The Bifurcation Problem Of Symmetric Double Homoclinic Loops With Orbit Flip

In this paper,we mainly study the bifurcation of a class of symmetric double ho-moclinic loops with orbit flip.We study it from two aspects.One is to consider the case that the orbit generated by the disturbance of the original system is R symmetric?the other is to consider the case that the orbit generated by the disturbance of the original system is asymmetric,then according to the symmetry of the system,we can get that there must be another orbit which is symmetric to this orbit.In the first chapter,we first give a brief analysis and overview of the research background and research status of the bifurcations of double homoclinic loops,as the basis of academic research.Then,we state the main research results and innovations of this paper.In the second chapter,we study a class of three-dimensional symmetric systems.By using the method of active coordinate frame,we construct Poincarémapping and derive the bifurcation equation.The different bifurcation situations are transformed into the existence of sufficiently small non-negative real number solutions of the bifurcation equation.In the third chapter,we obtain a series of results by using the bifurcation equation derived in Chapter 2.The related literature has used the Lyapunov-Schmidt simplifi-cation method to study the problem of1/2<?<1.In this case,the bifurcation of the homoclinic orbit,the existence region and existence condition of saddle node bifurca-tion surface of the periodic orbit and the bifurcation of period doubling are obtained.We use the method of the active frame to study the case of?<1,and obtain the co-existence of R symmetric double homoclinic loops and R symmetric periodic orbit,the existence region and existence condition of saddle node bifurcation surface of the R symmetric periodic orbit.Then,we study the coexistence conditions of asymmetric large homoclinic orbit and R symmetric periodic orbit,the coexistence conditions of asymmetric large periodic orbit and R symmetric periodic orbit,of course,asymmetric orbits appear in pairs.We also study the case of?=1,?>1,and get the existence of the R symmetric periodic orbit.