Twisted bifurcations and stability of homoclinic loop with higher dimensions

Abstract

By using the linear independent solutions of the linear variational equation along the homoclinic loop as the demanded local coordinates to construct the Poincare map, the bifurcations of twisted homoclinic loop for higher dimensional systems are studied. Under the nonresonant and resonant conditions, the existence, number and existence regions of the 1-homoclinic loop, 1-periodic orbit, 2-homoclinic loop, 2-periodic orbit and 2-fold 2-periodic orbit were obtained. Particularly, the asymptotic repressions of related bifurcation surfaces were also given. Moreover, the stability of homoclinic loop for higher dimensional systems and nontwisted homoclinic loop for planar systems were studied.