Problems Of Bifurcations Of Fine Heteroclinic Loops With Two Saddle Points

In the dissertation, we consider the problems of bifurcations of fine heteroclinicloops with two saddle points satisfying the condition ρ₁¹> λ₁¹, ρ₂¹<λ₂¹,（ρ₁¹ρ₂¹）/（λ₁¹λ₂¹）=11in the high dimensional systems. For the unperturbed system with two saddlepoints, and this two saddle points are connected by two heteroclinic loops, when thesmall perturbation is given, we will consider the bifurcation phenomenon producedby the original heteroclinic breaking. First, we give the basic assumptions and thestandard type of the perturbed system in some sufciently small neighborhood ofthe saddle points under some suitable transformation. Then, using the foundationalsolutions of the linear variational equation of the unperturbed system along the het-eroclinic orbits as the demanded local coordinate system of the system in the smalltubular neighborhood, in the small neighborhood of the saddle points, we selectthe P oincar′e sections of the heteroclinic orbits. Using the Silnikov coordinates, weconstruct the P oincar′e mapping by two parts. Following, the successor function andthe bifurcation equation are obtained. Last, by considering whether the bifurcationequation has sufciently small nonnegative solutions, we will get the main results ofthe reservation of the heteroclinic loop and the existence, coexistence, uniqueness,the existence regions and branch surfaces problems of the1homoclinic loop1-periodic orbit2-fold1-periodic orbit2-periodic orbit2-homoclinic loop ofthe system in twisted and distorted two cases.In the first chapter of this article will first introduce the preliminary knowledgeof the heteroclinic loop and homoclinic loop: basic concepts!development andresearch status of the bifurcations of the heteroclinic loops and the homoclinic loops,and gives the research objectives!difculty points of this thesis as well as one’s ownknack in innovation.The second chapter will introduce the specific practice to the problem of bifur-cations of fine heteroclinic loops with two saddle points in detail:In the first section, we will consider the Crsystem（z|˙）=f（z）+g（z, μ） and the unperturbed system（z|˙）=f（z）,where r≥5, z∈R^m+n, m≥2, n≥2, μ∈R^l, l≥3,0≤|μ|≤1, g（z,0）=0, andwill give five basic assumptions of the system and transform the perturbed systeminto the standard type under some suitable transformation.In the second section, we will establish the local coordinate system near theheteroclinic, using the Silnikov coordinates, P oincar′e mapping will be derived, andthe bifurcation equation will be got.In the third section, through considering whether the given bifurcation equationhas the nonnegative solutions, we will study the reservation of the heteroclinic loopand the existence!uniqueness!the existence regions and the expressions of thebranch surfaces of the1homoclinic loop of the system in twisted and distortedtwo cases.In the fourth and fifth section, for the given bifurcation equation, we will sep-arately study the existence, uniqueness, the existence regions and the expressionsof the branch surfaces of the1periodic orbit and2fold1periodic orbit, andtheir coexistence with the1homoclinic loop of the system in twisted and distortedtwo cases, and the branch graph will separately given.In the sixth section, using the same method, we will get the new bifurcationequation, and separately study the existence, uniqueness!the existence regions andthe expressions of the branch surfaces of2homoclinic loop of the system in twistedand distorted two cases.