1,280 research outputs found
Violation of hyperbolicity via unstable dimension variability in a chain with local hyperbolic chaotic attractors
We consider a chain of oscillators with hyperbolic chaos coupled via
diffusion. When the coupling is strong the chain is synchronized and
demonstrates hyperbolic chaos so that there is one positive Lyapunov exponent.
With the decay of the coupling the second and the third Lyapunov exponents
approach zero simultaneously. The second one becomes positive, while the third
one remains close to zero. Its finite-time numerical approximation fluctuates
changing the sign within a wide range of the coupling parameter. These
fluctuations arise due to the unstable dimension variability which is known to
be the source for non-hyperbolicity. We provide a detailed study of this
transition using the methods of Lyapunov analysis.Comment: 24 pages, 13 figure
Strange Nonchaotic Attractors
Aperiodic dynamics which is nonchaotic is realized on Strange Nonchaotic
attractors (SNAs). Such attractors are generic in quasiperiodically driven
nonlinear systems, and like strange attractors, are geometrically fractal. The
largest Lyapunov exponent is zero or negative: trajectories do not show
exponential sensitivity to initial conditions. In recent years, SNAs have been
seen in a number of diverse experimental situations ranging from
quasiperiodically driven mechanical or electronic systems to plasma discharges.
An important connection is the equivalence between a quasiperiodically driven
system and the Schr\"odinger equation for a particle in a related quasiperiodic
potential, giving a correspondence between the localized states of the quantum
problem with SNAs in the related dynamical system. In this review we discuss
the main conceptual issues in the study of SNAs, including the different
bifurcations or routes for the creation of such attractors, the methods of
characterization, and the nature of dynamical transitions in quasiperiodically
forced systems. The variation of the Lyapunov exponent, and the qualitative and
quantitative aspects of its local fluctuation properties, has emerged as an
important means of studying fractal attractors, and this analysis finds useful
application here. The ubiquity of such attractors, in conjunction with their
several unusual properties, suggest novel applications.Comment: 34 pages, 9 figures(5 figures are in ps format and four figures are
in gif format
Theory and computation of covariant Lyapunov vectors
Lyapunov exponents are well-known characteristic numbers that describe growth
rates of perturbations applied to a trajectory of a dynamical system in
different state space directions. Covariant (or characteristic) Lyapunov
vectors indicate these directions. Though the concept of these vectors has been
known for a long time, they became practically computable only recently due to
algorithms suggested by Ginelli et al. [Phys. Rev. Lett. 99, 2007, 130601] and
by Wolfe and Samelson [Tellus 59A, 2007, 355]. In view of the great interest in
covariant Lyapunov vectors and their wide range of potential applications, in
this article we summarize the available information related to Lyapunov vectors
and provide a detailed explanation of both the theoretical basics and numerical
algorithms. We introduce the notion of adjoint covariant Lyapunov vectors. The
angles between these vectors and the original covariant vectors are
norm-independent and can be considered as characteristic numbers. Moreover, we
present and study in detail an improved approach for computing covariant
Lyapunov vectors. Also we describe, how one can test for hyperbolicity of
chaotic dynamics without explicitly computing covariant vectors.Comment: 21 pages, 5 figure
Chaotic saddles in nonlinear modulational interactions in a plasma
A nonlinear model of modulational processes in the subsonic regime involving
a linearly unstable wave and two linearly damped waves with different damping
rates in a plasma is studied numerically. We compute the maximum Lyapunov
exponent as a function of the damping rates in a two-parameter space, and
identify shrimp-shaped self-similar structures in the parameter space. By
varying the damping rate of the low-frequency wave, we construct bifurcation
diagrams and focus on a saddle-node bifurcation and an interior crisis
associated with a periodic window. We detect chaotic saddles and their stable
and unstable manifolds, and demonstrate how the connection between two chaotic
saddles via coupling unstable periodic orbits can result in a crisis-induced
intermittency. The relevance of this work for the understanding of modulational
processes observed in plasmas and fluids is discussed.Comment: Physics of Plasmas, in pres
Covariant Lyapunov vectors
The recent years have witnessed a growing interest for covariant Lyapunov
vectors (CLVs) which span local intrinsic directions in the phase space of
chaotic systems. Here we review the basic results of ergodic theory, with a
specific reference to the implications of Oseledets' theorem for the properties
of the CLVs. We then present a detailed description of a "dynamical" algorithm
to compute the CLVs and show that it generically converges exponentially in
time. We also discuss its numerical performance and compare it with other
algorithms presented in literature. We finally illustrate how CLVs can be used
to quantify deviations from hyperbolicity with reference to a dissipative
system (a chain of H\'enon maps) and a Hamiltonian model (a Fermi-Pasta-Ulam
chain)
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
Linear And Nonlinear Arabesques: A Study Of Closed Chains Of Negative 2-Element Circuits
In this paper we consider a family of dynamical systems that we call
"arabesques", defined as closed chains of 2-element negative circuits. An
-dimensional arabesque system has 2-element circuits, but in addition,
it displays by construction, two -element circuits which are both positive
vs one positive and one negative, depending on the parity (even or odd) of the
dimension . In view of the absence of diagonal terms in their Jacobian
matrices, all these dynamical systems are conservative and consequently, they
can not possess any attractor. First, we analyze a linear variant of them which
we call "arabesque 0" or for short "A0". For increasing dimensions, the
trajectories are increasingly complex open tori. Next, we inserted a single
cubic nonlinearity that does not affect the signs of its circuits (that we call
"arabesque 1" or for short "A1"). These systems have three steady states,
whatever the dimension is, in agreement with the order of the nonlinearity. All
three are unstable, as there can not be any attractor in their state-space. The
3D variant (that we call for short "A1\_3D") has been analyzed in some detail
and found to display a complex mixed set of quasi-periodic and chaotic
trajectories. Inserting cubic nonlinearities (one per equation) in the same
way as above, we generate systems "A2\_D". A2\_3D behaves essentially as
A1\_3D, in agreement with the fact that the signs of the circuits remain
identical. A2\_4D, as well as other arabesque systems with even dimension, has
two positive -circuits and nine steady states. Finally, we investigate and
compare the complex dynamics of this family of systems in terms of their
symmetries.Comment: 22 pages, 12 figures, accepted for publication at Int. J. Bif. Chao
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