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Global Bifurcation Structure of Nonlinear Dynamics of Domain Motion

域运动非线性动力学的全局分岔结构

基本信息

批准号：
09640303
负责人：
IKEDA Tsutomu
金额：
$ 1.92万
依托单位：
Ryukoku University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640303/
关键词：
Bistable systems Piecewise-linear nonlinearity Internal transition layers Standing pulse solutions Traveling pulse solutions In-phase breathers Traveling breathers Hopf bifurcation ポップ分岐

项目摘要

In this research project, we deal with the dynamics of domain formed by pulse solutions of the fllowing bistable system of reaction-diffusion equations :epsilontauu_1=epsilon^2u_<xx>+f(u, v), v=v_<xx>+g(u, v), where 0 < epsilon << 1 is a layer parameter and 0 < tau a relaxation one.In a joint work with Prof. Hideo Ikeda (Toyama Univ.), we have studied the bifurcation phenomena of standing pulse solutions of the above system by adopting tau as a controllable parameter. It is shown that there exist two types of destabilization of standing pulse solutions when tau decreases. One is the appearance of traveling pulse solutions through the static bifurcation and the other is that of in-phase brealbers via the Hopf bifurcation. The order of these two destabilization is discussed for piecewise-linear nonlinearitics f and g.Another joint work with Prof. Hideo Ikeda and Prof. Masayasu Mimura (Hiroshima Univ.) is devoted to the study of global bifurcation structure of standing and traveling pulse solutions of the above system. The simplification of piecewise-linear nonlinearity enables us to reveal the global branch of traveling pulse solutions. Using the singular limit analysis as epsilon * 0, the appearance of the Hopf bifurcation point on the branch is shown, from which stable propagating pulse solutions with oscillating layers (travelling breathers) arise numerically. Moreover, we have shown that(1)All traveling pulse solutions unstably bifurcate from standing pulse solutions,(2)All traveling pulse solutions bifurcated from standing pulse solutions recover their stability through the saddle-node bifurcation and the Hopf bifurcation, and they become stable for sufficiently small tau> 0.

本文研究了反应扩散方程组的脉冲解所形成的区域的动力学性质：ε u_1 = ε u_2u_<xx>+f（u，v），v=v_<xx>+g（u，v），其中0 <τ << 1是层参数，0 <τ是弛豫参数。采用τ作为可控参数，研究了上述系统驻脉冲解的分岔现象。结果表明，当τ减小时，驻脉冲解存在两种类型的失稳。一种是通过静态分岔出现行波脉冲解，另一种是通过Hopf分岔出现同相脉冲解。这两个不稳定的顺序进行了讨论，分段线性非线性f和g。另一个联合工作与池田秀夫教授和三村教授（广岛大学）。研究了上述系统驻脉冲解和行波脉冲解的全局分歧结构。对分段线性非线性项的简化使我们能够揭示行波脉冲解的全局分支。利用奇异极限分析作为λ * 0，给出了在分支上的Hopf分岔点的出现，并从该分岔点数值地得到了具有振荡层（旅行呼吸子）的稳定传播脉冲解.此外，我们还证明了：（1）所有行波脉冲解都不稳定地从驻定脉冲解分叉出来;（2）所有从驻定脉冲解分叉出来的行波脉冲解都通过鞍结分叉和Hopf分叉恢复稳定，并且当τ> 0时，它们变得稳定.