Problems Of Homoclinic Flips Bifurcation In Four-Dimensional Systems

Homoclinic bifurcations have been greatly developed for around a semi-century (see the references therein), in which flip homoclinic phenomena, including orbit flip and inclination flip, have gradually become a spotlight during the last decade. In these instances, a period-doubling bifurcation is studied thoroughly as a central phenomenon that produces the homoclinic-doubling, which is a codimension-one bifurcation of a 2n-homoclinic orbit transformed from a n-homoclinic orbit, when it collides with the saddle. In papers [6, 43-45, 59, 80, 81, 109], the existences of cascades of homoclinic-doubling and period-doubling are discussed comprehensively because horseshoe dynamics may occur resulting from the appearances of arbitrary n-periodic orbit and n-homoclinic orbit. Up to now, the result about the flip homoclinic bifurcations, generally of codimension-two or three, such as the resonant orbit flips and the resonant inclination flips, is not complete yet due to its complexity.This thesis mainly considers some codimension-three and four flip homoclinic bifurcations in a four-dimensional vector field with a real saddle at the origin, in which the dimension is required to be four because it is necessary for the case of the orbit homoclinic to its two nonleading eigendirections, namely the strong stable direction and the strong unstable direction. This is a codimension-three homoclinic phenomenon with two orbit flips, which is introduced in detail in the first chapter to prove the existence and the distribution region of codimension-one saddle-node bifurcation, homoclinic-doubling bifurcation and period-doubling bifurcation. We point out that a codimension-two triple periodic orbit bifurcation may occur at the boundary of two saddle-node bifurcation surfaces, and also the appearance of a codimension-two homoclinic and double periodic orbit bifurcation if a homoclinic bifurcation intersects a saddle-node bifurcation. Furthermore, the existence, number and their coexistence of 1-periodic orbit, 1-homoclinic orbit, 2ⁿ-periodic orbit and 2ⁿ -homoclinic orbit are also proved based on these bifurcations. In the following two chapters we study two codimension-four bifurcations of homoclinic one orbit flip and two inclination flips under conditions of the principal eigenvalues resonance and the tangent directions resonance of homoclinic orbit respectively. We deduce that the system may have at most one 1-periodic orbit or one 1-homoclinic orbit and they could not coexist under a certain condition. If this condition is violated, the codimension-one saddle-node bifurcation, homoclinic-doubling bifurcation and period-doubling bifurcation; the codimension-two triple periodic orbit bifurcation and homoclinic and double periodic orbit bifurcation can still take place for the eigenvalues in accordance with the requirements of 2λ₁ > λ₂ >ρ₂ or 2λ₁ >ρ 2 >λ₂. In the last chapter we investigate a double homoclinic orbit flip, namely a "∞"-shape homoclinic orbit, at the same two resonances as above. The persistence of the original double homoclinic orbits is asserted when the small parameter lies in somecodimension-one surfaces. Moreover there may exist two large 1-periodic orbits, a large 1-homoclinic orbit, a large 2-homoclinic orbit or a large 2-periodic orbit near the double homoclinic orbits. As for above results, some partial bifurcation diagrams are displayed with the relevant bifurcation surfaces, the n-periodic orbits and n-homoclinic orbit. The method used here initially appeared in the papers [117, 121] and then was adapted extensively by the papers [48-50, 104]. It consists of constructing a Poincaré return map to obtain a successor function which is manageable to the research by means of combining a global transition map defined in a neighborhood of the homoclinic orbit with a local transition map in a neighborhood of the equilibrium, in which the global transition map is specially established by a fundamental solution matrix of a linear variational system and the local one by a linear approximation solution of a normal form.