DYNAMICAL SYSTEMS PERSPECTIVE OF WOLFRAM’S CELLULAR AUTOMATA

Real cellular automata (RCA) are time-discrete dynamical systems on ℝN. Like cellular automata they can be obtained from discretizing partial differential equations. Due to their structure RCA are ideally suited to implementation on parallel computers with a large number of processors. In a way similar to the Hénon mapping, the system we consider here embeds the logistic mapping in a system on ℝN, N>1. But in contrast to the Hénon system an RCA in general is not invertible. We present some results about the bifurcation structure of such systems, mostly restricting ourselves, due to the complexity of the problem, to the two-dimensional case. Among others we observe cascades of cusp bifurcations forming generalized crossroad areas and crossroad areas with the flip curves replaced by Hopf bifurcation curves.

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HIGHER ORDER CELLULAR AUTOMATA

Advances in Complex Systems ◽

10.1142/s0219525905000403 ◽

2005 ◽

Vol 08 (02n03) ◽

pp. 169-192 ◽

Cited By ~ 11

Author(s):

NILS A. BAAS ◽

TORBJØRN HELVIK

Keyword(s):

Dynamical Systems ◽

Cellular Automata ◽

Level Structure ◽

Higher Order ◽

New Concepts ◽

Multi Level

We introduce a class of dynamical systems called Higher Order Cellular Automata (HOCA). These are based on ordinary CA, but have a hierarchical, or multi-level, structure and/or dynamics. We present a detailed formalism for HOCA and illustrate the concepts through four examples. Throughout the article we emphasize the principles and ideas behind the construction of HOCA, such that these easily can be applied to other types of dynamical systems. The article also presents new concepts and ideas for describing and studying hierarchial dynamics in general.

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Almost all $S$-integer dynamical systems have many periodic points

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385798113378 ◽

1998 ◽

Vol 18 (2) ◽

pp. 471-486 ◽

Cited By ~ 4

Author(s):

T. B. WARD

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Cellular Automata ◽

Zeta Function ◽

Periodic Points ◽

Radius Of Convergence ◽

Dynamical Zeta Function ◽

Hyperbolic Dynamical Systems ◽

Almost All

We show that for almost every ergodic $S$-integer dynamical system the radius of convergence of the dynamical zeta function is no larger than $\exp(-\frac{1}{2}h_{\rm top})<1$. In the arithmetic case almost every zeta function is irrational.We conjecture that for almost every ergodic $S$-integer dynamical system the radius of convergence of the zeta function is exactly $\exp(-h_{\rm top})<1$ and the zeta function is irrational.In an important geometric case (the $S$-integer systems corresponding to isometric extensions of the full $p$-shift or, more generally, linear algebraic cellular automata on the full $p$-shift) we show that the conjecture holds with the possible exception of at most two primes $p$.Finally, we explicitly describe the structure of $S$-integer dynamical systems as isometric extensions of (quasi-)hyperbolic dynamical systems.

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Correlations Between Hysteretic Categorical and Continuous Judgments of Perceptual Stimuli Supporting a Unified Dynamical Systems Approach to Perception

Perception ◽

10.1177/0301006617731047 ◽

2017 ◽

Vol 47 (1) ◽

pp. 44-66 ◽

Cited By ~ 6

Author(s):

S. Kim ◽

T. D. Frank

Keyword(s):

Dynamical Systems ◽

Systems Approach ◽

Brain Activity ◽

Equation Model ◽

Measurement Scale ◽

Systems Perspective ◽

Within Subjects ◽

Scale Measure ◽

Positive Correlations ◽

Experimental Findings

We report from two variants of a figure-ground experiment that is known in the literature to involve a bistable perceptual domain. The first variant was conducted as a two-alternative forced-choice experiment and in doing so tested participants on a categorical measurement scale. The second variant involved a Likert scale measure that was considered to represent a continuous measurement scale. The two variants were conducted as a single within-subjects experiment. Measures of bistability operationalized in terms of hysteresis size scores showed significant positive correlations across the two response conditions. The experimental findings are consistent with a dualistic interpretation of self-organizing perceptual systems when they are described on a macrolevel by means of so-called amplitude equations. This is explicitly demonstrated for a Lotka–Volterra–Haken amplitude equation model of task-related brain activity. As a by-product, the proposed dynamical systems perspective also sheds new light on the anchoring problem of producing numerical, continuous judgments.

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