From Newton Fractals to Gradient Fractals in Addition to Zhang Fractals

The paper employs gradient dynamics (GD), in addition to Zhang dynamics (ZD), to generate new fractals which are quite different from the conventional Newton fractals (NF). The proposed gradient fractals (GF) have three types. Specifically, via appropriate iterative formulas derived in complex domain, the first-type and second-type fractals are generated, i.e., 1GF and 2GF. By expanding the conventional gradient formula from real domain to complex domain, the third-type fractals are thus obtained, i.e., 3GF. Besides, Zhang fractals (ZF) are yielded via solving static and time-variant equations using ZD. Numerical experiments substantiate the effectiveness of complex-valued GD and ZD for generating new fractals. Connections and differences among NF, GF and ZF are also revealed.