New Type Crisis, Hysteresis and Fractal in Coupled Logistic Map
Wang Xingyuan(1,2),Shi Qijiang(1)
(1 School of Electronic & Information Engineering, Dalian University of Technology , Dalian 116024, China)
(2 School of Computer Science & Technology, Dalian Maritime University, Dalian 116026, China)

Abstract: 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 reve aled, Chaotic patterns of the map may result from of Pomeau-Manneville route a nd 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.
Keywords: coupled Logistic map, bifurcation, crisis, hysteresis, fractal, Mandel brot-Julia set.