混沌系统周期轨道的最速下降方法研究

A steepest descent method for analyzing the periodic orbits of a chaotic system

提出了一种搜寻混沌系统不稳定周期解的新方法,首先应用泰勒展开将微分系统离散化,通过离散系统构造一个目标函数,并使其极小值点(0值点)对应该系统的不稳定周期解,再应用伪周期搜寻算法得到最优化算法的初始值,最后利用最速下降法对结果进行优化,得到系统的周期解。应用本文方法搜寻出Lorenz系统的多个不稳定周期点,包括具有朴素周期、超大周期以及倍周期的周期解,表明了本文方法的有效性和实用性。

A new method of searching for unstable periodic solutions in a chaotic system is presented in this paper.The Taylor expansion is applied to transform a differential dynamical system to a discrete dynamical system. A target function is then built such that its minimum（0 value） corresponds to an unstable periodic orbit for the differential system. A searching method for pseudo-periodic orbits is given to determine the initial value for an optimization method. The steepest descent method is then employed to find the minimum of the target function. This method is applied to the famous Lorenz system and the unstable periodic orbits obtained include a simple cycle,a super cycle and a double cycle. The results show that this method is effective and practical.