Bifurcation Of Double Homoclinic Loops

This thesis is devoted to investigating the bifurcation of some double homoclinic loops in four dimensional systems. The work is divided into two main parts including three chapters.In chapter 1, the background and research status of bifurcation theory are briefly given. Meanwhile, we introduce the main results achieved in this paper briefly.In chapter 2, firstly the local coordinates near the double homoclinic loops are set up in four dimensional vector fields, and then the Pomcare are constructed, and the bifurcation are induced. This method is initially used by [8][17] and then improved by [10][31]. Furthermore double homoclinic-loop bifurcations with resonant eigenvalues and inclination flip are studied. We proved the coexistence of large 1-hom orbit and large 1-per orbit nearΓ=Γ1∪Γ2. The condition and region of existence of saddle-node bifurcation surface are also obtained.In chapter 3, using the same method we studied codimension 1 reversible double ho-moclinic bifurcations. Firstly, we Found the condition of the persistence of R-symmetric double homoclinic loops. Furthermore, the existence and coexistence of R-symmetric homoclinic orbit and R-symmetric double homoclinic loops, R-symmetric homoclinic orbit and R-symmetric periodic orbit are proved. Finally, the condition and region of the existence of double R-symmetric homoclinic bifurcation are obtained.