consists in finding invariants that uniquely determine the equivalence and conjugacy classes. Since the set of nonwandering trajectories of a gradient-like flow is finite, it is reasonable to try to reduce the topological classification problem to a combinatorial problem. This was first done in [4, 5], where flows on a two-dimensional sphere with finitely many singular trajectories were classified, and further generalized in [6]–
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