On embedding of invariant manifolds of the simplest Morse-Smale flows with heteroclinical intersections

E. Ya. Gurevich ; D. A. Pavlova

Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 20 (2018) no. 4, pp. 378-383

Résumé

We study a structure of four-dimensional phase space decomposition on trajectories of Morse-Smale flows admitting heteroclinical intersections. More precisely, we consider a class $G(S^4)$ of Morse-Smale flows on the sphere $S^4$ such that for any flow $f\in G(S^4)$ its non-wandering set consists of exactly four equilibria: source, sink and two saddles. Wandering set of such flows contains finite number of heteroclinical curves that belong to intersection of invariant manifolds of saddle equilibria. We describe a topology of embedding of saddle equilibria’s invariant manifolds; that is the first step in the solution of topological classification problem. In particular, we prove that the closures of invariant manifolds of saddle equlibria that do not contain heteroclinical curves are locally flat 2-sphere and closed curve. These manifolds are attractor and repeller of the flow. In set of orbits that belong to the basin of attraction or repulsion we construct a section that is homeomoprhic to the direct product $\mathbb{S}^{2}\times \mathbb{S}^1$. We study a topology of intersection of saddle equlibria’s invariant manifolds with this section.

Détail

Keywords: topological equivalence, Morse-Smale flows, heteroclinic curves.