# Moduli of stability for heteroclinic cycles of periodic solutions

## TL;DR

This paper analyzes the stability properties of heteroclinic cycles between periodic solutions in 3D vector fields, identifying invariants that classify their local dynamics and applying these results to Bowen's example.

## Contribution

It provides a complete set of topological invariants for attracting heteroclinic cycles between periodic solutions in three-dimensional systems.

## Key findings

- Identifies invariants including periods, angular speeds, and transition maps.
- Characterizes the basin of attraction with historic behavior.
- Applies the theory to lifted Bowen cycles.

## Abstract

We consider $C^2$ vector fields in the three dimensional sphere with an attracting heteroclinic cycle between two periodic hyperbolic solutions with real Floquet multipliers. The proper basin of this attracting set exhibits historic behavior and from the asymptotic properties of its orbits we obtain a complete set of invariants under topological conjugacy in a neighborhood of the cycle. As expected, this set contains the periods of the orbits involved in the cycle, a combination of their angular speeds, the rates of expansion and contraction in linearizing neighborhoods of them, besides information regarding the transition maps and the transition times between these neighborhoods. We conclude with an application of this result to a class of cycles obtained by the lifting of an example of R. Bowen.