Solving Nonlinear Equations to Generate Fractals via Discrete Varying-Parameter Complex-Valued Dynamics

In this paper, the discrete complex-valued dynamics is used to solve nonlinear equations to generate fractals. This new class of fractals, called Zhang fractals, are different from the famous Newton Iteration. By introducing exponential functions in the basis of the iteration of Zhang fractals, generated fractals have wider dynamic range due to rapid growth or decrease of values during iterations. This great character also increases the diversity of fractals since it may lead to different iterative paths and results, which will result in multiple possible fractal mor-phologies. Moreover, the using of different activation functions will further improve the diversity of generated fractal. Actually, the selection and adjustment of activation functions provide more flexibility and creativity for the generation of fractals. A new way of fractal generation is opened up by introducing a exponential function in the traditional discrete complex-valued dynamics. The generation of fractals can be applied to many fields such as art and natural phenomena, and is of great significance to understanding the complex structure of nature.