Robust heteroclinic cycles in pluridimensions

TL;DR

This paper develops a new stability theory for robust heteroclinic cycles in higher dimensions, especially when traditional eigenvalue-based methods fail, and illustrates it with four-dimensional examples relevant to population dynamics.

Contribution

It introduces a novel stability analysis for heteroclinic cycles in pluridimensions without relying on contracting eigenvalues, expanding understanding of complex dynamical systems.

Findings

Developed stability criteria for heteroclinic cycles in pluridimensions.

Presented four explicit four-dimensional examples illustrating the theory.

Potential applications in modeling population transitions with multiple species.

Abstract

Heteroclinic cycles are sequences of equilibria along with trajectories that connect them in a cyclic manner. We investigate a class of robust heteroclinic cycles that does not satisfy the usual condition that all connections between equilibria lie in flow-invariant subspaces of equal dimension. We refer to these as robust heteroclinic cycles in pluridimensions. The stability of these cycles cannot be expressed in terms of ratios of contracting and expanding eigenvalues in the usual way because, when the subspace dimensions increase, the equilibria fail to have contracting eigenvalues. We develop the stability theory for robust heteroclinic cycles in pluridimensions, allowing for the absence of contracting eigenvalues. We present four new examples, each with four equilibria and living in four dimensions, that illustrate the stability calculations. Potential applications include modelling the dynamics of evolving populations when there are transitions between equilibria corresponding to mixed populations with different numbers of species.