Three heteroclinic orbits induce a countable family of equivalence classes of regular flows

TL;DR

This paper classifies smooth structurally stable flows on four-dimensional manifolds with two saddle points and heteroclinic curves, revealing a countable family of equivalence classes distinguished by the number of heteroclinic curves.

Contribution

It provides a topological classification of such flows, showing that the number of heteroclinic curves serves as a complete invariant on certain manifolds, unlike the finite classifications in three dimensions.

Findings

Number of heteroclinic curves is a complete invariant on $ ext{CP}^2$.

Countably many equivalence classes exist on $ ext{S}^4$ with arbitrary odd number of heteroclinic curves.

Contrast with three-dimensional case where only finitely many classes exist for each number of heteroclinic curves.

Abstract

We solve the problem of topological classification for smooth structurally stable flows on closed four-dimensional manifolds, the non-wandering set of which contains exactly two saddle equilibria, and the wandering set contains isolated trajectories connecting these saddle equilibria (heteroclinic curves). In particular, we show that for a flow of the class under consideration on $\mathbb{CP}^2$, the number of heteroclinic curves is a complete topological invariant, while on the sphere $\mathbb S^4$, there exists a countably many equivalence classes with an arbitrary odd number $\gamma\geq 3$ of heteroclinic curves. These results contrast with a three-dimensional case, where under similar conditions there exists only finite set of equivalence classes for each number of heteroclinic curves.