一般二维二次映射中的混沌与分形

研究了一般二维二次映射不动点的性质，给出了在参数空间中一般二维二次映射发生第一次分岔的边界方程，通过数值计算方法分析了一般二维二次映射非线性动态行为的普适特征，并利用Ｌｙａｐｕｎｏｖ指数作判据，构造了该映射的奇吸引子，又根据Ｌｙａｐｕｎｏｖ指数求出了奇怪吸引子的分维数。同时，对一般二维二次映射的分形图研究表明，控制参数不同，分形图互不相同，且它们的边界是分形的。

Chaos and Fractal of the General Two-Dimensional Quadratic Map

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.