Bifurcations Of Homoclinic And Heteroclinic Solutions With Degeneracy

The study of bifurcations of homoclinic(heteroclinic) orbits arise much in-teresting in history. One transfered the problem to consider the Melnicov func-tion since the zores of it correspended to the persitence of homoclinic(heteroclinic)orbit. Early, Melnikov functions was constructed from geometric viewpoint. Butit is difficult to generalize to higher dimension systems. In 80s, S. N. Chow,J. K. Hale, P. L. Holmes, J. Mallet-Parret, J. E. Marsden and K. L. Palmerused alternative principle to obtain the Melnicov-type function. The methodis so good that there is no restriction in the dimension. The perturbationsof parabolic equations are considered in chapter 2. In 1986, C. M. Blazquezdiscussed bifurcations for parabolic equations. But he did not consider the de-generate cases and give further discussion for the Poincar'e Map. We investigatethe bifurcations of degenerate homoclinic orbit for parabolic equation under pe-riodic perturbations. We not only extend the result of C. M. Blazquez but alsoconsider the chaotic motions for the periodic map.Many authors had studied the existence, uniqueness and smoothness fordegenerate Soblev-Galpern equations. But they did not consider bifurcationsof bounded solutions. In chapter 3, we consider the persistence of boundedsolutions for the degenerate Soblev-Galpern equations. Two projections is used.The first one seperates the equation into degenerate and non-degenerate part.The existence and uniqueness of the strong solution for the non-degenerate partare established. From it we define the solution operator, the evolution operator.Another one is to obtain exponential dichotomy and Fredholm alternative forthe evolution operator. From Lyapuonv-Schmidt reduction for the degenerateequations, we give a criterion for the persistence of bounded solution. Many works dealed with the existence of bounded solution for the per-turbed system. It is natural to ask how many bounded solutions can be bi-furcated from the degenerate one. In chapter 4, we give one of criterions toensure the co-existence of linearly independent homoclinic orbits bifurcatedfrom degenerate ones. Let d denote the number of the bounded solutions forthe variation equation along the degenerate homoclinic orbit. Then there areneighborhood,γ, containing 0, and d manifoldsΓ_k, which through origin withco-dimension kd, k=1, ..., d, such that when perturbation is taken from subsetγ∩(Γ_k\(Γ_k+1∪…∪Γ_d)), the perturbed system has k linearly independent ho-moclinic solutions. This result answers the co-existemce of bounded solutions.Recently, many researchers discussed the subharmonic bifurcations for Hamiltonian and singular system. The weakly coupled system is considered in chap-ter 5. We investigate subharmonic solutions with large period bifurcated frombolw-uped homoclinic orbit. From Fredholm alternatives, we obtain a criterionfor the appearence of a subharmonic solution for the perturbed system.It is important to find central configurations in celestial mechanics. Thepyramidal central configuration is found. It is not clear if the spacial centralconfiguration can be bifurcated from planar ones. In the last chapter, we find allof the planar and spatial central configurations of pyramid. And we obtain aninteresting phenomenon. The work of R. Moeckel and C. Simo imples that, forN≥473, the spatial central configurations can be bifurcated from planar ones,i.e., the non-planar central configurations can be almost planar ones. This is incontrast to the facts that there is no non-planar pyramidal central configurationsfor N≥473, and any spatial pyramidal central configurations are far fromplanar ones.