一类广义Logistic映射的分形特征与同步
Fractal features and synchronization of a class of generalized Logistic map

本文讨论了一类二维广义Logistic实映射的Julia集和Mandelbrot集. 首先采用盒维数计算法, 计算了实映 射Julia集的分形个维数, 并引入一种线性反馈控制的方法, 对实映射的Julia集进行了控制. 其次引入不同系统 间Julia集同步的概念, 通过非线性耦合控制的方法, 对具有不同参数两个实映射的Julia集进行了同步. 最后通过引 入实参数的方法构造了实映射的Mandelbrot集, 并通过梯度控制法实现了具有不同参数的两个实映射Mandelbrot集 的同步. 仿真结果表明了控制和同步方法的有效性.
In this paper, Julia and Mandelbrot sets of the two dimensional generalized real Logistic map are discussed. Firstly, the fractal dimension of a Julia set of this real map is obtained by using the box counting dimension method, and the control of Julia sets of this real map are accomplished by using the linear feedback control. Secondly, the concept of synchronization of Julia sets between different systems are introduced, then the synchronization of Julia sets of two real maps with different parameters are realized by using the nonlinear coupling control. Finally, the Mandelbrot set of this real map is constructed by introducing real parameters, and gradient control is taken to realizing the synchronization of Mandelbrot sets of two real maps which have different parameters. Simulation results show the effectiveness of the control and synchronization methods.