Bifurcation and fractal of the coupled logistic maps

The bifurcation of the coupled Logistic map is analyzedtheoretically. By using phase graphics, bifurcation graphics, power spectra,the computation of the fractal dimension and the Lyapunov exponent, thepaper reveals the general features of the coupled Logistic map transitionfrom regularity to chaos, the following conclusions are shown: (1) Chaoticpatterns of the map may emerge out of double-periodic bifurcation and Hopfbifurcation, respectively; (2) During the process of double-periodbifurcation, the system exhibits the self-similar structure and invariancewhich is under scale variety in both parameter space and phase space. Fromthe research on attractor basin of the coupled Logistic map andMandelbrot-Julia set, the following conclusions are indicated: (1) Theboundary between periodic and non-periodic regions is fractal, and thatindicates the impossibility to predict the moving end-result of the pointsin phase plane; (2) The structures of the Mandelbrot-Julia sets are determinedby the control parameters, and their boundaries have the fractal characteristic.