On a topological closeness of perturbed Julia sets

doi:10.1016/j.amc.2010.08.024

Applied Mathematics and Computation

Volume 217, Issue 6, 15 November 2010, Pages 2883-2890

https://doi.org/10.1016/j.amc.2010.08.024 Get rights and content

Abstract

In the present work we expand our previous work in [1] by introducing the Julia Deviation Distance and the Julia Deviation Plot in order to study the stability of the Julia sets of noise-perturbed Mandelbrot maps. We observe a power-law behaviour of the Julia Deviation Distance of the Julia sets of a family of additive dynamic noise Mandelbrot maps from the Julia set of the Mandelbrot map as a function of the noise level. Additionally, using the above tools, we support the invariance of the Julia set of a noise-perturbed Mandelbrot map under different noise realizations.

Introduction

In this work, we provide a mathematical framework for studying the stability and the changes of the morphology of the Julia sets of the Mandelbrot map under small changes of the parameters. This framework also applies for the study of the Julia sets of noise-perturbed Mandelbrot maps. For that aim, we define a metric, the Julia Deviation Distance, among the Julia sets of the same Mandelbrot map associated with different values of the parameters and perturbed Mandelbrot maps, as in [1], which is based on the finite escape algorithm [2], [3]. Hence, we provide a mathematical framework for the notion of similar shape of the Julia sets of perturbed Mandelbrot maps used in [4], [5], [6].

Furthermore we introduce a graphical tool the Julia Deviation Plot in order to investigate the way in which the Julia set is deformed due to the changes of the parameters of the Mandelbrot map or to the perturbations of the Mandelbrot map as in [1] for the study of the Mandelbrot sets of perturbed Mandelbrot maps. In fact, using this graphical tool, one can localize the regions of the Julia set that are affected by the changes of the parameter or by the effects of the noise. One of the interesting findings that comes out from this analysis is that the perturbations lead to creations of points belonging to the Julia set of a perturbed Mandelbrot map along with the points that lose their property of belonging to the Julia set of the original Mandelbrot map.

In addition, our analysis reveals the presence of a three-regime power-law behaviour of the Julia Deviation Distance of the Julia sets of a family of additive dynamic noise Mandelbrot maps from the Julia set of the Mandelbrot map as a function of the level of noise. Subsequently, using the new tools, we support the independence of the Julia set of a noise-perturbed Mandelbrot map [4], [5], [6] by different realizations of the noise used to create them.

Furthermore it would be of interest to consider the application of the tools introduced in this paper, the Julia Deviation Distance and the Julia Deviation plot, to the study of the Julia sets of the noise perturbed generalized Mandelbrot and Julia sets [7], [8], [9], [10], of the Noise-perturbed quaternionic Mandelbrot sets [11], and of the superior Julia sets [12]. These tools could provide further insight on the way the morphology of the generalized Mandelbrot and Julia sets introduced in the references [7], [8], [9], [10], [11], [12] is changing due to the application of noise.

Here, we consider a Gaussian noise with values in the interval [0, 1]. In this paper, using different noise time series, we construct a series of different noise-perturbed Mandelbrot maps. As the resulting Julia Deviation Distances of the Julia sets of the perturbed Mandelbrot maps from the Julia set of the Mandelbrot map are all equal within two decimal places, we support the independence of the Julia set of a noise-perturbed Mandelbrot map under different noise realizations.

Section snippets

On a Julia Deviation Distance and a Julia Deviation Plot of the Julia sets of a Mandelbrot map

We consider a Mandelbrot map, denoted by Q_C in this paper, which is also known as the complex logistic map [2], [3] and is defined as follows: $x_{n + 1} = (x_{n})^{2} - (y_{n})^{2} + c_{1},$ $y_{n + 1} = 2 x_{n} y_{n} + c_{2},$ where c₁, c₂ ∈ R.

Let us recall the definition of the Julia set [2], [3] of the Mandelbrot map Q_C which is denoted in this paper by J(Q_C) . We fix a value of the parameter C = (c₁, c₂), and consider variations of the initial condition (x₀, y₀) . The Julia set of the map Q_C , is the set of all the values of the initial condition (

On a topological stability of the Julia sets of perturbed Mandelbrot map

In this paragraph, we consider the Julia sets of an additive dynamic noise Mandelbrot map considered in [13], and defined as $x_{n + 1} = (x_{n})^{2} - (y_{n})^{2} + c_{1} + a_{1} w_{n},$ $y_{n + 1} = 2 x_{n} y_{n} + c_{2} + a_{2} w_{n},$ where w_n is a noise input and a₁, a₂ ∈ R, denote the strength of the additive dynamic noise. The special case a₁ = a₂ = α is denoted as ${AD}_{c}^{α}$ .

In this work, we will consider first the case of the additive dynamic noise Mandelbrot map ${AD}_{c}^{0.01}$ . In Fig. 3, we present its Julia sets of the maps, with C = C₁ = (−0.3904,−0.58769) (Fig. 3a)

The Julia Deviation Distance as a function of the level of noise

In this paragraph, we consider the Julia Deviation Distance of the Julia sets of a family of additive dynamic noise Mandelbrot Maps ${AD}_{c}^{α}$ from the Julia set of the original Mandelbrot map $Q_{C}, distJ (J ({AD}_{c}^{α}), J (Q_{C}))$ , as a function of the level of noise α , while fixing the values of the parameter C. For this aim, we consider values for the level of noise α starting from 0.0001 up to 1 for two different noise realizations.

The results appear in Fig. 5, where the $distJ (J ({AD}_{c}^{α}), J (Q_{C}))$ as a function of the

On the noise invariance of the Julia set of a noise-perturbed Mandelbrot map

In this paragraph, we consider the question of the independence of a Julia set of a noise-perturbed Mandelbrot map considered in [4], [5], [6] under different noise realizations, i.e. if there is any strong dependence on the set of random numbers employed to simulate noise perturbation. In this direction, we present results for the additive dynamic noise Mandelbrot Map ${AD}_{c}^{0.01}$ introduced above.

Firstly, we consider five different realizations of noise and we name the corresponding additive

Conclusions

In this paper, we expanded our results of [1] to the study of the morphology of the Julia sets either of the same Mandelbrot map while varying its parameter values or while fixing the parameter values and introducing noise-perturbation of the Mandelbrot map. These tools give the possibility to quantify the difference of the Julia set of the Mandelbrot map from the Julia set of the noise-perturbed Mandelbrot map; as well as to localize the regions of the set that are affected by noise. A first

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On a topological closeness of perturbed Julia sets

Abstract

Introduction

Section snippets

On a Julia Deviation Distance and a Julia Deviation Plot of the Julia sets of a Mandelbrot map

On a topological stability of the Julia sets of perturbed Mandelbrot map

The Julia Deviation Distance as a function of the level of noise

On the noise invariance of the Julia set of a noise-perturbed Mandelbrot map

Conclusions

Appl. Math. Comput.

Chaos Soliton. Fract.

Chaos Soliton. Fract.

Appl. Math. Comput.

Appl. Math. Comput.

Chaos Soliton. Fract.

The Beauty of Fractals