Bifurcations Of Heterodimensional Cycles And Heteroclinic Loop And BVPs Of Dynamic Equations On Time Scales

This paper is devoted to investigating the bifurcations of heterodimensional cycles, the bifurcations of heteroclinic loop with nonhyperbolic equilibrium, and three-point boundary value problems for p-Laplacian dynamic equations on time scales. The work is divided into four chapters.In Chapter 1, a basic summarization of the paper is given. Meanwhile, we introduce the main results achieved in this paper.As we know from [34], systems with heterodimensional cycles are normal, and the existence of heterodimensional cycles always implies extreme complexity of dynamical behaviors. So, the study of heterodimensional cycles has tremendous potential for applications and significant theoretical value. In Chapter 2, by establishing local active coordinate system in a small neighborhood of the heteroclinic orbits, we construct a Poincaré map and induce bifurcation equations to investigate the bifurcations of heterodimensional cycles in R~3 and R~4, respectively. Under some generic conditions and based on bifurcation equations, the persistence of heterodimensional cycles, the existence of homoclinic loops and periodic orbits, the coexistence of one heteroclinic loop and one periodic orbit are established. Also, bifurcation surfaces are given and some bifurcation patterns different to the case of non-heterodimensional heteroclinic loop are revealed. It is well known that, in the case of non-heterodimensional heteroclinic loop bifurcation problems, a persistent heteroclinic loop can coexist with a bifurcated periodic orbit only when the original cycle fulfils some nongeneric conditions (orbit flip, inclination flip etc).In fact, an increasing number of papers are devoted to studying bifurcations of homoclinic or heteroclinic orbits with hyperbolic equilibria. However, research works concerned with the corresponding problem of orbits joining nonhyperbolic equilibria are more difficult and relatively scarce. As is known, the nonhyperbolic equilibrium is unstable and always undergoes saddle-node (transcritical or pitchfork) bifurcation. References [37] and [38] deal with bifurcations of generic and nongeneric homoclinic loop with a nonhyperbolic equilibrium, respectively. Motivated by [37, 38], in Chapter 3, the bifurcations of heteroclinic loop connecting one hyperbolic saddle and one nonhyperbolic equilibrium are studied, and we assume the nonhyperbolic equilibrium undergoes trans-critical bifurcation. By means of local active coordinate system in a small neighborhood of the heteroclinic orbits, we construct a Poincaré map and induce bifurcation equations to analyse the bifurcations of generic and nongeneric heteroclinic loop, respectively. The persistence of heteroclinic loop, the existence of homoclinic loops, periodic orbits and heteroclinic orbits bifurcated from the heteroclinic loop are discussed. The difference of bifurcations between generic and nongeneric heteroclinic loop is discovered.The method adapted in this paper, which copes with bifurcation problems, is more applicable and the bifurcation equations achieved here are easy to compute.Dynamic equations on time scales is a fairly new subject, and research in this area is rapidly growing. With increasing attentions on time scales and p-Laplacian differential (or difference) equations, more and more people focus their interest on boundary value problems of p-Laplacian dynamic equations on time scales. In Chapter 4, based on some knowledge of nonlinear functional analysis, by virtue of a new fixed-point theory introduced by Avery and Peterson [80] and another fixed-point theorem in [92], we establish the existence of at least three solutions to a kind of three-point boundary value problem for p-Laplacian dynamic equations on time scales, which avoids the corresponding repeated investigations for continuous and discrete p-Laplacian dynamic equations. As an application, some examples are included to illustrate our main results.