二维Logistic映射的分岔与分形

理论分析了二维Logistic映射的分岔，并采用相图、分岔图、功率谱、Lyapunov指数和分维数计算的方法，揭示出：二维Logistic映射可按倍周期分岔和Hopf分岔走向混沌；在倍周期分岔过程中，系统在参数空间和相空间中都表现出自相似性和尺度变换下的不变性．对二维Logistic映射的吸引盆及其Mandelbrot-Julia集(简称M-J集)的研究表明：吸引盆中周期和非周期区域之间的边界是分形的，这意味着无法预测相平面上点运动的归宿；M-J集的结构由控制参数决定，且它们的边界是分形的．

Bifurcation and fractal of the coupled logistic maps

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.