The Study Of Bifurcations Of Double Homoclinic Loops

This thesis is devoted to study the bifurcation of double-homoclinic-loop withspecial hypothesis. This consists of three chapters:In chapter one, the background, significance of problems and research status ofbifurcation theory are briefly given; some definitions and notions used in this thesisare introduced; the main results achieved in this thesis are introduced.In chapter two, the bifurcation problem about one orbit twisted double-homoclinic-loop with resonant eigenvalues in high dimensional vector fields is discussed. In thischapter, consider the following Crsystem（z|˙） = f（z） + g（z,μ）,and the unperturbed vector field（z|˙） = f（z）,where z∈R^m+n+2, m 0, n 0, m + n 0,μ∈R^l, l≥2, 0≤μ1, f（0） =0, g（z, 0） = 0,·denotes the scalar product.Firstly, the singular map in some neighborhood of the equilibrium are struc-tured, secondly the local coordinates near the double homoclinic loops are struc-tured, thirdly the P oincare′mapping are structured, then the bifurcation equationsare induced. The existence and uniqueness of the the 1-1 large homoclinic loop,2-1 large homoclinic loop, 2-1 right homoclinic loop, 2-1 double homoclinic loop,2-1 large periodic orbit nearΓ=Γ₁∪Γ₂are proved. The corresponding existenceregions is obtained.In chapter three, using the same method above, we discussed the bifurcationproblem about double-homoclinic-loop with resonant eigenvalues in three dimen-sional vector fields. In this chapter, consider the following Crsystem and the un-perturbed vector field,where z∈R³,μ∈R², 0≤μ1, f（0） = 0, g（z, 0） = 0.At last, the existence and uniqueness of the the 1-1 large homoclinic loop and 1-1large periodic orbit nearΓ=Γ₁∪Γ₂are proved. The corresponding existence re-gions and bifurction diagrams are obtained.