Abstract
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.
Highlights
Mandelbrot sets and Julia sets are the classic sets in the fractal geometries
Experimental results show that quaternion J sets with multiplicative noise disturbance have various changes in the topology structures and periodicities
The experimental results show that in the evolution process of the M sets dynamical system additive noise disturbs the directions and positions of the periodic regions and multiplicative noise disturbs the speed for escaping and iteration, which corresponds to the changes of the M sets in spatial region and time space, respectively
Summary
Mandelbrot sets and Julia sets are the classic sets in the fractal geometries. The researches on the M–J sets. Argyris et al [8,9,10] analyzed the classification and influence of noise in a dynamical system and proposed a new scheme of analytic and non-analytic perturbations of the Mandelbrot map. The generalized quaternion M–J sets perturbed with dynamical noise are studied. The displacement of the stability regions of the M sets is theoretically analyzed, the characterization of the J sets is discussed, and the stable periodic points of the quaternion M– J sets are calculated.
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