1,200 research outputs found
Statistical-mechanical formulation of Lyapunov exponents
We show how the Lyapunov exponents of a dynamic system can in general be
expressed in terms of the free energy of a (non-Hermitian) quantum many-body
problem. This puts their study as a problem of statistical mechanics, whose
intuitive concepts and techniques of approximation can hence be borrowed.Comment: 10 pages, 3 figures, RevTex
LP-VIcode: a program to compute a suite of variational chaos indicators
An important point in analysing the dynamics of a given stellar or planetary
system is the reliable identification of the chaotic or regular behaviour of
its orbits. We introduce here the program LP-VIcode, a fully operational code
which efficiently computes a suite of ten variational chaos indicators for
dynamical systems in any number of dimensions. The user may choose to
simultaneously compute any number of chaos indicators among the following: the
Lyapunov Exponents, the Mean Exponential Growth factor of Nearby Orbits, the
Slope Estimation of the largest Lyapunov Characteristic Exponent, the Smaller
ALignment Index, the Generalized ALignment Index, the Fast Lyapunov Indicator,
the Othogonal Fast Lyapunov Indicator, the dynamical Spectra of Stretching
Numbers, the Spectral Distance, and the Relative Lyapunov Indicator. They are
combined in an efficient way, allowing the sharing of differential equations
whenever this is possible, and the individual stopping of their computation
when any of them saturates.Comment: 26 pages, 9 black-and-white figures. Accepted for publication in
Astronomy and Computing (Elsevier
Lyapunov Exponents of Two Stochastic Lorenz 63 Systems
Two different types of perturbations of the Lorenz 63 dynamical system for
Rayleigh-Benard convection by multiplicative noise -- called stochastic
advection by Lie transport (SALT) noise and fluctuation-dissipation (FD) noise
-- are found to produce qualitatively different effects, possibly because the
total phase-space volume contraction rates are different. In the process of
making this comparison between effects of SALT and FD noise on the Lorenz 63
system, a stochastic version of a robust deterministic numerical algorithm for
obtaining the individual numerical Lyapunov exponents was developed. With this
stochastic version of the algorithm, the value of the sum of the Lyapunov
exponents for the FD noise was found to differ significantly from the value of
the deterministic Lorenz 63 system, whereas the SALT noise preserves the Lorenz
63 value with high accuracy. The Lagrangian averaged version of the SALT
equations (LA SALT) is found to yield a closed deterministic subsystem for the
expected solutions which is found to be isomorphic to the original Lorenz 63
dynamical system. The solutions of the closed chaotic subsystem, in turn, drive
a linear stochastic system for the fluctuations of the LA SALT solutions around
their expected values.Comment: 19 pages, 4 figures, comments always welcome
Do Finite-Size Lyapunov Exponents Detect Coherent Structures?
Ridges of the Finite-Size Lyapunov Exponent (FSLE) field have been used as
indicators of hyperbolic Lagrangian Coherent Structures (LCSs). A rigorous
mathematical link between the FSLE and LCSs, however, has been missing. Here we
prove that an FSLE ridge satisfying certain conditions does signal a nearby
ridge of some Finite-Time Lyapunov Exponent (FTLE) field, which in turn
indicates a hyperbolic LCS under further conditions. Other FSLE ridges
violating our conditions, however, are seen to be false positives for LCSs. We
also find further limitations of the FSLE in Lagrangian coherence detection,
including ill-posedness, artificial jump-discontinuities, and sensitivity with
respect to the computational time step.Comment: 22 pages, 7 figures, v3: corrects the z-axis labels of Fig. 2 (left)
that appears in the version published in Chao
Achieving synchronization in arrays of coupled differential systems with time-varying couplings
In this paper, we study complete synchronization of the complex dynamical
networks described by linearly coupled ordinary differential equation systems
(LCODEs). The coupling considered here is time-varying in both the network
structure and the reaction dynamics. Inspired by our previous paper [6], the
extended Hajnal diameter is introduced and used to measure the synchronization
in a general differential system. Then we find that the Hajnal diameter of the
linear system induced by the time-varying coupling matrix and the largest
Lyapunov exponent of the synchronized system play the key roles in
