几类浅水波方程中孤立波及混沌分析

发布时间：2018-01-12 05:13

本文关键词：几类浅水波方程中孤立波及混沌分析　出处：《江苏大学》2017年硕士论文　论文类型：学位论文

【摘要】：本学位论文主要借助非线性动力学理论研究了几类浅水波方程孤立波解的稳定性问题以及外界干扰下系统产生混沌现象的理论证明和数值分析。进一步,通过Melnikov方法研究了受扰动力系统的混沌控制问题。本文首先研究了外部周期扰动对mKdV方程的孤立波的影响,通过改进Melnikov方法,理论证明了孤立波在任意周期扰动下均能转化为混沌状态。进一步研究发现更丰富的扰动频率、更快的传播速度以及更大的非线性参数需要更大的控制强度来抑制混沌。其次,研究了广义Camassa-Holm方程中孤立波的存在性和稳定性。非线性强度对孤立波的形状和稳定性有重要影响:当非线性项的强度是奇次方时,该方程被证明有正孤立波,并且当波速超过临界值时孤立波是轨道稳定的。当非线性项的强度是偶次方时,该方程被证明同时有正孤立波和负孤立波,并且在任何波速下孤立波都是轨道稳定的。最后,利用Menikov方法验证了在任意非线性强度下系统受外部周期扰动时孤立波都会转变成混沌状态。通过设计线性反馈控制器,混沌可以被控制到一个稳定的状态。
[Abstract]:This thesis is mainly through theoretical proof and numerical analysis of chaotic system under several kinds of shallow water wave equation stability of solitary wave solutions and the interference theory of nonlinear dynamics was studied. Further, through the Melnikov method to study the problem of chaos control of disturbed power system. This paper studies the external periodic perturbation of the mKdV equation, solitary the influence of wave, by using the improved Melnikov method is proved by the theory of solitary waves in arbitrary periodic perturbations can be transformed into a chaotic state. Further studies found that the disturbance frequency is more abundant, faster propagation speed and nonlinear parameters of the need for greater control of greater strength to suppress chaos. Secondly, study the existence and stability of the solitary wave generalized Camassa-Holm equation. Have important influence of shape and strength on the stability of nonlinear solitary wave: when the nonlinear term strength Is odd side, the equation is proved to have solitary waves, and when the velocity exceeds the critical value of solitary wave is stable. When the nonlinear term is the strength of second party, the equation is proved to be both positive and negative solitary wave and solitary wave, solitary wave at any wave velocities are orbit stable. Finally, using Menikov method to verify the external periodic perturbations in any nonlinear system by solitary wave intensity are converted into chaotic state. Through the design of linear feedback controller, chaos can be controlled to a steady state.

【学位授予单位】：江苏大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.29

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