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.

在这个研究项目中，我们处理反应扩散方程的流动双稳态系统的脉冲解形成的域的动力学：epsilontauu_1=epsilon^2u_<xx>+f(u, v), v=v_<xx>+ g(u, v)，其中 0 < epsilon << 1 是层参数，0 < tau 是松弛参数。与 Hideo Ikeda 教授合作（富山大学），我们采用tau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tau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进脉冲解，另一种是通过 Hopf 分岔出现同相断裂。对于分段线性非线性 f 和 g，讨论了这两种失稳的阶数。与 Hideo Ikeda 教授和 Masayasu Mimura 教授（广岛大学）的另一项合作致力于研究驻波和行波解的全局分岔结构上述系统的。分段线性非线性的简化使我们能够揭示行进脉冲解决方案的全局分支。使用 epsilon * 0 的奇异极限分析，显示了分支上 Hopf 分岔点的外观，从中可以在数值上产生具有振荡层（移动呼吸器）的稳定传播脉冲解。此外，我们还证明：(1)所有行波解都从驻波脉冲解不稳定分叉，(2)所有行波脉冲解从驻波脉冲解分叉，通过鞍结分叉和Hopf分叉恢复稳定性，并变得稳定对于足够小的 tau > 0。