Bifurcations and Stability of Nondegenerated Homoclinic Loops for Higher Dimensional Systems

By using the foundational solutions of the linear variational equation of the unperturbed system along the homoclinic orbit as the local current coordinates system of the system in the small neighborhood of the homoclinic orbit, we discuss the bifurcation problems of nondegenerated homoclinic loops. Under the nonresonant condition, existence, uniqueness, and incoexistence of 1-homoclinic loop and 1-periodic orbit, the inexistence of -homoclinic loop and -periodic orbit is obtained. Under the resonant condition, we study the existence of 1-homoclinic loop, 1-periodic orbit, 2-fold 1-periodic orbit, and two 1-periodic orbits; the coexistence of 1-homoclinic loop and 1-periodic orbit. Moreover, we give the corresponding existence fields and bifurcation surfaces. At last, we study the stability of the homoclinic loop for the two cases of non-resonant and resonant, and we obtain the corresponding criterions. 1. Introduction With the rapid development of nonlinear science, in the studies of many fields of research and application of medicine, life sciences and many other disciplines, there are a lot of variety high-dimensional nonlinear dynamical systems with complex dynamic behaviors. Homoclinic and heteroclinic orbits and the corresponding bifurcation phenomenons are the most important sources of complex dynamic behaviors, which occupy a very important position in the research of high-dimensional nonlinear systems. We know that in the study of high-dimensional dynamical systems of infectious diseases and population ecology we tend to ignore the stability switches and chaos when considered much more the nonlinear incidence rate, population momentum, strong nonlinear incidence rate, and so forth. The existence of transversal homoclinic orbits implies that chaos phenomenon occur; therefore, it is of very important significance to study the cross-sectional of homoclinic orbits and the preservation of homoclinic orbits for the system in small perturbation. In addition, in the study of infectious diseases and population ecology systems, we sometimes require the existence of periodic orbits. And, homoclinic and heteroclinic orbits bifurcate to periodic orbits in a small perturbation means that we can get the required periodic solution only by adding a small perturbation when using the similar system which exists homoclinic or heteroclinic orbits to represent the natural system. This also explains the importance of homoclinic and heteroclinic orbits bifurcating periodic orbits in real-world applications. Therefore, by using the research methods and theoretical