基于拓扑马蹄理论的双耦合振子系统混沌现象研究

Study of chaos in double coupled oscillators system based on topological horseshoe theory

目前,耦合振子网络中的群体混沌现象已经成为混沌研究的新兴热点。因为群体混沌的发现历史较短,缺少成熟的研究理论和方法,主要的研究手段还是集中在诸如数值计算、功率谱和Lyapunov指数等较为粗糙的方法,难以描述群体混沌发生机制,缺乏严格数学意义下的判定。借助拓扑马蹄理论,对一双耦合振子构成的四维连续系统中的群体混沌现象进行了深入研究,在其庞加莱映射的相空间中找到了一维拉伸的拓扑马蹄,不仅严格判定了双耦合振子系统中群体混沌,而且揭示了群体混沌行为发生的动力学机制。

Recently, collective chaos in coupled oscillators networks has become a new hot spot in the chaos study. On account of short growing history of collective chaos and lacking of mature theories and methods, the main means of research is still concentrated on the rough ones such as numerical computation, power spectrum and Lyapunov exponent, which are not strict judgment in mathematics and hard to describe the mechanism of chaos. By means of topological horseshoe theory, the authors studied deeply collective chaos of one four-dimensional continuous system consisting of a pair of coupled oscillators and found that topological horseshoe with expanding in one direction in the phase space of the corresponding Poincare map. It not only strictly demonstrates by numerical way that the coupled oscillator system is chaotic, but also reveals the dynamic mechanism of chaos.