scholarly journals COMPLEX DYNAMICS OF ELEMENTARY CELLULAR AUTOMATA EMERGING FROM CHAOTIC RULES

2012 ◽  
Vol 22 (02) ◽  
pp. 1250023 ◽  
Author(s):  
GENARO J. MARTÍNEZ ◽  
ANDREW ADAMATZKY ◽  
RAMON ALONSO-SANZ
Keyword(s):  
Cellular Automata ◽  
Complex Dynamics ◽  
Chaotic Behavior ◽  
Mean Field Theory ◽  
Mean Field ◽  
One Dimensional ◽  
Time Dynamics ◽  
Transition Rules ◽  
State Transition Rules ◽  
With Memory

We show techniques of analyzing complex dynamics of cellular automata (CA) with chaotic behavior. CA are well-known computational substrates for studying emergent collective behavior, complexity, randomness and interaction between order and chaotic systems. A number of attempts have been made to classify CA functions on their space-time dynamics and to predict the behavior of any given function. Examples include mechanical computation, λ and Z-parameters, mean field theory, differential equations and number conserving features. We aim to classify CA based on their behavior when they act in a historical mode, i.e. as CA with memory. We demonstrate that cell-state transition rules enriched with memory quickly transform a chaotic system converging to a complex global behavior from almost any initial condition. Thus, just in few steps we can select chaotic rules without exhaustive computational experiments or recurring to additional parameters. We provide an analysis of well-known chaotic functions in one-dimensional CA, and decompose dynamics of the automata using majority memory exploring glider dynamics and reactions.

scholarly journals DESIGNING COMPLEX DYNAMICS IN CELLULAR AUTOMATA WITH MEMORY

2013 ◽  
Vol 23 (10) ◽  
pp. 1330035 ◽  
Author(s):  
GENARO J. MARTÍNEZ ◽  
ANDREW ADAMATZKY ◽  
RAMON ALONSO-SANZ
Keyword(s):  
Cellular Automata ◽  
State Transition ◽  
Complex Dynamics ◽  
Transition Function ◽  
Systematic Analysis ◽  
Time Dynamics ◽  
Controversial Topic ◽  
State Transition Function ◽  
And Behavior ◽  
With Memory

Since their inception at Macy conferences in later 1940s, complex systems have remained the most controversial topic of interdisciplinary sciences. The term "complex system" is the most vague and liberally used scientific term. Using elementary cellular automata (ECA), and exploiting the CA classification, we demonstrate elusiveness of "complexity" by shifting space-time dynamics of the automata from simple to complex by enriching cells with memory. This way, we can transform any ECA class to another ECA class — without changing skeleton of cell-state transition function — and vice versa by just selecting a right kind of memory. A systematic analysis displays that memory helps "discover" hidden information and behavior on trivial — uniform, periodic, and nontrivial — chaotic, complex — dynamical systems.

scholarly journals DISSIPATIVE QUANTUM DISORDERED MODELS

2006 ◽  
Vol 20 (19) ◽  
pp. 2795-2804 ◽  
Author(s):  
LETICIA F. CUGLIANDOLO
Keyword(s):  
Glassy Phase ◽  
Quantum Critical Point ◽  
Mean Field ◽  
Static Properties ◽  
One Dimensional ◽  
Time Dynamics ◽  
Long Range Interactions ◽  
Disordered Models ◽  
Different Types ◽  
Dissipative Environments

This article reviews recent studies of mean-field and one dimensional quantum disordered spin systems coupled to different types of dissipative environments. The main issues discussed are: (i) The real-time dynamics in the glassy phase and how they compare to the behaviour of the same models in their classical limit. (ii) The phase transition separating the ordered – glassy – phase from the disordered phase that, for some long-range interactions, is of second order at high temperatures and of first order close to the quantum critical point (similarly to what has been observed in random dipolar magnets). (iii) The static properties of the Griffiths phase in random king chains. (iv) The dependence of all these properties on the environment. The analytic and numeric techniques used to derive these results are briefly mentioned.

scholarly journals A one-dimensional modified TASEP model on a track of variable length: analytical and computational results

2021 ◽  
Vol 2090 (1) ◽  
pp. 012025
Author(s):  
B. Reed ◽  
E. Aldrich ◽  
L. Stoleriu ◽  
D.A. Mazilu ◽  
I. Mazilu
Keyword(s):  
Monte Carlo ◽  
Molecular Motors ◽  
Mean Field Theory ◽  
Mean Field ◽  
Limited Range ◽  
Field Methods ◽  
One Dimensional ◽  
Emergent Features ◽  
Simulation Results ◽  
Mean Field Methods