synchronization analysis of LCODEs with the identity inner coupling matrix. As
an application, we obtain a general sufficient condition guaranteeing directed
time-varying graph to reach consensus. Example with numerical simulation is
provided to show the effectiveness the theoretical results.Comment: 22 pages, 4 figure
Analysis of Round Off Errors with Reversibility Test as a Dynamical Indicator
We compare the divergence of orbits and the reversibility error for discrete
time dynamical systems. These two quantities are used to explore the behavior
of the global error induced by round off in the computation of orbits. The
similarity of results found for any system we have analysed suggests the use of
the reversibility error, whose computation is straightforward since it does not
require the knowledge of the exact orbit, as a dynamical indicator. The
statistics of fluctuations induced by round off for an ensemble of initial
conditions has been compared with the results obtained in the case of random
perturbations. Significant differences are observed in the case of regular
orbits due to the correlations of round off error, whereas the results obtained
for the chaotic case are nearly the same. Both the reversibility error and the
orbit divergence computed for the same number of iterations on the whole phase
space provide an insight on the local dynamical properties with a detail
comparable with other dynamical indicators based on variational methods such as
the finite time maximum Lyapunov characteristic exponent, the mean exponential
growth factor of nearby orbits and the smaller alignment index. For 2D
symplectic maps the differentiation between regular and chaotic regions is well
full-filled. For 4D symplectic maps the structure of the resonance web as well
as the nearby weakly chaotic regions are accurately described.Comment: International Journal of Bifurcation and Chaos, 201
Stability Properties of 1-Dimensional Hamiltonian Lattices with Non-analytic Potentials
We investigate the local and global dynamics of two 1-Dimensional (1D)
Hamiltonian lattices whose inter-particle forces are derived from non-analytic
potentials. In particular, we study the dynamics of a model governed by a
"graphene-type" force law and one inspired by Hollomon's law describing
"work-hardening" effects in certain elastic materials. Our main aim is to show
that, although similarities with the analytic case exist, some of the local and
global stability properties of non-analytic potentials are very different than
those encountered in systems with polynomial interactions, as in the case of 1D
Fermi-Pasta-Ulam-Tsingou (FPUT) lattices. Our approach is to study the motion
in the neighborhood of simple periodic orbits representing continuations of
normal modes of the corresponding linear system, as the number of particles
and the total energy are increased. We find that the graphene-type model is
remarkably stable up to escape energy levels where breakdown is expected, while
the Hollomon lattice never breaks, yet is unstable at low energies and only
attains stability at energies where the harmonic force becomes dominant. We
suggest that, since our results hold for large , it would be interesting to
study analogous phenomena in the continuum limit where 1D lattices become
strings.Comment: Accepted for publication in the International Journal of Bifurcation
and Chao
Detecting chaos, determining the dimensions of tori and predicting slow diffusion in Fermi--Pasta--Ulam lattices by the Generalized Alignment Index method
The recently introduced GALI method is used for rapidly detecting chaos,
determining the dimensionality of regular motion and predicting slow diffusion
in multi--dimensional Hamiltonian systems. We propose an efficient computation
of the GALI indices, which represent volume elements of randomly chosen
deviation vectors from a given orbit, based on the Singular Value Decomposition
(SVD) algorithm. We obtain theoretically and verify numerically asymptotic
estimates of GALIs long--time behavior in the case of regular orbits lying on
low--dimensional tori. The GALI indices are applied to rapidly detect
chaotic oscillations, identify low--dimensional tori of Fermi--Pasta--Ulam
(FPU) lattices at low energies and predict weak diffusion away from
quasiperiodic motion, long before it is actually observed in the oscillations.Comment: 10 pages, 5 figures, submitted for publication in European Physical
Journal - Special Topics. Revised version: Small explanatory additions to the
text and addition of some references. A small figure chang
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