Languages, equicontinuity and attractors in cellular automata

1997 ◽  
Vol 17 (2) ◽  
pp. 417-433 ◽  
Author(s):  
PETR KŮRKA
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
State Space ◽  
Finite Type ◽  
Measure Zero ◽  
Initial Conditions ◽  
The State ◽  
Subshift Of Finite Type ◽  
Topologically Transitive ◽  
The Third

We consider three related classifications of cellular automata: the first is based on the complexity of languages generated by clopen partitions of the state space, i.e. on the complexity of the factor subshifts; the second is based on the concept of equicontinuity and it is a modification of the classification introduced by Gilman [9]. The third one is based on the concept of attractors and it refines the classification introduced by Hurley [16]. We show relations between these classifications and give examples of cellular automata in the intersection classes. In particular, we show that every positively expansive cellular automaton is conjugate to a one-sided subshift of finite type and that every topologically transitive cellular automaton is sensitive to initial conditions. We also construct a cellular automaton with minimal quasi-attractor, whose basin has measure zero, answering a question raised in Hurley [16].

scholarly journals Embedding Bratteli–Vershik systems in cellular automata

2009 ◽  
Vol 30 (5) ◽  
pp. 1561-1572 ◽  
Author(s):  
MARCUS PIVATO ◽  
REEM YASSAWI
Keyword(s):  
Dynamical Systems ◽  
Cellular Automata ◽  
Cellular Automaton ◽  
Finite Type ◽  
Two Dimensional ◽  
Time Diagram ◽  
Constant Length ◽  
Subshift Of Finite Type ◽  
One Dimensional ◽  
Toeplitz Systems

AbstractMany dynamical systems can be naturally represented as Bratteli–Vershik (or adic) systems, which provide an appealing combinatorial description of their dynamics. If an adic system X is linearly recurrent, then we show how to represent X using a two-dimensional subshift of finite type Y; each ‘row’ in a Y-admissible configuration corresponds to an infinite path in the Bratteli diagram of X, and the vertical shift on Y corresponds to the ‘successor’ map of X. Any Y-admissible configuration can then be recoded as the space-time diagram of a one-dimensional cellular automaton Φ; in this way X is embedded in Φ (i.e. X is conjugate to a subsystem of Φ). With this technique, we can embed many odometers, Toeplitz systems, and constant-length substitution systems in one-dimensional cellular automata.

scholarly journals On the Network Analysis of the State Space of Discrete Dynamical Systems

2017 ◽  
Vol 27 (04) ◽  
pp. 1750062 ◽  
Author(s):  
Cheng Xu ◽  
Chengqing Li ◽  
Jinhu Lü ◽  
Shi Shu
Keyword(s):  
Dynamical Systems ◽  
Network Analysis ◽  
Cellular Automata ◽  
State Space ◽  
Discrete Dynamical Systems ◽  
The State ◽  
Dynamical Complexity ◽  
Two Parameters ◽  
Theoretical Analyses

This paper discusses the letter entitled “Network analysis of the state space of discrete dynamical systems” by A. Shreim et al. [Phys. Rev. Lett. 98, 198701 (2007)]. We found that some theoretical analyses are wrong and the proposed indicators based on two parameters of the state-mapping network cannot discriminate the dynamical complexity of the discrete dynamical systems composed of a 1D cellular automata.

scholarly journals Characterization of the Evolution of Nonlinear Uniform Cellular Automata in the Light of Deviant States

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Pabitra Pal Choudhury ◽  
Sudhakar Sahoo ◽  
Mithun Chakraborty
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Boolean Functions ◽  
State Transition ◽  
The State ◽  
Transition Diagram ◽  
One Dimensional ◽  
State Transition Diagram ◽  
Linear Rule

