2,507 research outputs found
Predictability: a way to characterize Complexity
Different aspects of the predictability problem in dynamical systems are
reviewed. The deep relation among Lyapunov exponents, Kolmogorov-Sinai entropy,
Shannon entropy and algorithmic complexity is discussed. In particular, we
emphasize how a characterization of the unpredictability of a system gives a
measure of its complexity. Adopting this point of view, we review some
developments in the characterization of the predictability of systems showing
different kind of complexity: from low-dimensional systems to high-dimensional
ones with spatio-temporal chaos and to fully developed turbulence. A special
attention is devoted to finite-time and finite-resolution effects on
predictability, which can be accounted with suitable generalization of the
standard indicators. The problems involved in systems with intrinsic randomness
is discussed, with emphasis on the important problems of distinguishing chaos
from noise and of modeling the system. The characterization of irregular
behavior in systems with discrete phase space is also considered.Comment: 142 Latex pgs. 41 included eps figures, submitted to Physics Reports.
Related information at this http://axtnt2.phys.uniroma1.i
Measuring logarithmic corrections to normal diffusion in infinite-horizon billiards
We perform numerical measurements of the moments of the position of a tracer
particle in a two-dimensional periodic billiard model (Lorentz gas) with
infinite corridors. This model is known to exhibit a weak form of
super-diffusion, in the sense that there is a logarithmic correction to the
linear growth in time of the mean-squared displacement. We show numerically
that this expected asymptotic behavior is easily overwhelmed by the subleading
linear growth throughout the time-range accessible to numerical simulations. We
compare our simulations to the known analytical results for the variance of the
anomalously-rescaled limiting normal distributions.Comment: 10 pages, 4 figure
Oscillatory Finite-Time Singularities in Finance, Population and Rupture
We present a simple two-dimensional dynamical system where two nonlinear
terms, exerting respectively positive feedback and reversal, compete to create
a singularity in finite time decorated by accelerating oscillations. The power
law singularity results from the increasing growth rate. The oscillations
result from the restoring mechanism. As a function of the order of the
nonlinearity of the growth rate and of the restoring term, a rich variety of
behavior is documented analytically and numerically. The dynamical behavior is
traced back fundamentally to the self-similar spiral structure of trajectories
in phase space unfolding around an unstable spiral point at the origin. The
interplay between the restoring mechanism and the nonlinear growth rate leads
to approximately log-periodic oscillations with remarkable scaling properties.
Three domains of applications are discussed: (1) the stock market with a
competition between nonlinear trend-followers and nonlinear value investors;
(2) the world human population with a competition between a
population-dependent growth rate and a nonlinear dependence on a finite
carrying capacity; (3) the failure of a material subjected to a time-varying
stress with a competition between positive geometrical feedback on the damage
variable and nonlinear healing.Comment: Latex document of 59 pages including 20 eps figure
About the role of chaos and coarse graining in Statistical Mechanics
We discuss the role of ergodicity and chaos for the validity of statistical
laws. In particular we explore the basic aspects of chaotic systems (with
emphasis on the finite-resolution) on systems composed of a huge number of
particles.Comment: Summer school `Fundamental Problems in Statistical Physics' (Leuven,
Belgium), June 16-29, 2013. To be published in Physica
Applied Koopman Operator Theory for Power Systems Technology
Koopman operator is a composition operator defined for a dynamical system
described by nonlinear differential or difference equation. Although the
original system is nonlinear and evolves on a finite-dimensional state space,
the Koopman operator itself is linear but infinite-dimensional (evolves on a
function space). This linear operator captures the full information of the
dynamics described by the original nonlinear system. In particular, spectral
properties of the Koopman operator play a crucial role in analyzing the
original system. In the first part of this paper, we review the so-called
Koopman operator theory for nonlinear dynamical systems, with emphasis on modal
decomposition and computation that are direct to wide applications. Then, in
the second part, we present a series of applications of the Koopman operator
theory to power systems technology. The applications are established as
data-centric methods, namely, how to use massive quantities of data obtained
numerically and experimentally, through spectral analysis of the Koopman
operator: coherency identification of swings in coupled synchronous generators,
precursor diagnostic of instabilities in the coupled swing dynamics, and
stability assessment of power systems without any use of mathematical models.
Future problems of this research direction are identified in the last
concluding part of this paper.Comment: 31 pages, 11 figure
A Method to Study Relaxation of Metastable Phases: Macroscopic Mean-Field Dynamics
We propose two different macroscopic dynamics to describe the decay of
metastable phases in many-particle systems with local interactions. These
dynamics depend on the macroscopic order parameter through the restricted
free energy and are designed to give the correct equilibrium
distribution for . The connection between macroscopic dynamics and the
underlying microscopic dynamic are considered in the context of a projection-
operator formalism. Application to the square-lattice nearest-neighbor Ising
ferromagnet gives good agreement with droplet theory and Monte Carlo
simulations of the underlying microscopic dynamic. This includes quantitative
agreement for the exponential dependence of the lifetime on the inverse of the
applied field , and the observation of distinct field regions in which the
derivative of the lifetime with respect to depends differently on . In
addition, at very low temperatures we observe oscillatory behavior of this
derivative with respect to , due to the discreteness of the lattice and in
agreement with rigorous results. Similarities and differences between this work
and earlier works on finite Ising models in the fixed-magnetization ensemble
are discussed.Comment: 44 pages RevTeX3, 11 uuencoded Postscript figs. in separate file
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