一类广义Logistic映射的分形特征与同步

摘要

In this paper,Julia and Mandelbrot sets of a class of two-dimensional generalized real Logistic map are discussed.Firstly,the fractal dimension of Julia set of the real map is obtained by box-counting method,then the control of Julia sets of the real map is accomplished via using the linear feedback control.Secondly,we introduce the concept of synchronization of Julia sets between different systems and achieve the synchronization of Julia sets of two real maps with different parameters using the nonlinear coupling control.Finally,the Mandelbrot set of the 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 method and the synchronization method.%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.