Abstract We present analytical solutions and Monte Carlo simulation results for a one-dimensional modified TASEP model inspired by the interplay between molecular motors and their cellular tracks of variable lengths, known as microtubules. Our TASEP model incorporates rules for changes in the length of the track based on the occupation of the first two sites. Using mean-field theory, we derive analytical results for the particle densities and particle currents and compare them with Monte Carlo simulations. These results show the limited range of mean-field methods for models with localized high correlation between particles. The variability in length adds to the complexity of the model, leading to emergent features for the evolution of particle densities and particle currents compared to the traditional TASEP model.

scholarly journals Chaotic Behavior of One-Dimensional Cellular Automata Rule 24

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zujie Bie ◽  
Qi Han ◽  
Chao Liu ◽  
Junjian Huang ◽  
Lepeng Song ◽  
...  
Keyword(s):  
Cellular Automata ◽  
Characteristic Function ◽  
Symbolic Dynamics ◽  
Bernoulli Shift ◽  
Chaotic Behavior ◽  
Class Ii ◽  
Chaotic Attractors ◽  
One Dimensional ◽  
Dynamical Behaviors ◽  
Shift Map

Wolfram divided the 256 elementary cellular automata rules informally into four classes using dynamical concepts like periodicity, stability, and chaos. Rule 24, which is Bernoulliστ-shift rule and is member of Wolfram’s class II, is said to be simple as periodic before. Therefore, it is worthwhile studying dynamical behaviors of four rules, whether they possess chaotic attractors or not. In this paper, the complex dynamical behaviors of rule 24 of one-dimensional cellular automata are investigated from the viewpoint of symbolic dynamics. We find that rule 24 is chaotic in the sense of both Li-Yorke and Devaney on its attractor. Furthermore, we prove that four rules of global equivalenceε52of cellular automata are topologically conjugate. Then, we use diagrams to explain the attractor of rule 24, where characteristic function is used to describe the fact that all points fall into Bernoulli-shift map after two iterations under rule 24.

scholarly journals The 1/N-expansion as a perturbation about the mean field theory: A one-dimensional fermi model

1996 ◽  
Vol 179 (3) ◽  
pp. 623-646 ◽  
Author(s):  
D. H. U. Marchetti ◽  
P. A. Faria da Veiga ◽  
T. R. Hurd
Keyword(s):  
Field Theory ◽  
Mean Field Theory ◽  
Mean Field ◽  
Fermi Model ◽  
One Dimensional ◽  
The Mean

Mean-Field Theory of Quasi One-Dimensional Heisenberg Magnetγ-Phasep-NPNN

1992 ◽  
Vol 61 (10) ◽  
pp. 3745-3751 ◽  
Author(s):  
Minoru Takahashi ◽  
Minoru Kinoshita ◽  
Masayasu Ishikawa
Keyword(s):  
Field Theory ◽  
Mean Field Theory ◽  
Mean Field ◽  
One Dimensional

A NEW METHOD FOR MAPPING OPTIMIZATION PROBLEMS ONTO NEURAL NETWORKS

1989 ◽  
Vol 01 (01) ◽  
pp. 3-22 ◽  
Author(s):  
Carsten Peterson ◽  
Bo Söderberg
Keyword(s):  
Neural Network ◽  
Optimization Problems ◽  
Chaotic Behavior ◽  
Numerical Study ◽  
Mean Field Theory ◽  
Mean Field ◽  
Solution Space ◽  
Approximate Solutions ◽  
Modified Method ◽  
The Neural Network

A novel modified method for obtaining approximate solutions to difficult optimization problems within the neural network paradigm is presented. We consider the graph partition and the travelling salesman problems. The key new ingredient is a reduction of solution space by one dimension by using graded neurons, thereby avoiding the destructive redundancy that has plagued these problems when using straightforward neural network techniques. This approach maps the problems onto Potts glass rather than spin glass theories. A systematic prescription is given for estimating the phase transition temperatures in advance, which facilitates the choice of optimal parameters. This analysis, which is performed for both serial and synchronous updating of the mean field theory equations, makes it possible to consistently avoid chaotic behavior. When exploring this new technique numerically we find the results very encouraging; the quality of the solutions are in parity with those obtained by using optimally tuned simulated annealing heuristics. Our numerical study, which for TSP extends to 200-city problems, exhibits an impressive level of parameter insensitivity.

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