从退化同宿轨到多重拓扑马蹄

自Smale提出横截的同宿轨意味着“马蹄”型混沌以来，对同宿轨的研究一直是动力系统的重要研究内容。前人们主要研究非退化同宿轨的保持性。本项目主要研究退化同宿轨的分岔特性和混沌运动。首先，考虑连接非双曲平衡点的退化同宿轨，在自治扰动下的动力行为，这时平衡点和同宿轨可以同时发生分岔，扰动系统出现的多条同宿轨中，一部分来自于平衡点的分岔，另一部分来自于退化同宿轨的分岔；然后，研究奇异系统的异宿环分岔，通过引入尺度Banach空间和截断函数的办法，克服奇异性带来的实质性困难，给出扰动系统存在多条由异宿环分岔而来的同宿轨或周期解的各种参数条件；再次，在非周期小函数扰动下，考察高维系统在适当横截性条件下，拓扑马蹄的存在性，以及横截性条件消失后拓扑马蹄的分岔特性；最后，通过引入拓扑熵的办法，以给出固定周期的周期函数空间的剖分，使得扰动取自于同一剖分中时，周期扰动系统在马蹄上的动力行为共轭。

It is one of the important dynamical topics to study the dynamical behaviors near the homoclinic solution since Smale proposed that the transverse homoclinic solution implies the existence of chaotic motion. Many authors investigated the persistence of homoclinic solution under small perturbations. In this program, we will consider the bifurcations and chaotic motions of a periodic system near a degenerate homoclinic solution. Firstly, when an autonomous has a homoclinic solution to a degenerate equilibrium, we study the bifurcations of the homoclinic solution under small perturbation. The homoclinic bifurcations can be caused by both the homoclinic solution and the degenerate equilibrium. Then the dynamical behaviors near a heteroclinic loop consisting of two degenerate heteroclinic solutions of a singular system will be investigated. There are fast and slow variables in the singular system. And the system is not smooth at the origin in the parameter space. The scales of Banach spaces and the methods of truncation will be introduced to overcome the non-smoothness. Our goal is to get some conditions for which the homoclinic solution or periodic solution can be bifurcated from the degenerate heteroclinic loop. Thirdly, the topological horseshoe caused by a degenerate homoclinic solution for an n-dimensional system is considered. It is shown that the topological horseshoe exists for some transverse conditions. Furthermore, when the transverse conditions disappear, the saddle-node or pitchfork bifurcation of topological horseshoe will be studied. At last, we will introduce the entropy to consider the dynamical behaviors. If two periodic systems both have horseshoes from transverse homoclinic solutions, the topological entropy is proposed to study the homeomorphism between the two horseshoes.