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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
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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)