Dynamic analysis of the coupled logistic map redounds to know and predict the characteristics of high-dimension complex nonlinear system. Using the method combining calculation and experiment, the following conclusions are shown: (1) The boundary equation of the first bifurcation of the coupled logistic map in the parameter space is given out. (2) Chaotic patterns of the coupled logistic map may emerge out of double-periodic bifurcation and Hopf bifurcation, respectively. (3) The boundary between periodic and non-periodic regions in the attraction basin of the coupled logistic map is fractal, which indicates the impossibility to predict the moving result of the points in phase plane. (4) The structures of the Mandelbrot-Julia sets are determined by the control parameters, and their boundaries have the fractal characteristic.