不变集是动力系统理论最核心的研究对象之一. 该项目主要考虑紧不变集的存在性、Conley形指标和不变集分支理论及其应用中的一些问题．我们希望建立Banach空间中的单值系统在弱拓扑下和集值系统在强拓扑下的环绕定理与山路引理．对具有Lyapunov函数的系统，给出不变集的极小-极大定理和对称山路引理,并推广变分中的三解定理.受环绕定理中无穷远稳定概念和商流引理的启发，我们打算引入新的形指标偶概念，完整地建立非局部紧空间的Conley形指标理论. 此外，我们将利用Conley指标等工具讨论Banach空间中非线性发展方程定态解的不变集分支问题. 我们的方法主要是基于吸引子理论、变分思想和Conley-Morse理论等. 作为应用，我们将通过动力系统的方法研究薛定谔方程和非光滑椭圆变分问题非平凡解的存在性、非自治共振波方程有界整解的存在性和Cahn-Hillard方程平凡解的全局分支问题.

Invariant sets are of crucial importance in the theory of dynamical systems. This project is mainly devoted to the research of some problems concerning the existence of compact invariant sets, shape Conley index and invariant-set bifurcation theory as well as their applications. First, we establish some linking type results for semiflows in Banach spaces under the weak topology of the spaces by using some fundamental theory of weak attractors. We also extend some known results in this line on single-valued systems to set-valued ones and establish their linking theorem and mountain pass lemma which can be used to examine the existence of compact invariant sets of the systems. For those systems with some nice Lyapunov functions, we will give some mini-max type theorems and symmetric mountain pass lemma for invariant sets. Then, motivated by some ideas such as the concept of "stability at infinity"and the quotient flow lemma in the above work, we introduce some new types of shape index pairs and develop a complete shape Conley index theory for infinite dimensional dynamical systems. Finally, we discuss the invariant-set bifurcation of stationary solutions for nonlinear evolution equations in Banach spaces. Our methods are based on the basic theory of attractors, variational methods and the Conley-Morse theory, etc. As applications of the abstract results for invariant sets, we study the existence and multiplicity of nontrivial solutions for some typical smooth and nonsmooth variational problems of elliptic equations, the existence of bounded full solutions of nonautonomous resonant wave equations and the global bifurcation problem of the well-known Cahn-Hilliard system.