Bifurcations Of Homoclinic, Heteroclinic Loops And Oscillation Of Matrix Differential Equations

This paper is devoted to the study of the bifurcations of reversible heteoclinic loop with inclination.flips or orbit flips,heterodimensional cycles,homoclinic bellows configuration in general systems,as well as the oscillations of linear,superlinear matrix differential equations.The work is divided into six chapters.Chapter 1 serves as an introduction:the background and significance of our research work,the setting up of the problems,etc.are presented,a basic summarization of the paper is given.Meanwhile,the main results achieved in this paper are summarized.As we all known,the reversible system in mechanics,fluids and optics has tremendous potential applications and significant theoretical value.In 1998,Champneys A.R. [20]put forward an open problem:Can inclination flips occur for reversible systems? If so,are there examples and will an analysis reveal a similar bifurcation result to the reversible orbit flip? Motivated by[20],in Chapter 2,we adopt the method initiated in [114,115],improved by[50-52],(i.e.,by establishing a local moving flame system in a small neighborhood of the heteroclinic orbits,constructing a Poincarémap and deducing the bifurcation equations),we first study the bifurcations of codimensional 2 heteroclinic loop with inclination flips in reversible systems,then turn to investagate the bifucation of codimensional 3 reversible heteroclinic loop with orbit flips and inclination flips in Chapter 2.From[64],systems with heterodimensional cycles are normal,and the existence of heterodimensional cycles always implies the extreme complexity of dynamical behaviors. Accordingly,the study of heterodimensional cycles has many potential applications and theoretical value.In Chapter 3,we adopt the previous method introduced in[144,145]to investigate the heterodimensional cycles with orbit flip and inclination flip in four dimensional dynamical systems.Under some non-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,as well as the occurence of a family of periodic orbits are established.Also,bifurcation surfaces are given and some bifurcation patterns different from the case of non-inclination-flip heteroclinic loop are revealed.In fact,it has been known that,in the case of non-inclinationflip heteroclinic loop(excluding the heterodimensional cycls)bifurcation problems,a persistent heteroclinic loop cannot coexist with a bifurcated periodic orbit(homclinic orbit) even when the original cycle fulfils some degenerate conditions such as resonance,orbit flip,symmetry or orbit flip,etc.In recent years,a new phenomena named bellows configuration has drawn many scholars' attentions in several models.A fundamental characteristic of such configuration is that two homoclinic orbits approach the equilibrium along the same direction for positive and negative time.The famous fifth-order equation for gravity-capillary water waves possesses a homoclinic bellows,see[99]for details.In another different case,homoclinic bellows can also produced by degenerate homoclinic orbits,see,for example[34,43,57]. Under the assumption of orbit-flip,homoclinic bellows can even generated by a nongeneric homoclinic orbit[81,88].More interestingly,there is also a heteroclinic bellows configuration[100].Motivated by[42,43],in Chapter 4,we investigate the homoclnic bellows configuration in general systems,including the persistence of the primary loop, the existence of the periodic orbits with different routes and the shift-invariant sequences of curves.The method adapted in this paper,which copes with bifurcation problems,is more applicable and the bifurcation equations achieved here are easy to compute,and we can reveal more elaborate results.The oscillatory theory of matrix differential equations is one branch of the important differential equations.With the development in the research of the oscillation for scalar differential equations,many scholars have interest in its research.In Chapter 5,by introducing a linear integral operator,a positive linear functional and a monotone subhomogeneous functional,including the general means and Riccati technique,some new oscillation criteria are established for the second order linear matrix differential system with damping.In addition,some new Kamenev-type theorems of superlinear matrix differential equations are established,which improve many known results for superlinear matrix differential equations.Finally,some examples are included to illustrate our main results. Finally,in Chapter 6,we look ahead the future research and put forword some unresolved problems in dynamical systems.