9,279 research outputs found
Linear and nonlinear information flow in spatially extended systems
Infinitesimal and finite amplitude error propagation in spatially extended
systems are numerically and theoretically investigated. The information
transport in these systems can be characterized in terms of the propagation
velocity of perturbations . A linear stability analysis is sufficient to
capture all the relevant aspects associated to propagation of infinitesimal
disturbances. In particular, this analysis gives the propagation velocity
of infinitesimal errors. If linear mechanisms prevail on the nonlinear ones
. On the contrary, if nonlinear effects are predominant finite
amplitude disturbances can eventually propagate faster than infinitesimal ones
(i.e. ). The finite size Lyapunov exponent can be successfully
employed to discriminate the linear or nonlinear origin of information flow. A
generalization of finite size Lyapunov exponent to a comoving reference frame
allows to state a marginal stability criterion able to provide both in
the linear and in the nonlinear case. Strong analogies are found between
information spreading and propagation of fronts connecting steady states in
reaction-diffusion systems. The analysis of the common characteristics of these
two phenomena leads to a better understanding of the role played by linear and
nonlinear mechanisms for the flow of information in spatially extended systems.Comment: 14 RevTeX pages with 13 eps figures, title/abstract changed minor
changes in the text accepted for publication on PR
Simple deterministic dynamical systems with fractal diffusion coefficients
We analyze a simple model of deterministic diffusion. The model consists of a
one-dimensional periodic array of scatterers in which point particles move from
cell to cell as defined by a piecewise linear map. The microscopic chaotic
scattering process of the map can be changed by a control parameter. This
induces a parameter dependence for the macroscopic diffusion coefficient. We
calculate the diffusion coefficent and the largest eigenmodes of the system by
using Markov partitions and by solving the eigenvalue problems of respective
topological transition matrices. For different boundary conditions we find that
the largest eigenmodes of the map match to the ones of the simple
phenomenological diffusion equation. Our main result is that the difffusion
coefficient exhibits a fractal structure by varying the system parameter. To
understand the origin of this fractal structure, we give qualitative and
quantitative arguments. These arguments relate the sequence of oscillations in
the strength of the parameter-dependent diffusion coefficient to the
microscopic coupling of the single scatterers which changes by varying the
control parameter.Comment: 28 pages (revtex), 12 figures (postscript), submitted to Phys. Rev.
Nonlinearly driven transverse synchronization in coupled chaotic systems
Synchronization transitions are investigated in coupled chaotic maps.
Depending on the relative weight of linear versus nonlinear instability
mechanisms associated to the single map two different scenarios for the
transition may occur. When only two maps are considered we always find that the
critical coupling for chaotic synchronization can be predicted
within a linear analysis by the vanishing of the transverse Lyapunov exponent
. However, major differences between transitions driven by linear or
nonlinear mechanisms are revealed by the dynamics of the transient toward the
synchronized state. As a representative example of extended systems a one
dimensional lattice of chaotic maps with power-law coupling is considered. In
this high dimensional model finite amplitude instabilities may have a dramatic
effect on the transition. For strong nonlinearities an exponential divergence
of the synchronization times with the chain length can be observed above
, notwithstanding the transverse dynamics is stable against
infinitesimal perturbations at any instant. Therefore, the transition takes
place at a coupling definitely larger than and its
origin is intrinsically nonlinear. The linearly driven transitions are
continuous and can be described in terms of mean field results for
non-equilibrium phase transitions with long range interactions. While the
transitions dominated by nonlinear mechanisms appear to be discontinuous.Comment: 29 pages, 14 figure
Low dimensional behavior in three-dimensional coupled map lattices
The analysis of one-, two-, and three-dimensional coupled map lattices is
here developed under a statistical and dynamical perspective. We show that the
three-dimensional CML exhibits low dimensional behavior with long range
correlation and the power spectrum follows noise. This approach leads to
an integrated understanding of the most important properties of these universal
models of spatiotemporal chaos. We perform a complete time series analysis of
the model and investigate the dependence of the signal properties by change of
dimension.Comment: 7 pages, 6 figures (revised
Practical implementation of nonlinear time series methods: The TISEAN package
Nonlinear time series analysis is becoming a more and more reliable tool for
the study of complicated dynamics from measurements. The concept of
low-dimensional chaos has proven to be fruitful in the understanding of many
complex phenomena despite the fact that very few natural systems have actually
been found to be low dimensional deterministic in the sense of the theory. In
order to evaluate the long term usefulness of the nonlinear time series
approach as inspired by chaos theory, it will be important that the
corresponding methods become more widely accessible. This paper, while not a
proper review on nonlinear time series analysis, tries to make a contribution
to this process by describing the actual implementation of the algorithms, and
their proper usage. Most of the methods require the choice of certain
parameters for each specific time series application. We will try to give
guidance in this respect. The scope and selection of topics in this article, as
well as the implementational choices that have been made, correspond to the
contents of the software package TISEAN which is publicly available from
http://www.mpipks-dresden.mpg.de/~tisean . In fact, this paper can be seen as
an extended manual for the TISEAN programs. It fills the gap between the
technical documentation and the existing literature, providing the necessary
entry points for a more thorough study of the theoretical background.Comment: 27 pages, 21 figures, downloadable software at
http://www.mpipks-dresden.mpg.de/~tisea
Nonlinear time-series analysis revisited
In 1980 and 1981, two pioneering papers laid the foundation for what became
known as nonlinear time-series analysis: the analysis of observed
data---typically univariate---via dynamical systems theory. Based on the
concept of state-space reconstruction, this set of methods allows us to compute
characteristic quantities such as Lyapunov exponents and fractal dimensions, to
predict the future course of the time series, and even to reconstruct the
equations of motion in some cases. In practice, however, there are a number of
issues that restrict the power of this approach: whether the signal accurately
and thoroughly samples the dynamics, for instance, and whether it contains
noise. Moreover, the numerical algorithms that we use to instantiate these
ideas are not perfect; they involve approximations, scale parameters, and
finite-precision arithmetic, among other things. Even so, nonlinear time-series
analysis has been used to great advantage on thousands of real and synthetic
data sets from a wide variety of systems ranging from roulette wheels to lasers
to the human heart. Even in cases where the data do not meet the mathematical
or algorithmic requirements to assure full topological conjugacy, the results
of nonlinear time-series analysis can be helpful in understanding,
characterizing, and predicting dynamical systems
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