Global Bifurcation Structure of Nonlinear Dynamics of Domain Motion

在本研究项目中，我们研究了流动双稳反应扩散方程组的脉冲解所形成的区域动力学：epsilontauu_1=epsilon^2u_&lt；xx&gt；+f(u，v)，v=v_&lt；xx&gt；+g(u，v)，其中0&lt；epsilon&lt；&lt；1是层参数，0&lt；tau是松弛参数。结果表明，当T_u减小时，驻留脉冲解存在两种失稳。一种是通过静态分叉出现行波脉冲解，另一种是通过Hopf分叉出现同相酿酒人的行脉冲解。对于分段线性非线性f和g，讨论了这两个失稳的顺序。另一个与池田秀夫教授和广岛大学三村正治教授的合作。研究了上述系统的驻留脉冲和行波脉冲解的全局分叉结构。分段线性非线性的简化使我们能够揭示行波脉冲解的全局分支。利用奇异极限分析的方法，给出了分支上Hopf分岔点的出现，由此得到了具有振荡层(行进呼吸)的稳定传播脉冲解。此外，我们还证明了：(1)所有的行脉冲解都不稳定地从驻留脉冲解分叉出来；(2)所有从驻留脉冲解分叉出来的行脉冲解都通过鞍结分叉和Hopf分叉恢复了它们的稳定性，并且对于足够小的tau&gt；0，它们变得稳定。

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.