Fractal surfaces from simple arithmetic operations

Garcı́a-Morales, Vladimir

doi:10.1016/j.physa.2015.12.028

Cited by 12 publications

(17 citation statements)

References 37 publications

(75 reference statements)

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“…This type of decomposition is called the pλn decomposition. In [5] we find a definition of a generalized bitwise operator. To define this type of operators, the author uses the digit function.…”

Section: Discussionmentioning

confidence: 99%

Switching processes in polynomiography

Gdawiec

2016

Nonlinear Dyn

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Mandelbrot and Julia sets are examples of fractal patterns generated in the complex plane. In the literature, we can find many generalizations of those sets. One of such generalizations is the use of switching process. In this paper, we introduce some switching processes to another type of complex fractals, namely polynomiographs. Polynomiograph is an image presenting the visualization of the complex polynomial's root finding process. The proposed switching processes will be divided into four groups, i.e., switching of: the root finding methods, the iterations, the polynomials and the convergence tests. All the proposed switching processes change the dynamics of the root finding process and allow us to obtain new and diverse fractal patterns.

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Section: Discussionmentioning

confidence: 99%

Switching processes in polynomiography

Gdawiec

2016

Nonlinear Dyn

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“…for each j ∈ [0, N s − 1]. Here, we have introduced the digit function [12,14,15] which is defined, for p ∈ N, k ∈ Z and x ∈ R as…”

Section: A General Theorymentioning

confidence: 99%

“…In general, however, the layer values are nonlinearly coupled as seen in Eq. (15). In this general case we say that l R r p is a p-decomposable rule.…”

Section: A General Theorymentioning

confidence: 99%

“…Eq. (1) takes in this case the simple formx j t+1 = d p (x j−1 t + px j t , R) ≡ x j t * x j−1 t (70)where we have introduced the notation * to emphasize that d p (x j−1 t +px j t , R) acts as a binary operator S 2 → S. Thus, the integer R ∈ [0, p 2 − 1] lists all possible binary operators (magmas) on the set S[15]. If we tabulate the output x j t+1 as a function of the inputs x j t and x j−1 t in the rows and in the columns respectively, we obtain the Cayley table of the binary operator as[15]…”

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Diagrammatic approach to cellular automata and the emergence of form with inner structure

Garcı́a-Morales

2018

Communications in Nonlinear Science and Numerical Simulation

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We present a diagrammatic method to build up sophisticated cellular automata (CAs) as models of complex physical systems. The diagrams complement the mathematical approach to CA modeling, whose details are also presented here, and allow CAs in rule space to be classified according to their hierarchy of layers. Since the method is valid for any discrete operator and only depends on the alphabet size, the resulting conclusions, of general validity, apply to CAs in any dimension or order in time, arbitrary neighborhood ranges and topology. We provide several examples of the method, illustrating how it can be applied to the mathematical modeling of the emergence of order out of disorder. Specifically, we show how the the majority CA rule can be used as a building block to construct more complex cellular automata in which separate domains (with substructures having different dynamical properties) are able to emerge out of disorder and coexist in a stable manner.

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“…We first explain how to (approximately) map the dynamics on the torus T N to the shift space A N [61] of a CA. Here A denotes the set of integers in [0, p − 1] with p ∈ N (p ≥ 2) being the alphabet size.Since ϕ 2π ∈ [0, 1) is a real number, we can expand it in a base (radix) p ≥ 2, p ∈ N as ϕ j t 2π = lim D→∞ D m=1 p −m d p −m, ϕ j t 2π(2)where we have introduced the digit function [62][63][64]…”

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Cellular automaton for chimera states

Garcı́a-Morales

2016

EPL

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A minimalistic model for chimera states is presented. The model is a cellular automaton (CA) which depends on only one adjustable parameter, the range of the nonlocal coupling, and is built from elementary cellular automata and the majority (voting) rule. This suggests the universality of chimera-like behavior from a new point of view: Already simple CA rules based on the majority rule exhibit this behavior. After a short transient, we find chimera states for arbitrary initial conditions, the system spontaneously splitting into stable domains separated by static boundaries, ones synchronously oscillating and the others incoherent. When the coupling range is local, nontrivial coherent structures with different periodicities are formed. This motivates the need of simple models, reduced to the barest essentials, to describe the underlying mechanisms behind their formation. Cellular automata (CAs) [22,[50][51][52][53][54][55][56][57] hold promise for that goal. For example, chimera states were found in a three-level CA of ZykovMikhailov type [58], and Boolean phase oscillators, realized with electronic logic circuits, were also shown to support transient chimeras [44].In this letter we regard chimera states as an experimental fact of nature rather than a feature of certain systems of differential equations or maps. We then formulate a simple CA model for chimera states, describing a possible universal mechanism behind their spontaneous emergence out of any initial condition. We sketch a general mathematical approach to show how CAs can be regarded as approximations (shadowings) of maps of coupled oscillators. Although we do not attempt here to connect our specific CA model to any such map, we hypothesize that the latter should exist [59,60]. Chimeras are here modelled as specific instances of domain formation in spatially extended systems, a behavior that is statistically robust to small perturbations and which is ubiquitously found in nature. The chimera states encountered here are of the weak type [30,31] and are stable and coexist with synchronously oscillating domains separated by static walls. The model depends on one free parameter only, ξ ∈ N (ξ ≥ 1), whose physical meaning is the neighborhood radius (nonlocal coupling range). When ξ is small, nontrivial coherent structures are formed. However, when ξ is sufficiently large, incoherent domains of thickness d ≥ ξ + 1 arise.We first show how any map on a ring of N spatially coupled oscillators can be approximated by a CA. Let ϕ j t ∈ [0, 2π) denote the phase of the oscillator at location j, j ∈ [0, N − 1] and discrete time t. We assume that the evolution of the phases in the torus T

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Fractal surfaces from simple arithmetic operations

Cited by 12 publications

References 37 publications

Switching processes in polynomiography

Switching processes in polynomiography

Diagrammatic approach to cellular automata and the emergence of form with inner structure

Cellular automaton for chimera states

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