6,815 research outputs found
Effective Symbolic Dynamics
AbstractWe investigate computable subshifts and the connection with effective symbolic dynamics. It is shown that a decidable Î 10 class P is a subshift if and only if there is a computable function F mapping 2N to 2N such that P is the set of itineraries of elements of 2N. A Î 10 subshift is constructed which has no computable element. We also consider the symbolic dynamics of maps on the unit interval
Computational information for the logistic map at the chaos threshold
We study the logistic map on the unit square at the
chaos threshold. By using the methods of symbolic dynamics, the information
content of an orbit of a dynamical system is defined as the Algorithmic
Information Content (AIC) of a symbolic sequence. We give results for the
behaviour of the AIC for the logistic map. Since the AIC is not a computable
function we use, as approximation of the AIC, a notion of information content
given by the length of the string after it has been compressed by a compression
algorithm, and in particular we introduce a new compression algorithm called
CASToRe. The information content is then used to characterise the chaotic
behaviour.Comment: 23 pages, 3 figures, changed conten
Global and local Complexity in weakly chaotic dynamical systems
In a topological dynamical system the complexity of an orbit is a measure of
the amount of information (algorithmic information content) that is necessary
to describe the orbit. This indicator is invariant up to topological
conjugation. We consider this indicator of local complexity of the dynamics and
provide different examples of its behavior, showing how it can be useful to
characterize various kind of weakly chaotic dynamics. We also provide criteria
to find systems with non trivial orbit complexity (systems where the
description of the whole orbit requires an infinite amount of information). We
consider also a global indicator of the complexity of the system. This global
indicator generalizes the topological entropy, taking into account systems were
the number of essentially different orbits increases less than exponentially.
Then we prove that if the system is constructive (roughly speaking: if the map
can be defined up to any given accuracy using a finite amount of information)
the orbit complexity is everywhere less or equal than the generalized
topological entropy. Conversely there are compact non constructive examples
where the inequality is reversed, suggesting that this notion comes out
naturally in this kind of complexity questions.Comment: 23 page
The Algorithmic Information Content for randomly perturbed systems
In this paper we prove estimates on the behaviour of the Kolmogorov-Sinai
entropy relative to a partition for randomly perturbed dynamical systems. Our
estimates use the entropy for the unperturbed system and are obtained using the
notion of Algorithmic Information Content. The main result is an extension of
known results to study time series obtained by the observation of real systems.Comment: 17 pages, 1 figur
Compression and diffusion: a joint approach to detect complexity
The adoption of the Kolmogorov-Sinai (KS) entropy is becoming a popular
research tool among physicists, especially when applied to a dynamical system
fitting the conditions of validity of the Pesin theorem. The study of time
series that are a manifestation of system dynamics whose rules are either
unknown or too complex for a mathematical treatment, is still a challenge since
the KS entropy is not computable, in general, in that case. Here we present a
plan of action based on the joint action of two procedures, both related to the
KS entropy, but compatible with computer implementation through fast and
efficient programs. The former procedure, called Compression Algorithm
Sensitive To Regularity (CASToRe), establishes the amount of order by the
numerical evaluation of algorithmic compressibility. The latter, called Complex
Analysis of Sequences via Scaling AND Randomness Assessment (CASSANDRA),
establishes the complexity degree through the numerical evaluation of the
strength of an anomalous effect. This is the departure, of the diffusion
process generated by the observed fluctuations, from ordinary Brownian motion.
The CASSANDRA algorithm shares with CASToRe a connection with the Kolmogorov
complexity. This makes both algorithms especially suitable to study the
transition from dynamics to thermodynamics, and the case of non-stationary time
series as well. The benefit of the joint action of these two methods is proven
by the analysis of artificial sequences with the same main properties as the
real time series to which the joint use of these two methods will be applied in
future research work.Comment: 27 pages, 9 figure
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