非一致指数二分与伪轨跟踪

It is said that a dynamical system is hyperbolic, if the phase space admits an invariant decomposition such that the dynamical system restricted on one of the invariant subspace is a contraction and on the other one is an expansion. Dichotomy is the concept for describing the hyperbolicity of dynamical systems generated by differential equations. The exponential dichotomy has been extensively studied, because it plays an important role in the study of linearization, the existence of invariant manifolds and invariant foliations, and homoclinic bifurcations of differential equations. There is an extensive literature on the relationship between the existence of dichotomy and the admissibility, which has been used to study the existence of homoclinic orbit. Using the results on existence and roughness of dichotomy one can get the shadowing lemma. It is well known that one can use the shadowing lemma to prove the Smale’s theorem. In order to investigate the problems of nonuniformity and weaker growth in nonautonomous differential equations and random differential equations, recently, efforts have been made to extend the concept of exponential dichotomy. In this subject, we study on the problems of the relationships between nonuniform exponential dichotomy, nonuniform weak exponential dichotomy and the admissibility respectively, the roughness of nonuniform weak exponential dichotomy, and give the shadowing lemma for infinite-dimensional random dynamical systems by using the new results on the nonuniform exponential dichotomies.

动力系统的双曲性问题是动力系统的重要问题，涉及到系统的结构稳定性。在微分方程形式下双曲性是用指数二分来刻画的。在此基础上人们进一步研究微分方程的线性化、不变流形和不变叶层的存在性、同宿轨和异宿轨分岔等问题。在已有二分的应用当中，人们已获得二分的存在性与函数类的相容性的关系并据此研究同宿解；以及利用二分的存在性以及粗糙性等性质来解决伪轨跟踪问题。而伪轨跟踪引理可以用来证明著名的Smale横截同宿定理。随着非自治微分方程和随机微分方程理论的发展，近十多年来, 二分的非一致性问题和弱增长问题得到越来越多的关注。前人的工作已证明了非一致指数二分的存在性。我们将克服非一致性、弱增长以及不可逆性带来的困难给出非一致指数二分和非一致弱指数二分的存在性与函数类相容性的关系；证明非一致弱指数二分的粗糙性并给出扰动的具体估计；进而利用缓和指数二分的存在性以及粗糙性等结论解决无穷维随机系统的伪轨跟踪问题。

称系统的不变集具有双曲性的意思是相空间的切丛限制在不变集上存在不变分解, 且系统的切映射限制在其中一个不变子丛上是压缩的, 而限制在另外一个不变子丛上是扩张的。微分方程和差分方程的双曲性是用指数二分来刻画的。前人对一致指数二分的粗糙性问题和一致指数二分的存在性与相容性关系的问题进行广泛和深入的研究。近年来，随着非自治微分方程和随机微分方程理论的发展，这些系统产生的非一致性问题得到越来越多的关注。在本项目，我们研究了非一致指数二分的粗糙性问题和非一致指数二分的存在性与相容性的关系问题, 对于有限维非自治差分方程，我们用相容性分别给出了在单个半轴、两个半轴同时以及全轴上具有非一致指数二分的充分条件，利用该结论我们证明了在单个半轴、两个半轴同时以及全轴上具有非一致指数二分的粗糙性，该结果减弱了前人结果中线性扰动的条件；对于有限维微分方程,我们证明了一类Lyapunov有界函数类的相容性与非一致指数二分的存在性是等价的，我们不仅对单个半轴上的非一致指数二分证明了这个等价关系，同时还对在两个半轴同时以及在全轴上的非一致指数二分证明了该等价关系。进一步，我们把以上粗糙性和相容性结果推广到无穷维随机系统，并利用这些结论证明无穷维随机系统的伪轨跟踪引理。