A Novel Route to Oscillations via non-central SNICeroclinic Bifurcation: Unfolding the Separatrix Loop Between a Saddle-Node and a Saddle

TL;DR

This paper introduces a new bifurcation scenario called non-central SNICeroclinic bifurcation, which acts as a generic route to oscillations through global bifurcations, expanding understanding of transition mechanisms in dynamical systems.

Contribution

The paper uncovers and analyzes the minimal unfolding of non-central SNICeroclinic bifurcations in planar systems, highlighting its role as an organizing center for complex bifurcation transitions.

Findings

Identified how parameter variations lead to heteroclinic, homoclinic, and saddle-node on invariant circle bifurcations.

Demonstrated that non-central SNICeroclinic bifurcation can generate and destroy oscillations via global bifurcations.

Provided a minimal set of perturbations capturing all qualitative behaviors near the bifurcation.

Abstract

In this paper, we investigate saddle-node to saddle separatrix--loops that we term SNICeroclinic bifurcations. They are generic codimension-two bifurcations involving a heteroclinic loop between one non-hyperbolic and one hyperbolic saddle. A particular codimension-three case is the non-central SNICeroclinic bifurcation. We unfold this bifurcation in the minimal dimension (planar) case where the non-hyperbolic point is assumed to undergo a saddle-node bifurcation. Applying the method of Poincar\'e return maps, we present a minimal set of perturbations that captures all qualitatively distinct behaviors near a non-central SNICeroclinic loop. Specifically, we study how variation of the three unfolding parameters leads to transitions from heteroclinic and homoclinic loops, saddle-node on an invariant circle (SNIC), and periodic orbits as well as equilibria. We show that although the bifurcation has been largely unexplored in applications, it can act as an organizing center for transitions between various types of saddle-node and saddle separatrix loops. It is also a generic route to oscillations that are both born and destroyed via global bifurcations, compared to the commonly observed scenarios involving local (Hopf) and in some cases global (homoclinic or SNIC) bifurcations.