Bifurcations Of Homoclinic And Double Homoclinic Connecting Nonhyperbolic Equilibrium

The existence of homoclinic and heteroclinic orbits often imply the complexity of dynamics behavior. So the study of homoclinic and heteroclinic orbits has important application background.This paper studies the bifurcations of a nongeneric homoclinic orbit and a double homoclinic orbit connecting a nonhyperbolic equilibrium.The main content is divided into two parts. The first part studies bifurcations of a nongeneric homoclinic loop connecting a nonhyperbolic equilibrium in 4-dimensional dynamic system. By extending local activities coordinate frame method introduced by Zhu, the Poincare mapping and bifurcation equation is obtained, based on the analysis of bifurcation equation we give the sufficient conditions for the existence of homoclinic orbit and a periodic orbit when pitchfork bifurcation does not happen; While for the nonhyperbolic equilibrium undergoes a pitchfork bifurcation, we achieve the existence of homoclinic orbits and three heteroclinic orbits, where we may know the difference of bifurcations between the nongeneric homoclinic orbit and the generic one.The second part we discuss a class of half folded double homoclinic bifurcation connecting non-hyperbolic equilibrium in a 4-dimensional system, where we assum the nonhyperbolic equilibrium undergoes a transcritical bifurcation. By extending local activities coordinate frame method, the Poincare mapping and bifurcation equation is obtained. Firstly, we study the sufficient conditions for the existence of the double homoclinic bifurcation when the transcritical bifurcation does not happen, and for the new bifurcated(1,2)(or(2,1)) homoclinic orbits or periodic orbits and the homoclinic orbits and periodic orbits coexist or not. Then we give the sufficient conditions for the existence of the double homoclinic bifurcation when the equilibrium undergoes a transcritical bifurcation in the perturbed system, and for the bifurcation surface or parameters area of bifurcating out(1,2)(or(2,1)) homoclinic orbits periodic orbits, moreover we give the sufficient conditions for the existence of the bifurcated three heteroclinic orbit.