解析法提取分段线性混沌系统周期轨道的研究

Study on analytic approach for extracting periodic orbits of piecewise linear chaotic systems

采用解析法研究提取分段线性混沌系统周期轨道．分段线性混沌系统的状态空间被数个切换面分割成若干个线性子区间．联合求解周期轨道在各子区间的解方程,可得该周期轨道在各切换面的坐标及在各子区间的运行时间,从而得到所提取周期轨道的分段解析表达式．与传统的数值提取法相比,所提出的解析法具有提取的周期轨道精确度高、周期轨道信息的存储量小、便于实时应用、易于进行稳定性分析和控制等优点．应用该解析方法分别求得三阶蔡氏电路和四阶蔡氏电路混沌吸引子中许多周期轨道,验证了该方法的可行性和实用性．

This paper studies the extraction of periodic orbits of piecewise linear chaotic systems with analytic approach. The state space of piecewise linear chaotic system was cut into several linear subspaces by several exchanging surfaces. By solving all the equations of periodic orbit in these subspaces, the coordinates of the periodic orbit on each exchanging surface and the running time of the periodic orbit in each subspace were obtained, from which the analytic expressions in sections of periodic orbit can be derived further. Compared with the traditional numerical method, the periodic orbit extracted by the analytic approach is more accurate, requires smaller memory size and is more applicable in real time systems ; Furthermore, it is more suitable for stable analysis and control of the systems. By this analytic approach, many periodic orbits in the chaotic attractors of 3^rd-order and 4^th-order Chau＇s circuits can be obtained, which shows that the approach proposed by the authors is correct and practicable.