同宿环或异宿环分支是非线性系统分支与混沌研究中一项重要而又困难的课题，高维系统中存在的同宿环或异宿环的特征远比平面系统中的复杂，本项目拟首先构造几类包含具体特征与余维数的同宿环或异宿环的3维或4维非线性系统，其中的具体特征是指定连接同宿环或异宿环的平衡点类型，如双曲鞍点、鞍焦点、不变流形或某些特殊的非双曲平衡点，余维数主要指同宿环或异宿环由共振、轨道翻转或倾斜翻转等退化性产生，然后利用活动坐标架的方法分析以上构造的几类系统在某些自治小扰动或特殊的非自治小扰动后在原同宿环或异宿环附近能够产生同宿轨、异宿轨及周期轨的参数条件及其相应的存在个数，最后再分析在某些具体的小扰动表达式下扰动系统除以上讨论之外的其它共存的非双曲平衡点所扰动出的周期轨情况，期望突破3维或4维系统中现有的周期轨数目的结果，为高维动力系统的分支研究提供有力的理论支撑。

Bifurcations of homoclinic loop or heteroclinic loop are the important and difficult topics in the study of bifurcations and chaos in nonlinear dynamical system. The characteristics of homoclinic loop or heterolinic loop in higher dimensional systems are complex than that of quadratic systems. In this project, firstly, we will construct some three or four dimensional nonlinear systems with homoclinic loop or heteroclinic loop connecting hyperbolic saddles、saddle-focus、invariant manifolds or some special nonhyperbolic equilibria, particularly accompanied by some degeneracy condition including resonance、orbit flip and inclination flip et al, then study the parameters conditions and existence numbers of homoclinic orbits, heteroclinic orbits and periodic orbits in the small neighborhood of the old homoclinic loop and heteroclinic loop under some autonomous or special nonautonomous small perturbations using the moving coordiantes frame. Lastly, we will analyze other existence numbers of periodic orbits caused by other coexistence nonhyperbolic equilibria under some concrete small perturbations, which will break through the existing numbers of periodic orbits in three or four dimensional system and enrich the study on bifurcations in higher dimensional dynamical systems.