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In this study, the period-n steady regions of McMullen family M sets are discussed. The calculation methods of the numbers of period-n steady regions are presented and the center points and boundaries of period-1 are studied by complex dynamical system theory. In addition, the problem about the free critical points is discussed. The experimental results show that the free critical points do not influence the distributions of period-n steady regions when m = d, which is proved theoretically.

Time delays widely exist in the complex dynamical systems. In this paper, time-delay complex dynamical systems are studied by analyzing the properties of corresponding Julia sets. Experimental results show that the stability of complex dynamical system changes greatly due to the influence of time delay. The variations in time-delay Julia sets are studied. In addition, the conditions holding the dynamical systems stable are found. Time delay affects the stability of complex dynamical systems, and the existence of the fixed and periodic points is also discussed.

In this study, the quaternion Mandelbrot sets and Julia sets (abbreviated as M-J sets) with additive and multiplicative noise perturbations are constructed and the changes of their fractal characteristics are explored. The experimental results show that the quaternion M sets with additive noise perturbations move in the direction of the noise, while the structures remain unchanged. The quaternion J sets with additive noise perturbations change dramatically both in the structures and in the periodicities. The quaternion M sets with multiplicative noise perturbations present scaling and rotation, while the stable regions maintain the same distributions as the non-perturbed M sets. The structures and the periodicities of the J sets change under multiplicative noise perturbations, keeping different sensitivity to the noise parameters. The M sets and J sets still share the same stable points under both the additive and multiplicative noise perturbations.