Numerical unfoldings of codimension-three resonant homoclinic flip bifurcations

Abstract

Resonant homoclinic flip bifurcations are codimension-three phenomena that act as organizing centres for codimension-two inclination flip, orbit flip and eigenvalue-resonance bifurcations of homoclinic orbits to a real saddle. In a recent paper by Homburg and Krauskopf unfoldings for several cases of resonant homoclinic flip bifurcations were proposed as bifurcation diagrams on a sphere around the central singularity.This paper presents a comprehensive numerical investigation into these unfoldings in a specific three-dimensional vector field, which was constructed by Sandstede to explicitly contain inclination flip and orbit flip bifurcations. For both orbit and inclination flips, different cases can be classified according to the eigenvalues of the saddle point. All possible cases are treated including complicated ones involving homoclinic-doubling cascades and chaos. In each case, by choosing a sufficiently small sphere around the codimension-three point in parameter space, the conjectured unfoldings are largely confirmed. However, for larger spheres interesting new codimension-three bifurcations occur, leading to a more complicated bifurcation structure. The results suggest an important trade-off between finding bifurcation curves numerically and introducing new bifurcations by enlarging the sphere too much.

Bifurcations of double homoclinic flip orbits with resonant eigenvalues

“…Then authors in Refs. [8,9] used the unfolding techniques and numerical methods to study the resonant homoclinic flips with codimension 3, respectively, and obtained some initiative bifurcation results. Recently, Zhang, et al [10,11] took into account two cases of resonant bifurcations of the flip homoclinic orbit and Shui, et al [12,13] researched on some inclination and orbit flips without resonance.…”

Codimension-4 resonant homoclinic bifurcations with orbit flips and inclination flips

“…Flips bifurcation phenomena are comprehensively investigated during the last decade (see [1,3,5,8,9,[11][12][13]), amongst them the study is mainly focused on homoclinic flips, and the period-doubling bifurcations are found to exist widely, which is a central phenomenon within dynamical systems and may produce the homoclinic-doubling bifurcations. In recent years, the single flip of the heterodimensional cycle or accompanied by transcritical bifurcation is discussed more (see [2,6,7]), the double and triple periodic orbit bifurcation are proved to exist, and also some coexistence conditions for the homoclinic orbit and the periodic orbit are given.…”

Heterodimensional cycle bifurcation with two orbit flips

“…Sandstede [12] investigated codimension-2 bifurcations of homoclinic orbits with an orbit flip. For other references, see [13][14][15][16][17][18] and the references cited therein.…”