Real and complex behavior for networks of coupled logistic maps

Rǎdulescu, Anca; Pignatelli, Ariel

doi:10.1007/s11071-016-3115-4

Cited by 10 publications

(8 citation statements)

References 24 publications

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“…Can an encrypted image still be decrypted in the presence of signal interference in the transmission process [15]? Can the networks described by fractals systems remain symmetry with the occurrence of network interference [31,32]. We hope that the preliminary algorithm results in this work can provide insight into the applications mentioned above.…”

Section: Discussionmentioning

confidence: 83%

The Symmetry in the Noise-Perturbed Mandelbrot Set

Sun

2019

Symmetry

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This paper investigates the destruction of the symmetrical structure of the noise-perturbed Mandelbrot set (M-set). By applying the “symmetry criterion” method, we quantitatively compare the damages to the symmetry of the noise-perturbed Mandelbrot set resulting from additive and multiplicative noises. Because of the uneven distribution between the core positions and the edge positions of the noise-perturbed Mandelbrot set, the comparison results reveal a paradox between the visual sense and quantified result. Thus, we propose a new “visual symmetry criterion” method that is more suitable for the measurement of visual asymmetry.

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Section: Discussionmentioning

confidence: 83%

The Symmetry in the Noise-Perturbed Mandelbrot Set

Sun

2019

Symmetry

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“…Our prior work [13] suggests that even basic results from the case of a single iterated quadratic maps may have to be rediscovered in the context of networks (one yet needs to prove, for example, even the existence of an escape radius). In our study of dynamics in small quadratic networks, we redefined extensions of some of the traditional concepts: multi-orbits, Julia and Mandelbrot sets, as well as their one-dimensional complex slices, which we called uni-Julia and equi-Mandelbrot sets.…”

Section: Prior Results In Small Networkmentioning

confidence: 99%

“…For example, connectedness fails in general for network equi-Mandelbrot sets. To fix out ideas, we illustrate and prove disconnectedness for an example network in a three-dimensional family considered previously [13] (see Figure 2). This family (which we called the "self-drive model") is interesting and easy to analyze, since it is a feed-forward network (each node depends only on the ones with smaller indices):…”

Section: Connectedness Of the Mandelbrot Setmentioning

confidence: 99%

“…In our previous work [13], when suggesting possible ties of our results with broader applications to the life sciences, we interpreted iterated orbits as describing the temporal evolution of an evolving system (e.g., learning neural network). An escaping initial condition (whether in the complex plane C, for a single iterated map, or in C n , for an iterated network) may be seen as an eventually unsustainable feature of the system, while a prisoner may represent a trivial, or inefficient feature.…”

Section: Introduction 1network Of Complex Quadratic Mapsmentioning

confidence: 97%

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Asymptotic sets in networks of coupled quadratic nodes

Rǎdulescu

Evans

2018

Journal of Complex Networks

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We study asymptotic dynamics in networks of coupled quadratic nodes. While single map complex quadratic iterations have been studied over the past century, considering ensembles of such functions, organized as coupled nodes in a network, generate new questions with potentially interesting applications to the life sciences.We investigate how traditional Fatou-Julia results may generalize in the case of networks. We discuss extensions of concepts like escape radius, Julia and Mandelbrot sets (as parameter loci in C n , where n is the size of the network). We study topological properties of these asymptotic sets and of their two-dimensional slices in C (defined in previous work). We find that, while network Mandelbrot sets no longer have a hyperbolic bulb structure, some of their geometric landmarks are preserved (e.g., the cusp always survives), and other properties (such as connectedness) depend on the network structure. We investigate possible extensions of the relationship between the Mandelbrot set and the Julia set connectedness loci in the case of network dynamics.We discuss possible classifications of asymptotic behavior in networks based on their underlying graph structure, using the geometry of Julia or Mandelbrot sets as a classifier. Finally, we propose a method for book-keeping asymptotic dynamics simultaneously over many networks with a common graph-theoretical property. Core Julia and Mandelbrot sets describe statistically average asymptotic behavior of orbits over an entire collection of configurations.

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“…Based on the Julia set's image generated by ETA, the authors proposed a novel image with visually meaningful cryptography, which has stronger anti-interference ability [26]. In [27], the dynamic behaviors of the network whose nodes are logistic maps was well-described by the M-J set formulated by ETA .…”

Section: Introductionmentioning

confidence: 99%

On the Connectivity Measurement of the Fractal Julia Sets Generated from Polynomial Maps: A Novel Escape-Time Algorithm

Zhao

Zhang

et al. 2021

Fractal Fract

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In this paper, a novel escape-time algorithm is proposed to calculate the connectivity’s degree of Julia sets generated from polynomial maps. The proposed algorithm contains both quantitative analysis and visual display to measure the connectivity of Julia sets. For the quantitative part, a connectivity criterion method is designed by exploring the distribution rule of the connected regions, with an output value Co in the range of [0,1]. The smaller the Co value outputs, the better the connectivity is. For the visual part, we modify the classical escape-time algorithm by highlighting and separating the initial point of each connected area. Finally, the Julia set is drawn into different brightnesses according to different Co values. The darker the color, the better the connectivity of the Julia set. Numerical results are included to assess the efficiency of the algorithm.

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Real and complex behavior for networks of coupled logistic maps

Cited by 10 publications

References 24 publications

The Symmetry in the Noise-Perturbed Mandelbrot Set

The Symmetry in the Noise-Perturbed Mandelbrot Set

Asymptotic sets in networks of coupled quadratic nodes

On the Connectivity Measurement of the Fractal Julia Sets Generated from Polynomial Maps: A Novel Escape-Time Algorithm

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