Chaotic bands in the Mandelbrot set

Pastor, G.; Romera, M.; Álvarez, Gonzalo; Montoya, F.

doi:10.1016/j.cag.2004.06.015

Cited by 16 publications

(7 citation statements)

References 13 publications

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“…Property 4 indicates that the multiplicative perturbed generalized M-sets for even index number constructed from Eq. (8) are symmetrical about x-axis (shown in Fig. 3(a), (d), (e) and (g)) and the generalized M-sets for odd index number are both symmetrical about x-axis and y-axis (shown in Fig.…”

Section: Propertymentioning

confidence: 96%

Noise perturbed generalized Mandelbrot sets

Wang

Lang

et al. 2008

Journal of Mathematical Analysis and Applications

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Adopting the experimental mathematics method of combining the theory of analytic function of one complex variable with computer aided drawing, in this paper on the structure characteristics and the discontinuity evolution law of the additive noise perturbed generalized Mandelbrot sets (M-sets) was studied. On the influence of stochastic perturbed parameters of the structure of generalized M-sets was analyzed. The physical meaning of the additive noise perturbed generalized M-sets was expounded. IntroductionIn recent 20 years, people have lucubrated on the generalized Mandelbrot-sets (M-sets in short) constructed from theand found there existed orderly structure in it [1,2]. For example, Gujar et al. proposed several assumptions based on the visually structural characteristics of generalized M-sets [3]; Glynn found the symmetrical evolution of generalized M-sets when the phase angle θ ∈ [−π , π) [4]; the authors put forward the embedded topological distribution theorem of the generalized M-sets and discussed the different selections of the principal value range of the phase angle θ and the fission-evolution rule of the generalized M-sets for decimal index number [5]; Sasmor analyzed the fission of generalized M-sets for rational index number when the phase angle θ ∈ [−π , π) [6]; Romera and Pastor et al. researched on the embedded-layer relationship of bud of generalized M-sets in the Misiurewicz points [7,8]; Geum and the authors both studied the structure and distribution of the periodic bud of generalized M-sets and the topological rule of their periodic trajectories [9,10]; Beck and the authors both discussed the physical meaning of generalized M-sets [11,12]. Furthermore, Argyris, Lasota, Kapitaniak, et al. discussed the classification and affection of noise in complex dynamical system [13-15]; Argyris et al. studied the structural characteristic of M-sets containing noise after importing additive noise and multiplicative noise into the complex map z n+1 = z 2 n + c [16-18]. Based on above researches, this paper studied the structural characteristic and fission-evolution law of stochastic perturbed generalized M-sets, analyzed the influence of stochastic perturbed parameters of the structure of generalized M-sets and expatiated on the physical meaning of a type of generalized M-sets.

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Section: Propertymentioning

confidence: 96%

Noise perturbed generalized Mandelbrot sets

Wang

Lang

et al. 2008

Journal of Mathematical Analysis and Applications

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“…In this paper we have used harmonics in order to calculate for the first time the last appearance hyperbolic components of the chaotic bands of any Mandelbrot set hyperbolic component. In some manner this paper finishes papers [11] and [16]. Indeed, in [15] harmonics are used in order to calculate the last appearance hyperbolic components of the chaotic bands of the real axis, but here we enlarge it to all the Mandelbrot set.…”

Section: Introductionmentioning

confidence: 99%

“…Likewise, we also have studied the chaotic bands in the Mandelbrot set [15,16]. In [15] we have used the shrubs in order to study the chaotic region of the Mandelbrot set, and in [16] we focus on the study of chaotic bands but with no calculation of their hyperbolic components.…”

Section: Introductionmentioning

confidence: 99%

“…In [16] we analyze the chaotic bands in all the Mandelbrot set with no calculation, but here we calculate the last appearance hyperbolic components. Therefore, the use of harmonics as a tool for the calculation of the last appearance hyperbolic components of the chaotic bands in all the Mandelbrot set can be considered the main contribution of this paper.…”

Section: Introductionmentioning

confidence: 99%

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External arguments for the chaotic bands calculation in the Mandelbrot set

Pastor

Romera

Álvarez

et al. 2005

Physica A: Statistical Mechanics and its Applications

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In this paper we study the chaotic bands of any Mandelbrot set hyperbolic component. We use external arguments in order to identify the hyperbolic components. If we use harmonics as a tool, we can calculate the chaotic bands. Indeed, as we clearly show here, the harmonics of the external arguments of a given hyperbolic component (gene) are the external arguments of the last appearance hyperbolic components of the chaotic bands corresponding to the gene. Likewise, we show that only if the hyperbolic component is a cardioid the infinite number of the chaotic bands of such a cardioid fills up all its chaotic region.

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“…1 In the last 20 years, many have carried out in-depth studies of Mandelbrot and Julia sets and have found their regularity structures. [2][3][4][5][6][7][8][9][10][11][12][13][14][15] However, these studies ignored the inner and exterior (unboundary region) regional and structural properties of generalized M-J sets. Extreme modulus escaping time algorithm, 16 decomposition algorithm 17 and fisheye algorithm 18 can show the structural properties of un-boundary regions of generalized M-J sets.…”

Section: Introductionmentioning

confidence: 99%

Research Fractal Structures of Generalized M-J Sets Using Three Algorithms

Wang

2008

Fractals

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Extreme modulus escaping time algorithm, decomposition algorithm and fisheye algorithm are analyzed in this thesis, and we construct a series of generalized Mandelbrot-Julia (M-J) sets using these three algorithms. By studying the structural character of generalized M-J sets, we find: (1) extreme modulus escaping time algorithm and decomposition algorithm are simple modifications of classic escaping time algorithm, they can both construct the structure of nonboundary areas of generalized M-J sets; (2) non-boundary areas of generalized M-J sets have fractal characters; (3) generalized M-J sets have symmetry, and the process of evolvement depends on range of phase angle; (4) we can observe not only the whole structure of generalized Mandelbrot sets but also the details of some parts by using fisheye algorithm.

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Chaotic bands in the Mandelbrot set

Cited by 16 publications

References 13 publications

Noise perturbed generalized Mandelbrot sets

Noise perturbed generalized Mandelbrot sets

External arguments for the chaotic bands calculation in the Mandelbrot set

Research Fractal Structures of Generalized M-J Sets Using Three Algorithms

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