Dynamics of a nonlinear cellular automaton (CA) is, in general asymmetric, irregular, and unpredictable as opposed to that of a linear CA, which is highly systematic and tractable, primarily due to the presence of a matrix handle. In this paper, we present a novel technique of studying the properties of the State Transition Diagram of a nonlinear uniform one-dimensional cellular automaton in terms of its deviation from a suggested linear model. We have considered mainly elementary cellular automata with neighborhood of size three, and, in order to facilitate our analysis, we have classified the Boolean functions of three variables on the basis of number and position(s) of bit mismatch with linear rules. The concept of deviant and nondeviant states is introduced, and hence an algorithm is proposed for deducing the State Transition Diagram of a nonlinear CA rule from that of its nearest linear rule. A parameter called the proportion of deviant states is introduced, and its dependence on the length of the CA is studied for a particular class of nonlinear rules.

scholarly journals ‘Intractable and unsolved’: some thoughts on statistical data assimilation with uncertain static parameters

2013 ◽  
Vol 371 (1991) ◽  
pp. 20120297 ◽  
Author(s):  
Jonathan Rougier
Keyword(s):  
Numerical Methods ◽  
Data Assimilation ◽  
State Space ◽  
Stochastic Models ◽  
Environmental Science ◽  
Statistical Data ◽  
Initial Conditions ◽  
The State ◽  
Likelihood Functions ◽  
Sensitive Dependence

If you seem to be able to do data assimilation with uncertain static parameters then you are probably not working in environmental science. In this field, applications are often characterized by sensitive dependence on initial conditions and attracting sets in the state-space, which, taken together, can be a major challenge to numerical methods, leading to very peaky likelihood functions. Inherently stochastic models and uncertain static parameters increase the challenge.

On Dynamic Tooth Load and Stability of a Spur-Gear System Using the State-Space Approach

1985 ◽  
Vol 107 (1) ◽  
pp. 54-60 ◽  
Author(s):  
A. S. Kumar ◽  
T. S. Sankar ◽  
M. O. M. Osman
Keyword(s):  
State Space ◽  
Dynamic Load ◽  
Initial Conditions ◽  
Gear Tooth ◽  
Spur Gear ◽  
The State ◽  
Gear System ◽  
Computational Time ◽  
Steady State Condition ◽  
The Stability

In this study, a new approach using the state-space method is presented for the dynamic load analysis of spur gear systems. This approach gives the dynamic load on gear tooth in mesh as well as information on the stability of the gear system. Also a procedure is given for the selection of proper initial conditions that enable the steady-state condition to be reached faster, conditions that result in considerable savings in computational time. The variations in the dynamic load with respect to changes in contact position, operating speed, backlash, damping, and stiffness are also investigated. In addition, the stability of the gear system is studied, using the Floquet theory and the well-known stability conditions of difference systems.

Emergent Structures

2010 ◽  
pp. 83-113
Author(s):  
Eleonora Bilotta ◽  
Pietro Pantano
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Initial Conditions ◽  
Natural Sciences ◽  
Forward Problem ◽  
Second Step ◽  
Large Collection ◽  
Parameter Values

There are two classes of problem in the study of Cellular Automata. The forward. problem is the problem of determining the properties of the system. Solutions often consist of finding quantities that are computable on a rules table and characterizing the behavior of the rule upon repeated iterations, starting from different initial conditions. Solutions to the backwards problem begin with the properties of a system and find a rule or a set of rules which have these properties. This is especially useful in the application of Cellular Automata to the natural sciences, when researchers deal with a large collection of phenomena (Gutowitz, 1989). Another approach is to identify the basic structures of a Cellular Automaton (Adamatzky, 1995). Once these are known it becomes possible to develop specific models for particular systems and to detect general principles applicable to a wide variety of systems (Wolfram, 1984; Lam, 1998). According to Adamatzky, the identification of a system consists of two related steps, namely specification and estimation. In specification we choose a useful and efficient description of the system: perhaps an equation and a set of parameters. The second step involves the estimation of parameter values for the equation: exploiting measures of similarity.

