Homoclinic saddle to saddle-focus transitions in 4D systems

TL;DR

This paper analyzes a unique 4D homoclinic bifurcation involving eigenvalue exchange, revealing complex bifurcation structures and providing an example in a perturbed Lorenz-Stenflo model.

Contribution

It introduces the 3DL bifurcation, a novel type of saddle to saddle-focus transition with distinct eigenvalue behavior in 4D systems.

Findings

Identifies codimension 1 and 2 bifurcation curves near the 3DL point.

Shows the bifurcation structure differs from the standard Belyakov case.

Provides an example in a perturbed Lorenz-Stenflo 4D model.

Abstract

A saddle to saddle-focus homoclinic transition when the stable leading eigenspace is 3-dimensional (called the 3DL bifurcation) is analyzed. Here a pair of complex eigenvalues and a real eigenvalue exchange their position relative to the imaginary axis, giving rise to a 3-dimensional stable leading eigenspace. This transition is different from the standard Belyakov bifurcation, where a double real eigenvalue splits either into a pair of complex-conjugate eigenvalues or two distinct real eigenvalues. In the wild case, we obtain sets of codimension 1 and 2 bifurcation curves and points that asymptotically approach the 3DL bifurcation point and have a structure that differs from that the standard Belyakov case. We also give an example of this bifurcation in the wild case occuring in a perturbed Lorenz-Stenflo 4D ODE model.