二维Logistic映射中的一种新型激变、回滞和分形

New Type Crisis,Hysteresis and Fractal in Coupled Logistic Map

研究了二维Logistic映射不动点的性质,给出了在参数空间中二维Logistic映射发生第一次分岔的边界方程。采用相图、分岔图、功率谱、Lyapunov指数计算和分维数计算方法,揭示出具有二次耦合项的二维Logistic映射从规则运动转化到混沌运动所具有的普适特征:①系统是按Pomeau-Manneville途径通向混沌的,且其间歇性与Hopf分岔有关;②系统中存在一种新型循环激变:当参数连续变化时,不稳定周期轨道按固定顺序循环与奇怪吸引子的几个小部分相遇,并导致小部分两两合并,产生出较大的奇怪吸引子;③最大Lyapunov指数的曲线具有“回滞”特征,且回滞现象常伴随循环激变的出现。同时,作者对二维Logistic映射的Mandelbrot-Julia集(简称M-J集)的研究表明:M-J集的结构由控制参数决定,且它们的边界是分形的。

The nature of the fixed points of the coupled Logistic map is investigated analytically, and the boundary equation of the first bifurcation of the map in the parameter space is derived. With the aid of phase plot, bifurcation plot, power spectra, Lyapunov exponent and fractal dimension, the general features of coupled Logistic map transforming from regularity to chaos are revealed, Chaotic patterns of the map may result from of Pomeau-Manneville route and intermittency is associated with Hopf bifurcation; a new type of crisis in the system indicates that when the parameter varies continually, the unstable periodic trajectories circulate in some fixed way, encountering several groups of little strange attractors, merge into several bigger ones; the curve of maximal Lyapunov exponent has hysteretic behaviors usually accompanied by cyclic crisis. The research of coupled Logistic map confirms that the structures of the Mandelbrot-Julia sets are determined by control parameters with fractal boundaries.