Spherical flow diagram with finite hyperbolic chain-recurrent set

Abstract

In this paper, authors examine flows with a finite hyperbolic chain-recurrent set without heteroclinic intersections on arbitrary closed n-manifolds. For such flows, the existence of a dual attractor and a repeller is proved. These points are separated by a (n−1)-dimensional sphere, which is secant for wandering trajectories in a complement to attractor and repeller. The study of the flow dynamics makes it possible to obtain a topological invariant, called a spherical flow scheme, consisting of multi-dimensional spheres that are the intersections of a secant sphere with invariant saddle manifolds. It is worth known that for some classes of flows spherical scheme is complete invariant. Thus, it follows from G. Fleitas results that for polar flows (with a single sink and a single source) on the surface, it is the spherical scheme that is complete equivalence invariant.