On a topological classification of multidimensional polar flows

E. Ya. Gurevich ; N. S. Denisova

Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 24 (2022) no. 1, pp. 31-39

Résumé

The work solves the classification problem for structurally stable flows, which goes back to the classical works of Andronov, Pontryagin, Leontovich and Mayer. One of important examples of such flows is so-called Morse-Smale flow, whose non-wandering set consists of a finite number of fixed points and periodic trajectories. To date, there are exhaustive classification results for Morse-Smale flows given on manifolds whose dimension does not exceed three, and a very small number of results for higher dimensions. This is explained by increasing complexity of the topological problems that arise while describing the structure of the partition of a multidimensional phase space into trajectories. In this paper authors investigate the class $G(M^n)$ of Morse-Smale flows on a closed connected orientable manifold $M^n$ whose non-wandering set consists of exactly four points: a source, a sink, and two saddles. For the case when the dimension $n$ of the supporting manifold is greater or equal than four, it is additionally assumed that one of the invariant manifolds for each saddle equilibrium state is one-dimensional. For flows from this class, authors describe the topology of the supporting manifold, estimate minimum number of heteroclinic curves, and obtain necessary and sufficient conditions of topological equivalence. Authors also describe an algorithm that constructs standard representative in each class of topological equivalence. One of the surprising results of this paper is that while for $n=3$ there is a countable set of manifolds that admit flows from class $G(M^3)$, there is only one supporting manifold (up to homeomorphism) for dimension $n>3$.

Détail

Keywords: Morse-Smale flows, polar flow, topological classification, topology of ambient manifold, heteroclinic curve.