Kuramoto-Sivashinsky方程的稳态解研究

针对研究Kuramoto-Sivashinsky(K-S)方程的稳态解时遇到的多数轨道快速逃逸困难,应用变分法对该混沌系统的不稳定周期轨道开展了系统计算。当静态K-S方程取很小的积分常数值时,提出利用多尺度平均微扰方法分析对应系统相空间不动点和轨道的分布情况。结果表明,小积分常数值的动力系统行为是极其复杂的,同时存在有多条异宿轨道和周期轨道;当取固定的积分常数c=0.352 1时,可以根据四条周期轨道的拓扑结构建立合适的符号动力学,从而实现对全部短周期轨道的系统搜寻。

A study on the steady solutions of the Kuramoto-Sivashinsky equation

In order to overcome the difficulties for the fast escape of most orbits encountered in the steady solutions of the Kuramoto-Sivashinsky (KS) equation, a systematic calculation of the unstable periodic orbits of the chaotic system is carried out based on the variational method. When the steady solutions of the KS equation takes a small integral constant, the multi-scale averaging perturbation method is proposed to analyze the distribution of the fixed points and orbits in the corresponding phase space. The results indicate that the behavior of the dynamical system with small integral constant is very complicated, and there are many heteroclinic orbits and periodic orbits simultaneously. At a fixed integral constant c=0.3521, the appropriate symbolic dynamics can be established according to the topological structure of four periodic orbits, so as to realize the search systematically for all short cycles.