Bifurcation and fractal of the coupled logistic maps
二维Logistic映射的分岔与分形

The bifurcation of the coupled Logistic map is analyzed theoretically. By using phase graphics, bifurcation graphics, power spectra, the computation of the fractal dimension and the Lyapunov exponent, the paper reveals the general features of the coupled Logistic map transition from regularity to chaos, the following conclusions are shown: (1) Chaotic patterns of the map may emerge out of double-periodic bifurcation and Hopf bifurcation, respectively; (2) During the process of double-period bifurcation, the system exhibits the self-similar structure and invariance which is under scale variety in both parameter space and phase space. Prom the research on attractor basin of the coupled Logistic map and Mandelbrot-Julia set, the following conclusions are indicated: (1) The boundary between periodic and non-periodic regions is fractal, and that indicates the impossibility to predict the moving end-result of the points in phase plane; (2) The structures of the Mandelbrot-Julia sets are determined by the control parameters, and their boundaries have the fractal characteristic.