Coloring periodicities of two-dimensional mappings

Welstead, Stephen T.; Cromer, Thomas L.

doi:10.1016/0097-8493(89)90016-2

Cited by 16 publications

(4 citation statements)

References 3 publications

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“…Hence, we define the Mandelbrot set as follows: In 1989, Welstead and Cromer put forward the combination algorithm of escapetime and the scanning of periodic points to construct the classical Mandelbrot set. 17 In this paper, we extend this algorithm to construct the Mandelbrot-Julia sets of the complex C-K map. The details of the extended algorithm are as follows:…”

Section: Dynamic Analysis Of the Carotid-kundalini Map 249mentioning

confidence: 99%

Dynamic Analysis of the Carotid–kundalini Map

2008

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The nature of the fixed points of the Carotid–Kundalini (C–K) map was studied and the boundary equation of the first bifurcation of the C–K map in the parameter plane is presented. Using the quantitative criterion and rule of chaotic system, the paper reveals the general features of the C–K Map transforming from regularity to chaos. The following conclusions are obtained: (i) chaotic patterns of the C–K map may emerge out of double-periodic bifurcation; (ii) the chaotic crisis phenomena are found. At the same time, the authors analyzed the orbit of critical point of the complex C–K Map and put forward the definition of Mandelbrot–Julia set of the complex C–K Map. The authors generalized the Welstead and Cromer's periodic scanning technique and using this technology constructed a series of the Mandelbrot–Julia sets of the complex C–K Map. Based on the experimental mathematics method of combining the theory of analytic function of one complex variable with computer aided drawing, we investigated the symmetry of the Mandelbrot–Julia set and studied the topological inflexibility of distribution of the periodic region in the Mandelbrot set, and found that the Mandelbrot set contains abundant information of the structure of Julia sets by finding the whole portray of Julia sets based on Mandelbrot set qualitatively.

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Section: Dynamic Analysis Of the Carotid-kundalini Map 249mentioning

confidence: 99%

Dynamic Analysis of the Carotid–kundalini Map

2008

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“…The comparative analysis between experiments and simulations shows that the level of performance and agreement is excellent. Moreover, we support this by analyzing the numerical and experimental data using Lyapunov exponents [11], orbit's periodicity [30], and phase portraits for the uncoupled and coupled cases.…”

Section: Introductionmentioning

confidence: 56%

“…Another useful characterization for the coupled maps behavior is done by quantifying the periodicity of their orbits [30]. In other words, looking at the periodic properties of the resultant orbits for every dynamical regime.…”

Section: Dynamical Behavior Characterizationmentioning

confidence: 99%

Electronically-implemented coupled logistic maps

et al. 2016

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The logistic map is a paradigmatic dynamical system originally conceived to model the discretetime demographic growth of a population, which shockingly, shows that discrete chaos can emerge from trivial low-dimensional non-linear dynamics. In this work, we design and characterize a simple, low-cost, easy-to-handle, electronic implementation of the logistic map. In particular, our implementation allows for straightforward circuit-modifications to behave as different one-dimensional discrete-time systems. Also, we design a coupling block in order to address the behavior of two coupled maps, although, our design is unrestricted to the discrete-time system implementation and it can be generalized to handle coupling between many dynamical systems, as in a complex system. Our findings show that the isolated and coupled maps' behavior has a remarkable agreement between the experiments and the simulations, even when fine-tuning the parameters with a resolution of ∼ 10 −3 . We support these conclusions by comparing the Lyapunov exponents, periodicity of the orbits, and phase portraits of the numerical and experimental data for a wide range of coupling strengths and map's parameters.PACS. 0 5.45.-a -0 7.05.Fb -0 7.50.Ek

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“…Nowadays, a lot of research has been done on the improvement of ETA and proposal of other algorithms. In [16], the Mandelbrot set was displayed by a variety of colours to reveal patterns of finite attracting orbits. Liu et al [17] proposed a new escape-time algorithm that could accelerate the construction process.…”

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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Coloring periodicities of two-dimensional mappings

Cited by 16 publications

References 3 publications

Dynamic Analysis of the Carotid–kundalini Map

Dynamic Analysis of the Carotid–kundalini Map

Electronically-implemented coupled logistic maps

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

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