J-M方程的行波解分支及一类PLL方程的混沌与次谐波分支

【作者】 冯大河；

【导师】 李继彬；

【作者基本信息】 昆明理工大学 ， 应用数学， 2003， 硕士

【摘要】 全文分为两部分，第一部分利用动力系统分支理论研究了J-M方程，在一类特定曲面上得出了该方程的所有精确行波解。本部分由六节组成，第一节介绍了该系统的研究现状并给出了其行波方程在特定曲面上的两个参数条件；第二、三节分别讨论了行波系统(19)在这两个条件下的分支集与相图；第四、五节根据第二、三节的情形分别求得了J-M方程在给定条件下的所有精确显式行波解；第六节给出了主要结果第二部分利用Melnikov方法研究了一类PLL方程，证明了该系统Smale马蹄型混沌及次谐波的存在性，并并分别给出了其混沌区域及次谐波分支区域本部分由四节组成，第一节介绍了Melnikov方法及所要研究的系统；第二节讨论了其未扰动系统的定性性质；第三、四节分别证明了混沌及次谐波的存在性，并给出了混沌区域及次谐波分支区域更多还原

【Abstract】 The paper includes two parts. In the first part, J-M equation is considered by using the bifurcation theory of dynamical system. Also, all the exact traveling wave solutions in a family of special curved surfaces are obtained. This part consists of six sections. In section 1, the introduction is stated and two parameter conditions are obtained. In sections 2 and 3 ,under conditions (1) or (2), the bifurcation sets and phase portraits of (1.9) are given, respectively. In sections 4 and 5, corresponding to sections 2 and 3, all exact and explicit formulas of traveling wave solutions under given parameter conditions are showen. In section 6, the main results are given.In the second part, a family of PLL equations are investigated by using Melnikov method. the existence of chaos in the sense of Smale horseshoes is showen and the partitions of the regions of chaos and subharmonic bifurcations are obtained. This part includes four sections. In section 1, the Melnikov method and the system which will be discussed are narrated. In section 2, the qualititative property of the unperturbed system is studied. In section 3 and 4, the existence of chaos and subharmonic bifurcations is proved and their existent regions are given.更多还原