Heteroclinic cycles arising in generic unfoldings of nilpotent singularities

TL;DR

This paper investigates how generic unfoldings of nilpotent singularities in low-dimensional systems lead to heteroclinic and homoclinic cycles, which are key to understanding complex dynamics like strange attractors.

Contribution

It demonstrates that nilpotent singularities of specific codimensions generically produce bifurcation structures leading to complex dynamics, extending the understanding of singularity unfoldings.

Findings

Unfoldings of codimension four nilpotent singularities produce bifocal homoclinic orbits.

Unfoldings of codimension three nilpotent singularities produce heteroclinic cycles between saddle-focus points.

These cycles imply the existence of homoclinic bifurcations and complex dynamics.

Abstract

In this paper we study the existence of heteroclinic cycles in generic unfoldings of nilpotent singularities. Namely we prove that any nilpotent singularity of codimension four in $\mathbb{R}^4$ unfolds generically a bifurcation hypersurface of bifocal homoclinic orbits, that is, homoclinic orbits to equilibrium points with two pairs of complex eigenvalues. We also prove that any nilpotent singularity of codimension three in $\mathbb{R}^3$ unfolds generically a bifurcation curve of heteroclinic cycles between two saddle-focus equilibrium points with different stability indexes. Under generic assumptions these cycles imply the existence of homoclinic bifurcations. Homoclinic orbits to equilibrium points with complex eigenvalues are the simplest configurations which can explain the existence of complex dynamics as, for instance, strange attractors. The proof of the arising of these dynamics from a singularity is a very useful tool, particularly for applications.