复杂动力系统的势函数分解与初始值敏感问题研究

The Study on Potential Decomposition and Initial Value Sensitive Problem in a Complex Dynamic System

随着蝴蝶效应的提出,混沌理论逐渐发展为世界性的热点难题。而初始值敏感现象,作为混沌动力系统的一个重要特征,也仍未被人们完全认识和了解。发现并描述了在著名的Kuramoto模型中的初始敏感现象,它是一种特殊的轨迹跳变分离现象,传统的李雅普诺夫指数等方法不能对此给出一个清晰的解释。运用敖平教授近年提出的势函数分解理论,构造出模型势函数的解析形式,然后运用独特的分析方法再结合一些传统的计算,对这种现象给出了一个直观的解释,这也是对混沌系统和初始值敏感问题研究的一个重要突破。

With the proposal of the butterfly effect, chaos theory gradually developed into a worldwide hotspot problem. The initial value sensitive phenomenon, as an important feature of chaotic dynamical system, has not yet been fully understood. The initial value sensitive phenomenon in the well-known Kuramoto model is found and de- scribed. The phenomenon is a special jump separation of trajectories, which cannot be explained directly by tradi- tional methods such as Lyapunov exponents. The potential decomposition theory proposed by Prof. Ao recently is use. Constructing the analytic form of model＇ s potential decomposition, classical calculation and novel analysis on the problem are combined. Finally an intuitive interpretation of the phenomenon is obtained, which makes a pro- gress on the research of initial va｜ue sensitive problem and chaos theory.