The generalized Mandelbrot set perturbed by composing noise of additive and multiplicative

Wang, Xingyuan; Jia, Ruihong; Zhang, Zhenfeng

doi:10.1016/j.amc.2008.12.017

Cited by 23 publications

(3 citation statements)

References 18 publications

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“…(iv) Extension of the algorithms and implementations used in this paper to functions of the form a 1 x α 1 y β 1 +a 2 x α 2 y β 2 +· · ·+a n x α n y β n with complex indexes numbers α 1 , β 1 , · · · , α n , β n ∈ C. This requires the study of the dynamics of new homogeneous maps defined on products of projective lines. The analysis of the new fractal structures of the boundary of attraction basins should incorporate some of the techniques developed in [31,34,35,37,41,[44][45][46]. (v) Study of the changes and implementations of algorithms that would allow us to analyze basins of attracting roots of two bivariate equations formulated with more general functions: complex exponential functions, trigonometric functions, etc.…”

Section: Conclusion and Further Workmentioning

confidence: 99%

Bivariate Newton-Raphson method and toroidal attraction basins

Paricio

2015

Numer Algor

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When the numerical Newton-Raphson method is applied to find the intersections of two algebraic curves (that is, the roots of a pair of bivariate polynomials), some difficulties appear when the value of a denominator of the corresponding bivariate rational functions is zero. In this paper we give a solution to these problems by using adequate homogeneous coordinates and extending the domain of the iteration function. The iteration of a map given by a pair of bivariate rational maps is analyzed by taking a canonical extension, which is defined on the product of two copies of an (real or complex) augmented projective line. This method gives a global description of the basins of attraction of fixed points of an iteration. In particular, the attraction basins of fixed points associated to the intersection points of the two algebraic curves is obtained. As a consequence of our techniques, we are able to plot the attraction basins of a real root either in a torus (considered as a compactification of the real plane) or in an open square, which is homeomorphic to the global real plane containing the two algebraic curves.Keywords Roots of polynomial equations · Bivariate Newthon-Raphson method · Discrete semi-flow · Basin of attraction · Bivariate polynomials · Intersection of algebraic curves · Toroidal fractals · Toroidal basins of attraction Mathematics Subject Classification (2010) 65H04 · 65S05 · 68W25The author acknowledges the financial help given by the project PROFAI13/15.

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Section: Conclusion and Further Workmentioning

confidence: 99%

Bivariate Newton-Raphson method and toroidal attraction basins

Paricio

2015

Numer Algor

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“…Immediately afterward, the Julia set of the perturbed Mandelbrot map was investigated in detail [15,16]. Moreover, the noiseperturbed generalized Mandelbrot sets were considered by [17], and by composing the additive and multiplicative noise, the perturbations of the generalized Mandelbrot set were also searched in [18]. With the aid of developing computer drawing tools, this realm of study has blossomed rapidly and attracted significant interest in recent years [19][20][21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

Fuzzy Mandelbric Set and Its Perturbations by Dynamical Noises

Popović,

Ersoy,

İnce

et al. 2024

Fractal Fract

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In this paper, we introduce a membership function used to form the fuzzy Mandelbric set and investigate the structural effects of additive and multiplicative dynamic noises on it. The newly defined membership function of this fuzzy set and its perturbations is a generalization of the indicator function for the classical Mandelbric set. We present an algorithm for detecting each complex number’s fuzzy membership degree. Through the use of the membership degrees of each complex number and experimental mathematics based on the visualizations of a variety of versions by utilizing computer-aided design, we gain a deep foresight for the structure characteristics of the additive and multiplicative perturbed fuzzy Mandelbric sets. Our novel approach allows us to identify the symmetry states of the Mandelbric set and its perturbations by the membership degrees of complex numbers, unlike the existing methods described in the literature.

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“…Fractal theory has attracted considerable research interest [1][2][3][4][5][6][7][8], with the diffusion-limited aggregation (DLA) [9] model proposed by Witten and Sander receiving particular attention. The DLA model successfully simulates fractal growth processes under far-from-equilibrium conditions.…”

Section: Introductionmentioning

confidence: 99%

Control of the Thermal Fractal Diffusion of Tightly Compressed Heterogeneous Layers of Thin Plates

Wang

Liu

Sun

2014

Mathematical Problems in Engineering

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As the thermal conductivity of thin plates composed of tightly compressed heterogeneous layers varies continuously in the form of an exponential function, we present a nonlinear dynamical model of the fractal growth of thermal diffusion. We also analyze the quantitative relationship between the probability of growth and the disturbance term, predict the control action of the environmental disturbance term on fractal growth, and use Matlab simulation to verify the control effectiveness of thermal fractal diffusion. The results facilitate the selection of appropriate control areas and control parameters for the thermal diffusion variable coefficients. In addition, variation in the fractal dimension reflects the influence of environmental disturbance on the complex process of thermal fractal diffusion.

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The generalized Mandelbrot set perturbed by composing noise of additive and multiplicative

Cited by 23 publications

References 18 publications

Bivariate Newton-Raphson method and toroidal attraction basins

Bivariate Newton-Raphson method and toroidal attraction basins

Fuzzy Mandelbric Set and Its Perturbations by Dynamical Noises

Control of the Thermal Fractal Diffusion of Tightly Compressed Heterogeneous Layers of Thin Plates

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