Bifurcations Of Homoclinic Loop And Heterodimensional Cycles In High Dimensional Space

This thesis considers bifurcations of homoclinic loops with two orbit flips in a four dimensional vector field, bifurcations of heterodimensional cycles in a reversible vector field in R~3 with an involution whose fixed point subspace is one-dimensional, and the existence of heteroclinic orbits for the kinetic system associated to the near-integrable coupled nonlinear Schrodinger (NLS) equations. Furthermore, the persistence of solitions and the accompanied chaotic behavior for the perturbed NLS equations with even periodic boundary conditions are also investigated.Chapter 1 mainly reviews the interrelated concepts of the homoclinic and heteroclinic bifurcations, and the applications and research status of bifurcation thory. Meanwhile, the main results of this thesis are presented.Bifurcations of the homoclinic orbit connecting the strong stable and strong unstable directions are introduced in detail in the second chapter. The homoclinic bifurcations have complicated dynamics phenomena due to the violation of the genericity conditions. We get not only bifurcation surfaces of the 2~n-periodic orbit and 2~n-homoclinic orbit for each positive integer n, but also a codimension-2 triple periodic orbit bifurcation surface which is a part of the boundary of two double periodic orbit bifurcation surfaces, and a codimension-2 homoclinic and double periodic orbit bifurcation surface which is the intersection of a homoclinic bifurcation surface and a double periodic orbit bifurcation surface. At the same time the numbers, co-existence and incoexistence of 1-homoclinic orbit, 2~n-homoclinic orbit, 1-periodic orbit and 2~n -periodic orbit are also obtained. Finally, the bifurcation diagrams are depicted containing all of the results.Many dynamic systems with practical background possess heterodimensional cycles, and the significance for the research of heterodimensional cycles means more than practical applications([47, 48]). In chapter 3, we study bifurcations of heterodimensional cycles in a three dimensional reversible system. Under some generic conditions, we establish the existence of homoclinic loops, periodic orbits, 2-fold periodic orbits and the coexistence of one homoclinic loop and one periodic orbit. Bifurcation surfaces are also given.The method used in the second and third chapers is more applicable and the bifur- cation equations achieved here are easy to calculate. The main point of this method is to construct the Poincare map and obtain a successor function first by establishing a local moving frame system in a small neighborhood of the homoclinic (or heteroclinic) orbits and introducing several local coordinate changes to straighten invariant manifolds in a neighborhood of the equilibrium (or quilibria), and is initially used by [29, 33] and then improved by [35, 36].In Chapter 4, geometrical perturbation method and Melnikov analysis are employed to prove the existence of many heteroclinic orbits and a two dimensional heteroclinic manifold for the kinetic system of near-integrable coupled NLS equations. The Baklund transformation and Lax pair are then used to explicitly construct global homoclinic orbits of a plane wave solution and a family of heteroclinic orbits to two points situated in a torus consisting of equilibria for the unperturbed nonlinear Schroinger equations. In addition, we derive explicit representations of the Melnikov function for the system of coupled NLS equations. Finally, we duduce the persistence of homoclinic orbits for the perturbed NLS equations with even periodic boundary conditions. Chaos is also investigated under generic conditions.