Abstract: The nature of the fixed points of the general two-dimensional quadratic map is considered analytically,and the boundary equation of the first bifurcation of the map in the parameter space is given out.The general feature of the nonlinear dynamic activities of the map is analyzed by the method of numeral computation.By utilizing the Lyapunov exponent as criterion, this paper constructs the strange attractors of the general two-dimensional quadratic map,and calculates fractal dimension of the strange attractors according to the Lyapunov exponents.At the same time,the researches on the fractal images of the general two-dimensional quadratic map make it clear that when the control parameters are different,the fractal images are different from each other and these fractal images exhibit the fractal property of self-similarity.