平面微分系统极限环分支与扰动波动方程的同(异)宿分支及混沌

项目名称： 平面微分系统极限环分支与扰动波动方程的同(异)宿分支及混沌

项目编号： No.11301455

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 周宏宪

作者单位： 许昌学院

项目金额： 22万元

中文摘要： 随着动力系统分支理论的深入研究和广泛应用,利用动力系统分支与混沌基本理论和方法研究非线性波动方程的特殊精确解及复杂动力学性质成为关注热点.本项目首先利用动力系统分支方法研究平面微分系统的局部及大范围极限环分支,讨论微分系统周期解的存在性以及稳定性问题;其次,将相关分支理论和方法拓广,研究非线性波动方程的行波解分支及其稳定性,参数扰动作用下波动方程所对应的动力学系统的解的渐近行为、同(异)宿轨的存在性与保持性及其相应的混沌性态问题;同时研究探索时滞、噪声诱发的波动方程的次谐分支、同(异)宿分支、混沌运动及普适特征,揭示时滞和噪声在混沌运动中的内在关联和作用机理,发展Melnikov方法的应用条件和形式.本项目的研究将丰富动力系统分支理论和方法,为波动方程及其复杂动力学性质研究奠定理论和技术基础.

中文关键词： 非线性波方程；行波解；同宿分支；全局吸引子；正则性

英文摘要： The bifurcation theory of dynamical system is substantially studied and widely applied, much attention was paid to applying the bifurcation and chaos theory of dynamical systems to seek the special exact solutions and investigate the complicated dynamical properties of nonlinear wave equations. In the project, we firstly study the local and global bifurcations of limit cycle for planar differential systems by using the bifurcation theory of dynamical systems, discuss the existence and stability of periodic solutions; secondly, generalizing the use of related bifurcation theory and methods in nonlinear wave equation, we study the bifurcation and stability of travelling wave of nonlinear wave equations, analyze asymptotic properties of solutions, the existence, persistence and the related chaotic behavior of the homoclinic and heteroclinic orbits of corresponding dynamical system under parameter perturbation. Simultaneously, we explore subharmonic bifurcation, homoclinic (heteroclinic) bifurcation, chaotic motion and universal characteristic induced by time delay and noise in wave equations. Moreover, internal connection and mechanism of action are revealed in chaotic motion. Application conditions and form of Melnikov method are also expanded. The research project will enrich the bifurcation theory and method of

英文关键词： Nonlinear wave equations；Travelling wave solution；Homoclinic bifurcation；Global attractor；Regularity