ANALYSIS OF COLD-STANDBY SYSTEMS — CUTTING AND CLUSTERING THE STATE SPACE

1995 ◽  
Vol 02 (03) ◽  
pp. 327-340 ◽  
Author(s):  
M.N. GOPALAN ◽  
U. DINESH KUMAR
Keyword(s):  
State Space ◽  
Integral Equations ◽  
Initial Conditions ◽  
Approximate Solutions ◽  
The State ◽  
Convolution Integral ◽  
Cold Standby ◽  
Convolution Integral Equations ◽  
Systems Of Integral Equations ◽  
Repair Facility

In this paper two methods, namely cutting and clustering the state space are discussed to find approximate solutions to n-unit cold-standby system with a single repair facility. It has been assumed that the failure rate is constant and the repair time is arbitrarily distributed. A mathematical model is developed using semi-regenerative phenomena and systems of convolution integral equations satisfied by various state probabilities corresponding to different initial conditions are obtained. Explicit expressions for the availability and the reliability of the system are obtained. An iterative method is used to solve the systems of integral equations obtained and a comparative study has been carried out between exact and approximate solutions.

Models and Paradigms of Cellular Automata With an Organized Set of Active Cells

2021 ◽  
pp. 301-322
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Colony Formation ◽  
The State ◽  
Time Step ◽  
Asynchronous Cellular Automata ◽  
Main Cell ◽  
Image Contour

The chapter presents the principles of functioning of asynchronous cellular automata with a group of cells united in a colony. The rules of the formation of colonies of active cells and methods to move them along the field of a cellular automaton are considered. Each formed colony of active cells has a main cell that controls the movement of the entire colony. If several colonies of identical cells meet and combine, then the main cell is selected according to the priority, which is evaluated by the state of the cells of their neighborhoods. Colonies with different active cells can interact, destroying each other. The methods of interaction of colonies with different active states are described. An example of colony formation for solving the problem of describing contour images is presented. The image is described by moving the colony through the cells belonging to the image contour and fixing the cell sectors of the colony, which include the cells of the contour at each time step.

Pseudorandom Number Generators Based on Cellular Automata With the Hexagonal Coverage

2017 ◽  
pp. 139-156
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Random Number ◽  
The State ◽  
Time Step ◽  
Random Number Generators ◽  
Asynchronous Cellular Automata ◽  
Local Functions ◽  
Initial Settings ◽  
The Right

The seventh chapter describes approaches to constructing pseudo-random number generators based on cellular automata with a hexagonal coating. Several variants of cellular automata with hexagonal coating are considered. Asynchronous cellular automata with hexagonal coating are used. To simulate such cellular automata with software, a hexagonal coating was formed using an orthogonal coating. At the same time, all odd lines shifted to the cell floor to the right or to the left. The neighborhood of each cell contains six neighboring cells that have one common side with one cell of neighborhood. The chapter considers the behavior of cellular automata for different sizes and different initial settings. The behavior of cellular automata with various local functions is described, as well as the behavior of the cellular automaton with an additional bit inverting the state of the cell in each time step of functioning.

THE EFFECTIVE POTENTIAL AND TRANSSHIPMENT IN THERMODYNAMIC FORMALISM AT ZERO TEMPERATURE

2012 ◽  
Vol 13 (01) ◽  
pp. 1250009 ◽  
Author(s):  
EDUARDO GARIBALDI ◽  
ARTUR O. LOPES
Keyword(s):  
Finite Type ◽  
Effective Potential ◽  
Transition Matrix ◽  
Thermodynamic Formalism ◽  
Absolute Temperature ◽  
Zero Temperature ◽  
Hölder Continuous ◽  
Subshift Of Finite Type ◽  
Topologically Transitive ◽  
Unique Eigenvalue

For a topologically transitive subshift of finite type defined by a symmetric transition matrix, we introduce a temperature-based problem related to the usual thermodynamic formalism. This problem is described by an operator acting on Hölder continuous observables which is actually superlinear with respect to the max-plus algebra. We thus show that, for each fixed absolute temperature, such an operator admits a unique eigenfunction and a unique eigenvalue. We also study the convergence as the temperature goes to zero and we relate the limit objects to an ergodic version of Kantorovich transshipment problem.

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