1,991 research outputs found
Complex Dynamics and Synchronization of Delayed-Feedback Nonlinear Oscillators
We describe a flexible and modular delayed-feedback nonlinear oscillator that
is capable of generating a wide range of dynamical behaviours, from periodic
oscillations to high-dimensional chaos. The oscillator uses electrooptic
modulation and fibre-optic transmission, with feedback and filtering
implemented through real-time digital-signal processing. We consider two such
oscillators that are coupled to one another, and we identify the conditions
under which they will synchronize. By examining the rates of divergence or
convergence between two coupled oscillators, we quantify the maximum Lyapunov
exponents or transverse Lyapunov exponents of the system, and we present an
experimental method to determine these rates that does not require a
mathematical model of the system. Finally, we demonstrate a new adaptive
control method that keeps two oscillators synchronized even when the coupling
between them is changing unpredictably.Comment: 24 pages, 13 figures. To appear in Phil. Trans. R. Soc. A (special
theme issue to accompany 2009 International Workshop on Delayed Complex
Systems
Regular and chaotic interactions of two BPS dyons at low energy
We identify and analyze quasiperiodic and chaotic motion patterns in the time
evolution of a classical, non-Abelian Bogomol'nyi-Prasad-Sommerfield (BPS) dyon
pair at low energies. This system is amenable to the geodesic approximation
which restricts the underlying SU(2) Yang-Mills-Higgs dynamics to an
eight-dimensional phase space. We numerically calculate a representative set of
long-time solutions to the corresponding Hamilton equations and analyze
quasiperiodic and chaotic phase space regions by means of Poincare surfaces of
section, high-resolution power spectra and Lyapunov exponents. Our results
provide clear evidence for both quasiperiodic and chaotic behavior and
characterize it quantitatively. Indications for intermittency are also
discussed.Comment: 22 pages, 6 figures (v2 contains a few additional references, a new
paragraph on intermittency and minor stylistic corrections to agree with the
published version
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
Lagrangian Descriptors: A Method for Revealing Phase Space Structures of General Time Dependent Dynamical Systems
In this paper we develop new techniques for revealing geometrical structures
in phase space that are valid for aperiodically time dependent dynamical
systems, which we refer to as Lagrangian descriptors. These quantities are
based on the integration, for a finite time, along trajectories of an intrinsic
bounded, positive geometrical and/or physical property of the trajectory
itself. We discuss a general methodology for constructing Lagrangian
descriptors, and we discuss a "heuristic argument" that explains why this
method is successful for revealing geometrical structures in the phase space of
a dynamical system. We support this argument by explicit calculations on a
benchmark problem having a hyperbolic fixed point with stable and unstable
manifolds that are known analytically. Several other benchmark examples are
considered that allow us the assess the performance of Lagrangian descriptors
in revealing invariant tori and regions of shear. Throughout the paper
"side-by-side" comparisons of the performance of Lagrangian descriptors with
both finite time Lyapunov exponents (FTLEs) and finite time averages of certain
components of the vector field ("time averages") are carried out and discussed.
In all cases Lagrangian descriptors are shown to be both more accurate and
computationally efficient than these methods. We also perform computations for
an explicitly three dimensional, aperiodically time-dependent vector field and
an aperiodically time dependent vector field defined as a data set. Comparisons
with FTLEs and time averages for these examples are also carried out, with
similar conclusions as for the benchmark examples.Comment: 52 pages, 25 figure
Experimental realization of strange nonchaotic attractors in a quasiperiodically forced electronic circuit
We have identified the three prominent routes, namely Heagy-Hammel,
fractalization and intermittency routes, and their mechanisms for the birth of
strange nonchaotic attractors (SNAs) in a quasiperiodically forced electronic
system constructed using a negative conductance series LCR circuit with a diode
both numerically and experimentally. The birth of SNAs by these three routes is
verified from both experimental and their corresponding numerical data by
maximal Lyapunov exponents, and their variance, Poincar\'e maps, Fourier
amplitude spectrum, spectral distribution function and finite-time Lyapunov
exponents. Although these three routes have been identified numerically in
different dynamical systems, the experimental observation of all these
mechanisms is reported for the first time to our knowledge and that too in a
single second order electronic circuit.Comment: 21 figure
Dynamics of Coupled Maps with a Conservation Law
A particularly simple model belonging to a wide class of coupled maps which
obey a local conservation law is studied. The phase structure of the system and
the types of the phase transitions are determined. It is argued that the
structure of the phase diagram is robust with respect to mild violations of the
conservation law. Critical exponents possibly determining a new universality
class are calculated for a set of independent order parameters. Numerical
evidence is produced suggesting that the singularity in the density of Lyapunov
exponents at is a reflection of the singularity in the density of
Fourier modes (a ``Van Hove'' singularity) and disappears if the conservation
law is broken. Applicability of the Lyapunov dimension to the description of
spatiotemporal chaos in a system with a conservation law is discussed.Comment: To be published in CHAOS #7 (31 page, 16 figures
Mutual information rate and bounds for it
The amount of information exchanged per unit of time between two nodes in a
dynamical network or between two data sets is a powerful concept for analysing
complex systems. This quantity, known as the mutual information rate (MIR), is
calculated from the mutual information, which is rigorously defined only for
random systems. Moreover, the definition of mutual information is based on
probabilities of significant events. This work offers a simple alternative way
to calculate the MIR in dynamical (deterministic) networks or between two data
sets (not fully deterministic), and to calculate its upper and lower bounds
without having to calculate probabilities, but rather in terms of well known
and well defined quantities in dynamical systems. As possible applications of
our bounds, we study the relationship between synchronisation and the exchange
of information in a system of two coupled maps and in experimental networks of
coupled oscillators
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