Research On Fractal Characteristics Of Generalized M-J Sets Under Noise Perturbation

In1975, American mathematician B.B.Mandelbrot researched the geometrical structure of the complex dynamical system with f(z)=z2+c, and proposed fractal theory, which is a new theory swept the world quickly. And over the years, more and more scientists participated in the study of fractal, confirming the definition, completing the theory, popularizing the knowledge and developing the application. Now, fractal theory has already penetrated into every aspect of our social life.Complex analytic dynamics f(z)=z2+c may generate two kinds of figures in the complex plane space, which are the classic Mandelbrot set and Julia sets in the fractal theory. In this paper, we introduce composing noise perturbation into dynamic system, and research the fractal character of the M-J sets under diffenent disturbances.The main content of this paper is as follows:(1) Considering the additive noise disturbance:Study the fractal character of complex analytic dynamics with f(z)=za+c+m in the quaternion space. The experiment results show that the generalized quaternion M set under additive noise disturbance move along the direction of disturbance, keep stable topology at the same time; the generalized quaternion J sets under additive noise disturbance occur dramatic changes on topology, periodicity for example, and every two J sets can transform into another via additive noise disturbance.(2) Considering the multiplicative noise disturbance:Study the fractal character of complex analytic dynamics with f(z)=k*z"+c in the quaternion space. The experiment results show that the generalized quaternion M set under multiplicative noise disturbance occurs scaling, rotation and other phenomena along the direction of disturbance, while the relative position of every period domain remains unchanged substantially; the generalized quaternion J sets under multiplicative noise disturbance occur dramatic change from the topology to periodicity, and the sensitivity to disturbance are different.(3) Considering the rational disturbance:Study the fractal character of complex analytic dynamics with f(z)=zn+γ/zd+c in the complex space. This paper mainly discusses when c=0, at which point the dynamics system is also known as McMullen maps. The experiment results show that the M-J sets corresponding to McMullen maps distribute within a closed area surrounding the original point, and numerous similar but smaller junior M-J sets distribute in this area separately. The order of period of junior Mandelbrot sets corresponding to McMullen maps may jump according their lelves. When the Julia sets corresponding to McMullen maps are sierpinski curve, a series of new Julia sets appear on it, we call them J-S-Julia sets. Moreover, Mandelbrot set remains the catalog of Julia sets under the three kind of disturbances above.