From Newton Fractals to Zhang Fractals Yielded via Solving nonlinear equations in Complex Domain

A novel type of fractals (i.e., Zhang fractals) is yielded via solving time-varying or static nonlinear equations in complex domain by discrete-time complex-valued Zhang dynamics (DTCVZD). The DTCVZD model that uses different types of activation functions can generate various Zhang fractals. These fractals are different from the conventional Newton fractals discovered 30 years ago (since 1983) because they are generated via solving static nonlinear equations and time-varying nonlinear equations. Newton fractals cannot be obtained via solving time-varying nonlinear equations (i.e., Newton iteration cannot converge to the theoretical solutions of time-varying nonlinear equations). DTCVZD model can reduce to Newton iteration when suitable parameters are used. This reduction implies that the novel Zhang fractals generated by the proposed DTCVZD model incorporate Newton fractals as special cases. Computer simulation results demonstrate that the DTCVZD model can serve as a new algorithm to generate new fractals.