67 research outputs found
Synchronisation of time--delay systems
We present the linear-stability analysis of synchronised states in coupled
time-delay systems. There exists a synchronisation threshold, for which we
derive upper bounds, which does not depend on the delay time. We prove that at
least for scalar time-delay systems synchronisation is achieved by transmitting
a single scalar signal, even if the synchronised solution is given by a
high-dimensional chaotic state with a large number of positive
Lyapunov-exponents. The analytical results are compared with numerical
simulations of two coupled Mackey-Glass equations
Estimation of Lyapunov spectra from space-time data
A method to estimate Lyapunov spectra from spatio-temporal data is presented,
which is well-suited to be applied to experimental situations. It allows to
characterize the high-dimensional chaotic states, with possibly a large number
of positive Lyapunov exponents, observed in spatio-temporal chaos. The method
is applied to data from a coupled map lattice
The identification of continuous, spatiotemporal systems
We present a method for the identification of continuous, spatiotemporal
dynamics from experimental data. We use a model in the form of a partial
differential equation and formulate an optimization problem for its estimation
from data. The solution is found as a multivariate nonlinear regression problem
using the ACE-algorithm. The procedure is successfully applied to data,
obtained by simulation of the Swift-Hohenberg equation. There are no
restrictions on the dimensionality of the investigated system, allowing for the
analysis of high-dimensional chaotic as well as transient dynamics. The demands
on the experimental data are discussed as well as the sensitivity of the method
towards noise
Identifying and modelling delay feedback systems
Systems with delayed feedback can possess chaotic attractors with extremely
high dimension, even if only a few physical degrees of freedom are involved. We
propose a state space reconstruction from time series data of a scalar
observable, coming along with a novel method to identify and model such
systems, if a single variable is fed back. Making use of special properties of
the feedback structure, we can understand the structure of the system by
constructing equivalent equations of motion in spaces with dimensions which can
be much smaller than the dimension of the chaotic attractor. We verify our
method using both numerical and experimental data
Chaos Synchronization of delayed systems in the presence of delay time modulation
We investigate synchronization in the presence of delay time modulation for
application to communication. We have observed that the robust synchronization
is established by a common delay signal and its threshold is presented using
Lyapunov exponents analysis. The influence of the delay time modulation in
chaotic oscillators is also discussed.Comment: 9 pages, 6 figure
Symbolic dynamics and synchronization of coupled map networks with multiple delays
We use symbolic dynamics to study discrete-time dynamical systems with
multiple time delays. We exploit the concept of avoiding sets, which arise from
specific non-generating partitions of the phase space and restrict the
occurrence of certain symbol sequences related to the characteristics of the
dynamics. In particular, we show that the resulting forbidden sequences are
closely related to the time delays in the system. We present two applications
to coupled map lattices, namely (1) detecting synchronization and (2)
determining unknown values of the transmission delays in networks with possibly
directed and weighted connections and measurement noise. The method is
applicable to multi-dimensional as well as set-valued maps, and to networks
with time-varying delays and connection structure.Comment: 13 pages, 4 figure
A Tool to Recover Scalar Time-Delay Systems from Experimental Time Series
We propose a method that is able to analyze chaotic time series, gained from
exp erimental data. The method allows to identify scalar time-delay systems. If
the dynamics of the system under investigation is governed by a scalar
time-delay differential equation of the form ,
the delay time and the functi on can be recovered. There are no
restrictions to the dimensionality of the chaotic attractor. The method turns
out to be insensitive to noise. We successfully apply the method to various
time series taken from a computer experiment and two different electronic
oscillators
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