微分方程的退化同宿轨附近的动力性态

From the last 60s, many famous mathematicians, such as V. K. Melnikov、L. P. Silnikov、S.-N. Chow、J. K. Hale and so on, studied the persistence of a known homoclinic (heteroclinic) solutions under small perturbations. Their results implied that the homoclinic (heteroclinic) solutions can exist under suitable transverse conditions. Based on their work, we will consider the following problems: 1, For an autonomous differential equation with degenerate homoclinic solution, consider the existence of a homoclinic manifold consisting of a family of homoclinic solutions under perturbations. The radius of the manifold along the normal directions will be given. 2, Investigate the periodic solution bifurcated from a known heteroclinic loop consisting of degenerate heteroclinic solution. By the co-diagonalization method, the conditions of the coexistence of periodic solutions are considered. 3, When the unperturbed system has a bounded solution which is homoclinic to a nonhyperbolic equilibrium. We investigate that the perturbed system has multiple homoclinic solutions which are homoclinic to different equilibria, bifurcated from the nonhyperbolic equilibrium. 4, For the homoclinic bifurcations, the known results show that the solvabilitis of the Melnikov function are two-degree homogeneous polynomials without the tangent parameter. We will consider the new solvabilitis which are nonhomogeneous polynomials with the tangent parameter.

自上世纪60年代以来，有很多著名数学家，如V. K. Melnikov、L. P. Silnikov、S.-N. Chow、J. K. Hale等，研究过有同宿轨的自治发展方程，在小扰动下发生的动力行为：在适当的非退化条件下，扰动系统的非退化同宿轨能够保持。本项目将研究：1，研究从退化同宿轨分岔出由一族同宿轨构成的同宿流形，给出同宿流形沿同宿轨法向上的半径估计；2，考虑由退化异宿轨构成的异宿环分岔出周期解的问题，找到决定扰动系统存在周期解的分岔函数，通过多个矩阵同时对角化的方法，给出系统存在多条不同周期解的判据；3，考察同宿于非双曲平衡点的同宿轨的分岔的问题，得到平衡点和同宿轨同时发生分岔的条件，给出系统有多个平衡点和多条同宿轨的参数条件；4，对于同宿轨的保持性，前人得到不含切向参数的二次齐次型分岔函数，本项目研究含有切向参数的高次项的非齐次可解性条件，得到扰动系统存在同宿轨的新判据。

早在上个世纪60年代，著名的数学家Melnikov就用几何的办法研究了平面自治系统在周期扰动下同宿轨的保持性问题，到上个世纪80年代，美国著名数学家Hale提出人们应该关注由退化同宿轨引起的动力性态。本项目沿着这个问题进行研究，主要考虑了带退化同宿轨或异宿轨的自治系统，在一般小扰动或周期扰动下的动力性态，比如多重同宿轨分叉、多重同宿不变流形的存在性等，本项目主要寻找到一些充分或必要条件，使得扰动系统出现分叉现象，还构造出一维扰动系统，使得这些现象得以实现；还研究了在顺从群作用下的序列拓扑熵测度熵与拓扑熵测度熵的若干性质以及它们的关系；在序列熵为无穷时，研究了序列均值维数、序列度量均值维数等问题。