Complete set of invariants for a Bykov attractor

TL;DR

This paper identifies a complete set of topological invariants for a heteroclinic cycle called a Bykov attractor, based on eigenvalues and transition maps, enhancing understanding of its local dynamics.

Contribution

It provides the first comprehensive set of invariants for Bykov attractors, linking eigenvalue properties and transition maps to topological conjugacy classifications.

Findings

Invariants depend on eigenvalue ratios and transition maps.

The basin exhibits historic behavior affecting asymptotic averages.

Complete invariants classify local dynamics near the cycle.

Abstract

In this paper we consider an attracting heteroclinic cycle made by a 1-dimensional and a 2-dimensional separatrices between two hyperbolic saddles having complex eigenvalues. The basin of the global attractor exhibits historic behaviour and, from the asymptotic properties of these non-converging time averages, we obtain a complete set of invariants under topological conjugacy in a neighborhood of the cycle. These invariants are determined by the quotient of the real parts of the eigenvalues of the equilibria, a linear combination of their imaginary components and also the transition maps between two cross sections on the separatrices.