3,839 research outputs found
Undecidable Properties of Limit Set Dynamics of Cellular Automata
Cellular Automata (CA) are discrete dynamical systems and an abstract model
of parallel computation. The limit set of a cellular automaton is its maximal
topological attractor. A well know result, due to Kari, says that all
nontrivial properties of limit sets are undecidable. In this paper we consider
properties of limit set dynamics, i.e. properties of the dynamics of Cellular
Automata restricted to their limit sets. There can be no equivalent of Kari's
Theorem for limit set dynamics. Anyway we show that there is a large class of
undecidable properties of limit set dynamics, namely all properties of limit
set dynamics which imply stability or the existence of a unique subshift
attractor. As a consequence we have that it is undecidable whether the cellular
automaton map restricted to the limit set is the identity, closing, injective,
expansive, positively expansive, transitive
Limit sets of stable Cellular Automata
We study limit sets of stable cellular automata standing from a symbolic
dynamics point of view where they are a special case of sofic shifts admitting
a steady epimorphism. We prove that there exists a right-closing
almost-everywhere steady factor map from one irreducible sofic shift onto
another one if and only if there exists such a map from the domain onto the
minimal right-resolving cover of the image. We define right-continuing
almost-everywhere steady maps and prove that there exists such a steady map
between two sofic shifts if and only if there exists a factor map from the
domain onto the minimal right-resolving cover of the image. In terms of
cellular automata, this translates into: A sofic shift can be the limit set of
a stable cellular automaton with a right-closing almost-everywhere dynamics
onto its limit set if and only if it is the factor of a fullshift and there
exists a right- closing almost-everywhere factor map from the sofic shift onto
its minimal right- resolving cover. A sofic shift can be the limit set of a
stable cellular automaton reaching its limit set with a right-continuing
almost-everywhere factor map if and only if it is the factor of a fullshift and
there exists a factor map from the sofic shift onto its minimal right-resolving
cover. Finally, as a consequence of the previous results, we provide a
characterization of the Almost of Finite Type shifts (AFT) in terms of a
property of steady maps that have them as range.Comment: 18 pages, 3 figure
Spatial organization and evolutional period of the epidemic model using cellular automata
We investigate epidemic models with spatial structure based on the cellular
automata method. The construction of the cellular automata is from the study by
Weimar and Boon about the reaction-diffusion equations [Phys. Rev. E 49, 1749
(1994)]. Our results show that the spatial epidemic models exhibit the
spontaneous formation of irregular spiral waves at large scales within the
domain of chaos. Moreover, the irregular spiral waves grow stably. The system
also shows a spatial period-2 structure at one dimension outside the domain of
chaos. It is interesting that the spatial period-2 structure will break and
transform into a spatial synchronous configuration in the domain of chaos. Our
results confirm that populations embed and disperse more stably in space than
they do in nonspatial counterparts.Comment: 6 papges,5 figures. published in Physics Review
Synchronization of non-chaotic dynamical systems
A synchronization mechanism driven by annealed noise is studied for two
replicas of a coupled-map lattice which exhibits stable chaos (SC), i.e.
irregular behavior despite a negative Lyapunov spectrum. We show that the
observed synchronization transition, on changing the strength of the stochastic
coupling between replicas, belongs to the directed percolation universality
class. This result is consistent with the behavior of chaotic deterministic
cellular automata (DCA), supporting the equivalence Ansatz between SC models
and DCA. The coupling threshold above which the two system replicas synchronize
is strictly related to the propagation velocity of perturbations in the system.Comment: 16 pages + 12 figures, new and extended versio
A self-organized model for cell-differentiation based on variations of molecular decay rates
Systemic properties of living cells are the result of molecular dynamics
governed by so-called genetic regulatory networks (GRN). These networks capture
all possible features of cells and are responsible for the immense levels of
adaptation characteristic to living systems. At any point in time only small
subsets of these networks are active. Any active subset of the GRN leads to the
expression of particular sets of molecules (expression modes). The subsets of
active networks change over time, leading to the observed complex dynamics of
expression patterns. Understanding of this dynamics becomes increasingly
important in systems biology and medicine. While the importance of
transcription rates and catalytic interactions has been widely recognized in
modeling genetic regulatory systems, the understanding of the role of
degradation of biochemical agents (mRNA, protein) in regulatory dynamics
remains limited. Recent experimental data suggests that there exists a
functional relation between mRNA and protein decay rates and expression modes.
In this paper we propose a model for the dynamics of successions of sequences
of active subnetworks of the GRN. The model is able to reproduce key
characteristics of molecular dynamics, including homeostasis, multi-stability,
periodic dynamics, alternating activity, differentiability, and self-organized
critical dynamics. Moreover the model allows to naturally understand the
mechanism behind the relation between decay rates and expression modes. The
model explains recent experimental observations that decay-rates (or turnovers)
vary between differentiated tissue-classes at a general systemic level and
highlights the role of intracellular decay rate control mechanisms in cell
differentiation.Comment: 16 pages, 5 figure
Breakdown and recovery in traffic flow models
Most car-following models show a transition from laminar to ``congested''
flow and vice versa. Deterministic models often have a density range where a
disturbance needs a sufficiently large critical amplitude to move the flow from
the laminar into the congested phase. In stochastic models, it may be assumed
that the size of this amplitude gets translated into a waiting time, i.e.\
until fluctuations sufficiently add up to trigger the transition. A recently
introduced model of traffic flow however does not show this behavior: in the
density regime where the jam solution co-exists with the high-flow state, the
intrinsic stochasticity of the model is not sufficient to cause a transition
into the jammed regime, at least not within relevant time scales. In addition,
models can be differentiated by the stability of the outflow interface. We
demonstrate that this additional criterion is not related to the stability of
the flow. The combination of these criteria makes it possible to characterize
commonalities and differences between many existing models for traffic in a new
way
Chaotic Scattering Theory, Thermodynamic Formalism, and Transport Coefficients
The foundations of the chaotic scattering theory for transport and
reaction-rate coefficients for classical many-body systems are considered here
in some detail. The thermodynamic formalism of Sinai, Bowen, and Ruelle is
employed to obtain an expression for the escape-rate for a phase space
trajectory to leave a finite open region of phase space for the first time.
This expression relates the escape rate to the difference between the sum of
the positive Lyapunov exponents and the K-S entropy for the fractal set of
trajectories which are trapped forever in the open region. This result is well
known for systems of a few degrees of freedom and is here extended to systems
of many degrees of freedom. The formalism is applied to smooth hyperbolic
systems, to cellular-automata lattice gases, and to hard sphere sytems. In the
latter case, the goemetric constructions of Sinai {\it et al} for billiard
systems are used to describe the relevant chaotic scattering phenomena. Some
applications of this formalism to non-hyperbolic systems are also discussed.Comment: 35 pages, compressed file, follow directions in header for ps file.
Figures are available on request from [email protected]
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