Lagrange系统的最小周期解和次调和解问题研究

Lagrange系统是数理科学、生命科学以及社会科学领域中一类非常重要的系统，其最小周期解和次调和解问题的研究关系到动力系统的动力学性态的发展。本项目拟使用临界点理论来研究Lagrange系统的最小周期解和次调和解问题。具体地，发展和应用临界点理论中的Minimax方法和指标理论来建立Lagrange系统所对应变分泛函的临界点的存在性、多重性、唯一性以及不存在性定理，并利用这些定理来研究Lagrange系统的最小周期解和次调和解的存在性、多重性、唯一性以及不存在性问题。. 本项目的研究不仅可以丰富和发展Lagrange系统理论，而且还将有助于临界点理论的补充和完善，具有较高的学术价值。

Lagrangian system is a very important system in the fields of mathematical sciences, life sciences and social sciences. Studies on minimal periodic solutions and subharmonic solutions of this system are related to the development of dynamical behavior for dynamical systems. In this project, we will use the critical point theory to study those problems on minimal periodic solutions and subharmonic solutions of Lagrangian system. To be precise, we will develop and apply those tools in critical point theory such as Minimax methods and Index theory to construct some existence, multiplicity, uniqueness and non-existence theorems on critical points of variational functions for Lagrangian system, and then use these theorems to study those problems on existence, multiplicity, uniqueness, and non-existence of minimal periodic solutions and subharmonic solutions for Lagrangian system. . The study of this project can not only enrich and develop the theory on Lagrangian system, but also contribute to the complement and perfection of critical point theory. It has the higher academic value.