Zhang fractals yielded via solving nonlinear equations by discrete-time complex-valued ZD

Abstract

In this paper, a novel kind of new fractals, named Zhang fractals, is yielded by using the discrete-time complex-valued Zhang dynamics (DTCVZD) to solve the nonlinear equations in the complex domain. Such a novel DTCVZD model is designed based on the elimination of an indefinite complex-valued error function, instead of a square-based non-negative energy function associated with the discrete-time complex-valued gradient-based dynamics (DTCVGD). Comparing with the well-known (generalized) Newton fractals (i.e., the famous fractals generated by the well-known Newton iteration), we find that the novel Zhang fractals synthesized by the proposed DTCVZD model incorporate such Newton fractals as special cases. The Zhang fractals generated by the novel DTCVZD model are completely different from Newton fractals. The DTCVZD model using different types of activation functions can be seen as a new iterative algorithm to generate new fractals, i.e., Zhang fractals.