COARSE-GRAINING INVARIANT ORBITS OF ONE-DIMENSIONAL ℤp-LINEAR CELLULAR AUTOMATA

1996 ◽  
Vol 06 (12a) ◽  
pp. 2237-2297 ◽  
Author(s):  
ANDRÉ M. BARBÉ
Keyword(s):  
Cellular Automata ◽  
Periodic Solutions ◽  
Time Evolution ◽  
Coarse Graining ◽  
One Dimensional ◽  
State Time ◽  
All Solutions ◽  
Invariance Problem ◽  
Self Similar ◽  
Existence Of Periodic Solutions

This paper studies the properties of state-time evolution patterns of one-dimensional linear cellular automata over the field ℤp (p prime) which are invariant under certain coarse-graining operations. A procedure is developed for finding all solutions to this invariance problem. The resulting patterns display a complexity which may range from periodic over self-similar, quasi-periodic, quasi-randomlike towards randomlike. Conditions for the existence of periodic solutions are derived.

COARSE-GRAINING INVARIANT PATTERNS OF ONE-DIMENSIONAL TWO-STATE LINEAR CELLULAR AUTOMATA

1995 ◽  
Vol 05 (06) ◽  
pp. 1611-1631 ◽  
Author(s):  
A. BARBÉ ◽  
F. V. HAESELER ◽  
H.-O. PEITGEN ◽  
G. SKORDEV
Keyword(s):  
Cellular Automata ◽  
Coarse Graining ◽  
Complex Nature ◽  
One Dimensional ◽  
Initial States

We consider one-dimensional two state cellular automata and their orbital patterns. We are particulary interested in initial states and their orbital pattern which are invariant under a certain coarse-graining operation. We show that invariant initial states are automatic. Moreover, we numerically study the complex nature of the initial states and their invariant orbits.

ON THE STRUCTURE OF REAL-VALUED ONE-DIMENSIONAL CELLULAR AUTOMATA

2011 ◽  
Vol 21 (05) ◽  
pp. 1265-1279 ◽  
Author(s):  
XU XU ◽  
STEPHEN P. BANKS ◽  
MAHDI MAHFOUF
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Fixed Points ◽  
Periodic Solutions ◽  
Cellular Systems ◽  
One Dimensional ◽  
Dynamical Behaviors ◽  
New Type ◽  
Complex Patterns

It is well-known that binary-valued cellular automata, which are defined by simple local rules, have the amazing feature of generating very complex patterns and having complicated dynamical behaviors. In this paper, we present a new type of cellular automaton based on real-valued states which produce an even greater amount of interesting structures such as fractal, chaotic and hypercyclic. We also give proofs to real-valued cellular systems which have fixed points and periodic solutions.

Stability of a Semi-Active Impact Damper: Part II—Periodic Solutions

1989 ◽  
Vol 56 (4) ◽  
pp. 930-940 ◽  
Author(s):  
M. P. Karyeaclis ◽  
T. K. Caughey
Keyword(s):  
Asymptotic Stability ◽  
Periodic Solutions ◽  
Impact Damper ◽  
All Solutions ◽  
Existence Of Periodic Solutions

All solutions of the semi-active impact damper described in Part I were shown to be bounded when the excitation is bounded. In this part, the existence of periodic solutions is investigated. Emphasis is placed on two impacts/cycle periodic solutions. Exact symmetric and nonsymmetric harmonic solutions are derived analytically and the region of asymptotic stability is determined.

Complex Order from Disorder and from Simple Order in Coarse-Graining Invariant Orbits of Certain Two-Dimensional Linear Cellular Automata

1997 ◽  
Vol 07 (07) ◽  
pp. 1451-1496 ◽  
Author(s):  
André Barbé
Keyword(s):  
Cellular Automata ◽  
Three Dimensional ◽  
Coarse Graining ◽  
Dimensional Case ◽  
Two Dimensional ◽  
One Dimensional ◽  
Complex Order ◽  
Simple Order ◽  
Problems And Solutions ◽  
The One

This paper considers three-dimensional coarse-graining invariant orbits for two-dimensional linear cellular automata over a finite field, as a nontrivial extension of the two-dimensional coarse-graining invariant orbits for one-dimensional CA that were studied in an earlier paper. These orbits can be found by solving a particular kind of recursive equations (renormalizing equations with rescaling term). The solution starts from some seed that has to be determined first. In contrast with the one-dimensional case, the seed has infinite support in most cases. The way for solving these equations is discussed by means of some examples. Three categories of problems (and solutions) can be distinguished (as opposed to only one in the one-dimensional case). Finally, the morphology of a few coarse-graining invariant orbits is discussed: Complex order (of quasiperiodic type) seems to emerge from random seeds as well as from seeds of simple order (for example, constant or periodic seeds).

