Codimension 2 reversible heteroclinic bifurcations with inclination flips

Abstract

In this paper, the heteroclinic bifurcation problem with real eigenvalues and two inclination-flips is investigated in a four-dimensional reversible system. We perform a detailed study of this case by using the method originally established in the papers "Problems in Homoclinic Bifurcation with Higher Dimensions" and "Bifurcation of Heteroclinic Loops," and obtain fruitful results, such as the existence and coexistence of R-symmetric homoclinic orbit and R-symmetric heteroclinic loops, R-symmetric homoclinic orbit and R-symmetric periodic orbit. The double R-symmetric homoclinic bifurcation (i.e., two-fold R-symmetric homoclinic bifurcation) for reversible heteroclinic loops is found, and the existence of infinitely many R-symmetric periodic orbits accumulating onto a homoclinic orbit is demonstrated. The relevant bifurcation surfaces and the existence regions are also located.

Nonresonant Bifurcations of Heteroclinic Loops With One Inclination Flip

“…Recently, some papers were devoted to the study on the bifurcations of the homoclinic or heteroclinic loops with an orbit flip or an inclination flip in higher dimensional vector fields, e.g. see Geng et al, 2009;Lu, 2009;Qiao et al, 2009;Shui & Zhu, 2004, 2005Wu & Sun, 2006;Xu et al, 2008;Xu et al, 2009;Zhang & Zhu, 2004, 2007, 2009. In this paper, we study the bifurcations of rough heteroclinic loops which are composed of two principal heteroclinic orbits but there is one stable foliation that involves the inclination flip in four-dimensional systems.…”