模式生成系统同宿环全局分岔问题研究

Study on the Homoclinic Orbit Bifurcations in Pattern Formation System

对一维Gray-Scott模型中脉冲自我复制的精细全局动力学结构进行了数值探索,分析了奇异同宿稳定解及其分岔问题。结果发现,与系统相应的常微分方程的解在余维2分岔时具有组织特征,并由其产生与偏微分方程孤波解对应的2环或n环同宿轨。对全局分岔图的分析发现,自我复制系统动力学特性与参数空间中折叠分岔的层次结构密切相关。数值结果表明,Bogdanov-Takens分岔点及对应于某一同宿轨的角状形参数域对系统的周期轨、同宿轨、全局分岔以及复杂混沌动力学具有决定性作用。数值仿真过程揭示反应扩散系统中存在调制的2脉冲及多脉冲解,并伴随有脉冲自我复制及分裂过程。

Singular homoclinic stationary solutions and its bifurcations in the one-dimensional Gray-Scott model were carried out, A careful analysis of the scenario of the global bifurcation diagram suggests that the dynamics of self-replicating system is related to a hierarchy structure of folding bifurcation branches in parameter regions. The numerics suggests the Bogdanov-Takens points together with a presence of critical points emanating from the particular codimension-two homoclinic orbit play a central role for global bifurcation of periodic obits and the homoclinic solutions and the complex chaotic dynamics. Numerical simulation also reveals the existence of the modulating two-pulse or multi-pulse, which companying the procedure of pulse self-replicating in reaction-diffusion systems.