几类非线性Hamilton系统周期解和多包同宿解的存在性与多重性研究

Hamiltonian system is an important branch of dynamical systems, and the existence and multiplicity of periodic solutions and homoclinic solutions of the system is of important dynamic implications. Since this system has variational structure, in recent decades, the mathematicians such as Ambrosetti, Rabinowitz and Chang have developed many variational methods, such as infinite dimensional Morse theory and minimax principle, which made significant progress on this issue. In this project, by using Morse theory we will study the existence and multiplicity of periodic solutions, multibump homoclinic solutions and rotating periodic solutions for several classes of Hamiltonian systems, the details are as follows: (1) using the Morse theory and a deformation technique to study the existence and multiplicity of nontrivial periodic solutions for second order Hamiltonian systems with damping term and sign-changing potential; (2) using the Morse theory and a technique of gluing translates of basic isolated homoclinic solution to study the multiplicity of multibump homoclinic solutions for second order Hamiltonian systems with damping term; (3) using the Morse theory and a critical point theorem for symmetric functionals to study the existence and multiplicity of nontrivial rotating periodic solutions for first order Hamiltonian systems under a more general twist condition.

Hamilton系统是动力系统的一个重要分支，而对该类系统周期解和同宿解的存在性与多重性研究有着重要的动力学意义。由于这类问题具有变分结构，近几十年来，Ambrosetti,Rabinowitz和张恭庆等数学家发展了无穷维Morse理论和极小极大原理等很多变分方法，从而使得对Hamilton系统周期解与同宿解的研究取得了重大进展。本项目拟利用Morse理论研究几类非线性Hamilton系统周期解，多包同宿解与旋转周期解的存在性与多重性，具体内容如下：(1)利用Morse理论和形变技巧研究具有变号位势的含阻尼项二阶Hamilton系统非平凡周期解的存在性与多重性；(2)利用Morse理论和粘合基本孤立同宿解的平移的技巧研究含阻尼项二阶Hamilton系统多包同宿解的多重性；(3)利用Morse理论和对称泛函临界点定理研究一般扭转条件下一阶Hamilton系统非平凡旋转周期解的存在性与多重性。

Hamilton系统是一类重要的动力系统，而对这类系统周期解和同宿解的存在性与多重性的研究具有非常重要的理论意义。本项目利用变分方法研究了几类含阻尼项二阶Hamilton系统（也称为阻尼振动系统）周期解和同宿解的存在性与多重性。首先，利用Morse理论和对称泛函多重临界点定理证明了一类具有一般线性增长非线性项的含阻尼项二阶Hamilton系统周期解的存在性与多重性；其次，利用喷泉定理证明了一类新的超二次含阻尼项二阶Hamilton系统无穷多非平凡周期解的存在性；再次，利用极小化方法证明了一类具有非正位势的含阻尼项二阶Hamilton系统非平凡同宿解的存在性；最后，利用Clark定理证明了一类部分次二次含阻尼项二阶Hamilton系统无穷多小范数非平凡同宿解的存在性。这些工作丰富和发展了Hamilton系统周期解和同宿解的存在性与多重性方面的相关研究。