Boundary crisis for degenerate singular cycles

TL;DR

This paper analyzes the dynamics near a heteroclinic cycle involving a hyperbolic equilibrium and a periodic solution, exploring boundary crises, bifurcation scenarios, and the emergence of complex shift dynamics.

Contribution

It provides a detailed bifurcation analysis of heteroclinic cycles with codimension-one connections and quadratic tangencies, revealing new types of shift dynamics and bifurcation structures.

Findings

Identification of the crisis region near the cycle

Analysis of multipulse homoclinic and heteroclinic solutions

Characterization of bifurcating periodic solutions

Abstract

The term boundary crisis refers to the destruction or creation of a chaotic attractor when parameters vary. The locus of a boundary crisis may contain regions of positive Lebesgue measure marking the transition from regular dynamics to the chaotic regime. This article investigates the dynamics occurring near a heteroclinic cycle involving a hyperbolic equilibrium point E and a hyperbolic periodic solution P, such that the connection from E to P is of codimension one and the connection from P to E occurs at a quadratic tangency (also of codimension one). We study these cycles as organizing centers of two-parameter bifurcation scenarios and, depending on properties of the transition maps, we find different types of shift dynamics that appear near the cycle. Breaking one or both of the connections we further explore the bifurcation diagrams previously begun by other authors. In particular, we identify the region of crisis near the cycle, by giving information on multipulse homoclinic solutions to E and P as well as multipulse heteroclinic tangencies from P to E, and bifurcating periodic solutions, giving partial answers to the problems (Q1)-(Q3) of E. Knobloch (2008), Spatially localised structures in dissipative systems: open problems, Nonlinearity, 21, 45-60. Throughout our analysis, we focus on the case where E has real eigenvalues and P has positive Floquet multipliers.