1,709 research outputs found
Celulární automat a CML systémy
The main aim of this thesis is the study of cellular automata and discrete dynamical systems on a lattice.
Both tools, cellular automata as well as dynamical systems on a lattice are introduced and elementary properties described.
The relation between cellular automata and dynamical system on lattice is derived.
The main goal of the thesis is also the use of the cellular automata as that mathematical tool of evolution visualization of discrete dynamical systems.
The theory of cellular automata is applied to the discrete dynamical systems on a lattice Laplacian type and implemented in Java language.Hlavním cílem práce je studium vztahu celulárních automatů a diskrétních dynamických systémů na mřížce. Oba nástroje, jak celulární automat tak dynamický systém na mřížce, jsou zavedeny a jejich základní vlastnosti popsány. Vztah mezi celulárními automaty a dynamickými systémy na mřížce je podrobně popsán. Hlavním cílem práce je dále použití nástroje celulárního automatu jako matematického vizualizačního prostředku evoluce diskrétních dynamických systémů. Teorie celulárních automatů je použita na dynamické systémy na mřížce Lamplaceova typu a implementována v prostředí Java.470 - Katedra aplikované matematikyvelmi dobř
Automatic Filters for the Detection of Coherent Structure in Spatiotemporal Systems
Most current methods for identifying coherent structures in
spatially-extended systems rely on prior information about the form which those
structures take. Here we present two new approaches to automatically filter the
changing configurations of spatial dynamical systems and extract coherent
structures. One, local sensitivity filtering, is a modification of the local
Lyapunov exponent approach suitable to cellular automata and other discrete
spatial systems. The other, local statistical complexity filtering, calculates
the amount of information needed for optimal prediction of the system's
behavior in the vicinity of a given point. By examining the changing
spatiotemporal distributions of these quantities, we can find the coherent
structures in a variety of pattern-forming cellular automata, without needing
to guess or postulate the form of that structure. We apply both filters to
elementary and cyclical cellular automata (ECA and CCA) and find that they
readily identify particles, domains and other more complicated structures. We
compare the results from ECA with earlier ones based upon the theory of formal
languages, and the results from CCA with a more traditional approach based on
an order parameter and free energy. While sensitivity and statistical
complexity are equally adept at uncovering structure, they are based on
different system properties (dynamical and probabilistic, respectively), and
provide complementary information.Comment: 16 pages, 21 figures. Figures considerably compressed to fit arxiv
requirements; write first author for higher-resolution version
The Computational Complexity of Symbolic Dynamics at the Onset of Chaos
In a variety of studies of dynamical systems, the edge of order and chaos has
been singled out as a region of complexity. It was suggested by Wolfram, on the
basis of qualitative behaviour of cellular automata, that the computational
basis for modelling this region is the Universal Turing Machine. In this paper,
following a suggestion of Crutchfield, we try to show that the Turing machine
model may often be too powerful as a computational model to describe the
boundary of order and chaos. In particular we study the region of the first
accumulation of period doubling in unimodal and bimodal maps of the interval,
from the point of view of language theory. We show that in relation to the
``extended'' Chomsky hierarchy, the relevant computational model in the
unimodal case is the nested stack automaton or the related indexed languages,
while the bimodal case is modeled by the linear bounded automaton or the
related context-sensitive languages.Comment: 1 reference corrected, 1 reference added, minor changes in body of
manuscrip
Topology regulates pattern formation capacity of binary cellular automata on graphs
We study the effect of topology variation on the dynamic behavior of a system
with local update rules. We implement one-dimensional binary cellular automata
on graphs with various topologies by formulating two sets of degree-dependent
rules, each containing a single parameter. We observe that changes in graph
topology induce transitions between different dynamic domains (Wolfram classes)
without a formal change in the update rule. Along with topological variations,
we study the pattern formation capacities of regular, random, small-world and
scale-free graphs. Pattern formation capacity is quantified in terms of two
entropy measures, which for standard cellular automata allow a qualitative
distinction between the four Wolfram classes. A mean-field model explains the
dynamic behavior of random graphs. Implications for our understanding of
information transport through complex, network-based systems are discussed.Comment: 16 text pages, 13 figures. To be published in Physica
Statistical Mechanics of Surjective Cellular Automata
Reversible cellular automata are seen as microscopic physical models, and
their states of macroscopic equilibrium are described using invariant
probability measures. We establish a connection between the invariance of Gibbs
measures and the conservation of additive quantities in surjective cellular
automata. Namely, we show that the simplex of shift-invariant Gibbs measures
associated to a Hamiltonian is invariant under a surjective cellular automaton
if and only if the cellular automaton conserves the Hamiltonian. A special case
is the (well-known) invariance of the uniform Bernoulli measure under
surjective cellular automata, which corresponds to the conservation of the
trivial Hamiltonian. As an application, we obtain results indicating the lack
of (non-trivial) Gibbs or Markov invariant measures for "sufficiently chaotic"
cellular automata. We discuss the relevance of the randomization property of
algebraic cellular automata to the problem of approach to macroscopic
equilibrium, and pose several open questions.
As an aside, a shift-invariant pre-image of a Gibbs measure under a
pre-injective factor map between shifts of finite type turns out to be always a
Gibbs measure. We provide a sufficient condition under which the image of a
Gibbs measure under a pre-injective factor map is not a Gibbs measure. We point
out a potential application of pre-injective factor maps as a tool in the study
of phase transitions in statistical mechanical models.Comment: 50 pages, 7 figure
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