Homoclinic Bifurcations And Strange Attractors

The Chaos is one of the focuses of nonlinear science, and strange attractors are typ-ical features which reflects the chaotic motion. So, the study on generating mechanism, existence conditions and the properties of strange attractors has great significance. S-ince 1963, the famous Lorenz equations were found to exist attractors in numerical solution and it was proved that the Lorenz equations really have a strange attractor by normal form for computer-assisted in 1999, the analysis and researches on attrac-tors have been gradually deepened. Among them, the most study was about geometric Lorenz attractor model proposed to simulate the behavior of Lorenz equation.The bifurcation theory of vector fields mainly studys the changes of topology struc-ture of orbits in dynamical systems with the change of parameters. It is structural un-stable when a system has a homoclinic orbit to a saddle point. Therefore, homoclinic bifurcation contains a wealth of dynamic behavior. In Lorenz model, corresponding to the bifurcation of a homoclinic butterfly, upon splitting the two symmetric homoclinic loops outward, a saddle periodic orbit is born from each loop. Furthermore, the stable manifold of one of the periodic orbits intersects transversely to the unstable manifold of the other one, and vice versa. This transverse homoclinic phenomenon leads to nu-merous complex phenomena.In this paper, we study two types of symmetric homoclinic bifurcation of codi-mension two which could produce Lorenz-like attractor:inclination flip and orbit flip. Specifically, we consider a Cr(r≥3) symmetric systems X=f(X), X∈R3 with a saddle. First, bring the system into a simple form for easy analysis in the neighbor-hood of a saddle equilibrium state. Thus the Poincare return map can be constructed. Based on the Poincare return map, detailed bifurcation analysis of inclination flip and orbit flip are discussed, and bifurcation curves are also given. Followed, the existence of Lorentz-like attractor in the unfoldings of the two homoclinic bifurcations is dis-cussed. By reducing the two-dimensinal Poincare return map to one-dimensional map, and with the topological properties of one-dimensional-map, we eventually verify that the Lorentz-like attractor exist in a certain parameter range.