Bifurcations of double homoclinic flip orbits with resonant eigenvalues

Abstract

Concerns double homoclinic loops with orbit flips and two resonant eigenvalues in a four-dimensional system. We use the solution of a normal form system to construct a singular map in some neighborhood of the equilibrium, and the solution of a linear variational system to construct a regular map in some neighborhood of the double homoclinic loops, then compose them to get the important Poincaré map. A simple calculation gives explicitly an expression of the associated successor function. By a delicate analysis of the bifurcation equation, we obtain the condition that the original double homoclinic loops are kept, and prove the existence and the existence regions of the large 1-homoclinic orbit bifurcation surface, 2-fold large 1-periodic orbit bifurcation surface, large 2-homoclinic orbit bifurcation surface and their approximate expressions. We also locate the large periodic orbits and large homoclinic orbits and their number.

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.…”