具有不定对称矩阵二阶Hamilton系统的同宿轨研究

It is well known that Hamiltonian systems exist extensively in the study of celestial mechanics, gas dynamics, aerospace science and so on. As one of the most important problems in the qualitative theory of Hamiltonian systems, homoclinic orbits play a key role in the study of nonlinear dynamical behavior of Hamiltonian systems. Using variational methods and critical point theory, this project aims at studying the problem of homoclinic orbits for second-order Hamiltonian systems and proving the existence and multiplicity of homoclinic orbits for second-order Hamiltonian systems, where L(t) is not necessarily positive definite and potential functions satisfy the different growth conditions. Our research consists of three parts: .1) potential function W(t,u) satisfies the condition of subquadratic growth and the growth rate of potential function W(t,u) can be in (1,3/2);.2) potential function W(t,u) satisfies the condition of superquadratic growth, but the Ambrosetti-Rabinowitz condition is false; .3) potential function W(t,u) satisfies some conditions of local growth.

众所周知，Hamilton系统广泛存在于天体力学、空气动力学和航天科学等多个研究领域。Hamilton系统的同宿轨道作为Hamilton系统定性理论中的一个重要方面，对于理解和研究Hamilton系统的非线性动力学行为有着非常重要的作用。本项目主要利用变分方法和临界点理论系统地研究L(t)为不定对称矩阵时二阶Hamilton系统的同宿轨道问题，证明位势函数满足不同增长条件时同宿轨道的存在性和多重性。主要研究内容有以下三个方面：.1) 位势函数W(t,u)满足次二次增长，并且增长指数小于3/2；.2) 位势函数W(t,u)满足超二次增长但是不满足Ambrosetti-Rabinowitz条件；.3) 位势函数W(t,u)满足非整体增长性条件。

在本项目中，我们讨论了具有非正定对称矩阵的二阶Hamiltonian系统同宿轨道的多重性，给出了判别准则；研究了位势函数满足局部增长性条件时二阶Hamiltonian系统同宿轨道的存在性；证明了二阶p-Laplace系统周期解的存在性；证明了带有乘法噪声和加法噪声的随机反馈系统的全局稳定性。