Generalized quaternion M sets and Julia sets perturbed by dynamical noises

Nonlinear Dynamics, May 2015

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.

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Generalized quaternion M sets and Julia sets perturbed by dynamical noises

Generalized quaternion M sets and Julia sets perturbed by dynamical noises Yuanyuan Sun Peng Li Zhixing Lu 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. Generalized M sets; Generalized J sets; Noise perturbations; Quaternion; Fractal - focus on the discussion of the structure stability of the iterative dynamic systems. In recent years, the highdimensional fractals have drawn peoples much attention. The quaternion MJ sets are one of the major areas people interested in, researches which help explore the 3-space dynamics [15]. The subject of dynamics is concerned with what happens to a physical or geometric system over time, when it is subjected to a force or undergoes some kind of manipulation. The dynamics of motion in 3-space is easily expressed in terms of quaternion operation. This relationship implies computational advantages in using quaternions to express 3-D spatial manipulation [1]. On the other hand, there is a variety of noise disturbance in nature. They produce various effects on the movements and the states of the dynamic systems. The noise is a random variable essentially. People utilized different models, such as Gaussian noise [6], sinusoidal noise [6] and time-delay model [7], to simulate the interference characteristics of the dynamic system evolution. Argyris et al. [810] 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. Andreadis and Karakasidis [1113] also discussed the topological closeness of perturbed MJ sets and proposed a definition for a probabilistic Mandelbrot map. Negi et al. introduced a new noise criterion and analyzed its effect on superior MJ maps, discussing the difference comparing the usual MJ maps with perturbation [1416]. Rani dynamical noise may take the form of an additive or multiplicative expression which illustrates the kind of parameters by which noise may appear in the equations of a dynamical system. The analog of additive noise is of the following form, where pn represents a random deviation from the deterministic orbit. The multiplicative dynamical noise perturbs the form of the nonlinear function or the parameters in the map. If we write the deterministic equation as with as the bifurcation parameter, then parametric fluctuations would be of the form and Agarwal [17] presented an integrated approach to study the additive and multiplicative noises with respect to perturbations in superior Julia sets. Wang et al. [18,19] researched on the structural characteristic and the fission-evolution law of perturbed generalized MJ sets on complex plane. Sun and Wang [20] analyzed the effect of noise perturbation on the quaternion M sets. Eliazar [21] laid more focus on the noise in relationship between noise and fractal, showing that classic shot noise is intrinsically fractal. On the other hand, the MJ sets of the other models are proposed and some non-trivial results are obtained. Wang and Song [22] studied the generalized MJ sets for bicomplex numbers. Liu [23] studied the M sets in coupled map lattice for control and synchronization. Danca et al. [24] obtained the connectivity domains of alternated Julia sets, defined by switching the dynamics of two quadratic Julia sets. In this paper, the generalized quaternion MJ sets perturbed with dynamical noise are studied. The quaternion MJ sets are constructed, and the additive and multiplicative noise perturbations are interfered on the sets. The topological structures, the stable regions and the periodicities of the disturbed MJ sets are explored. The rest of the paper is organized as follows. Section 1 introduces the definitions of the generalized quaternion MJ sets under dynamical noise perturbations. Section 2 describes the influence on the quaternion MJ sets with additive dynamic noises. 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. Sect (...truncated)


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Yuanyuan Sun, Peng Li, Zhixing Lu. Generalized quaternion M sets and Julia sets perturbed by dynamical noises, Nonlinear Dynamics, 2015, pp. 143-156, Volume 82, Issue 1-2, DOI: 10.1007/s11071-015-2145-7