scholarly journals Periodic Solutions of One-Dimensional Cellular Automata with Uniformly Chosen Random Rules

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Janko Gravner ◽  
Xiaochen Liu
Keyword(s):  
Cellular Automata ◽  
Periodic Solutions ◽  
Main Tool ◽  
Random Variable ◽  
Poisson Approximation ◽  
Spatial Period ◽  
Finite Range ◽  
Poisson Random Variable ◽  
One Dimensional ◽  
Stein Method

We study cellular automata whose rules are selected uniformly at random. Our setting are two-neighbor one-dimensional rules with a large number $n$ of states. The main quantity we analyze is the asymptotic distribution, as $n \to \infty$, of the number of different periodic solutions with given spatial and temporal periods. The main tool we use is the Chen-Stein method for Poisson approximation, which establishes that the number of periodic solutions, with their spatial and temporal periods confined to a finite range, converges to a Poisson random variable with an explicitly given parameter. The limiting probability distribution of the smallest temporal period for a given spatial period is deduced as a corollary and relevant empirical simulations are presented.

AUTOMATICITY OF COARSE-GRAINING INVARIANT ORBITS OF ONE-DIMENSIONAL LINEAR CELLULAR AUTOMATA

1999 ◽  
Vol 09 (01) ◽  
pp. 67-95 ◽  
Author(s):  
ANDRÉ BARBÉ ◽  
HEINZ-OTTO PEITGEN ◽  
GENCHO SKORDEV
Keyword(s):  
Cellular Automata ◽  
Prime Power ◽  
Coarse Graining ◽  
One Dimensional ◽  
Modulo P ◽  
Initial States ◽  
The Relationship

We consider one-dimensional linear cellular automata whose states are the integers modulo a prime power pd and their orbital patterns. We are particulary interested in initial states and their orbital patterns which are invariant under a certain coarse-graining operation. We show that these coarse-graining invariant initial states are p-automatic. The relationship between the solutions of a certain family of coarse-graining invariant problems concerning linear cellular automata over the integers modulo pn is investigated.

SHIFT-ADD CORRELATION PATTERNS OF ORBITS OF ONE-DIMENSIONAL LINEAR CELLULAR AUTOMATA

1996 ◽  
Vol 06 (12b) ◽  
pp. 2507-2530
Author(s):  
ANDRÉ BARBÉ
Keyword(s):  
Cellular Automata ◽  
Cellular Automaton ◽  
Commutative Ring ◽  
Time Evolution ◽  
Two Dimensional ◽  
Correlation Pattern ◽  
One Dimensional ◽  
Evolution Pattern ◽  
Finite Commutative Ring ◽  
Correlation Space

We consider the two-dimensional state-time evolution pattern (orbit) of a one-dimensional linear cellular automaton (CA) defined over a finite commutative ring R, and correlate this to shifted versions of itself by adding the original pattern and its shifted versions. The resulting pattern is called the shift-add (correlation) pattern. A dual counterpart of this pattern is also introduced: it forms a so-called pseudo-CA orbit. We show that the set of shift-add patterns for different shifts interrelate like a pseudo-CA whose states are two-dimensional patterns over R. The set of dual shift-add patterns forms likewise a regular CA. Complementary ways of viewing the shift-add correlation space are also presented. Finally, the nature of shift-add correlation patterns for the newly defined class of pseudo-CA is investigated.

scholarly journals Periodic Solutions for a Prescribed Mean Curvature Equation with Multiple Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Shiping Lu ◽  
Ming Lu
Keyword(s):  
Periodic Solutions ◽  
Mean Curvature ◽  
Multiple Delays ◽  
Mawhin’S Continuation Theorem ◽  
One Dimensional ◽  
Prescribed Mean Curvature ◽  
Mean Curvature Equation ◽  
Prescribed Mean Curvature Equation ◽  
Existence Of Periodic Solutions ◽  
The One

We study the existence of periodic solutions for the one-dimensional prescribed mean curvature delay equation(d/dt)(x'(t)/1+x't2) +∑i=1naitgxt-τit=pt. By using Mawhin's continuation theorem, a new result is obtained. Furthermore, the nonexistence of periodic solution for the equation is investigated as well.

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