556 research outputs found
Information-Geometric Indicators of Chaos in Gaussian Models on Statistical Manifolds of Negative Ricci Curvature
A new information-geometric approach to chaotic dynamics on curved
statistical manifolds based on Entropic Dynamics (ED) is proposed. It is shown
that the hyperbolicity of a non-maximally symmetric 6N-dimensional statistical
manifold M_{s} underlying an ED Gaussian model describing an arbitrary system
of 3N degrees of freedom leads to linear information-geometric entropy growth
and to exponential divergence of the Jacobi vector field intensity, quantum and
classical features of chaos respectively.Comment: 8 pages, final version accepted for publicatio
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
Covariant Lyapunov Vectors and Finite-Time Normal Modes for Geophysical Fluid Dynamical Systems
Dynamical vectors characterizing instability and applicable as ensemble
perturbations for prediction with geophysical fluid dynamical models are
analysed. The relationships between covariant Lyapunov vectors (CLVs),
orthonormal Lyapunov vectors (OLVs), singular vectors (SVs), Floquet vectors
and finite-time normal modes (FTNMs) are examined for periodic and aperiodic
systems. In the phase-space of FTNM coefficients, SVs are found to equate with
unit norm FTNMs at certain times. In the long-time limit, when SVs approach
OLVs, the Oseledec theorem and the relationships between OLVs and CLVs are used
to connect CLVs to FTNMs in this phase-space. The covariant properties of both
the CLVs, and the FTNMs, together with their phase-space independence, and the
norm independence of global Lyapunov exponents and FTNM growth rates,
establishes their asymptotic convergence. Conditions on the dynamical systems
for the validity of these results, particularly ergodicity, boundedness and
non-singular FTNM characteristic matrix and propagator, are documented. Systems
with nondegenerate OLVs, and with degenerate Lyapunov spectrum as is the rule
in the presence of waves such as Rossby waves, are examined, and efficient
numerical methods for the calculation of leading CLVs proposed. Norm
independent finite-time versions of the Kolmogorov-Sinai entropy production and
Kaplan-Yorke dimension are presented.Comment: 38 pages, no figure
Dynamical analysis of blocking events: spatial and temporal fluctuations of covariant Lyapunov vectors
One of the most relevant weather regimes in the midlatitude atmosphere is the persistent deviation from the approximately zonally symmetric jet stream leading to the emergence of so-called blocking patterns. Such configurations are usually connected to exceptional local stability properties of the flow which come along with an improved local forecast skills during the phenomenon. It is instead extremely hard to predict onset and decay of blockings. Covariant Lyapunov Vectors (CLVs) offer a suitable characterization of the linear stability of a chaotic flow, since they represent the full tangent linear dynamics by a covariant basis which explores linear perturbations at all time scales. Therefore, we assess whether CLVs feature a signature of the blockings. As a first step, we examine the CLVs for a quasi-geostrophic beta-plane two-layer model in a periodic channel baroclinically driven by a meridional temperature gradient ΔT. An orographic forcing enhances the emergence of localized blocked regimes. We detect the blocking events of the channel flow with a Tibaldi-Molteni scheme adapted to the periodic channel. When blocking occurs, the global growth rates of the fastest growing CLVs are significantly higher. Hence, against intuition, the circulation is globally more unstable in blocked phases. Such an increase in the finite time Lyapunov exponents with respect to the long term average is attributed to stronger barotropic and baroclinic conversion in the case of high temperature gradients, while for low values of ΔT, the effect is only due to stronger barotropic instability. In order to determine the localization of the CLVs we compare the meridionally averaged variance of the CLVs during blocked and unblocked phases. We find that on average the variance of the CLVs is clustered around the center of blocking. These results show that the blocked flow affects all time scales and processes described by the CLVs
Complexity Characterization in a Probabilistic Approach to Dynamical Systems Through Information Geometry and Inductive Inference
Information geometric techniques and inductive inference methods hold great
promise for solving computational problems of interest in classical and quantum
physics, especially with regard to complexity characterization of dynamical
systems in terms of their probabilistic description on curved statistical
manifolds. In this article, we investigate the possibility of describing the
macroscopic behavior of complex systems in terms of the underlying statistical
structure of their microscopic degrees of freedom by use of statistical
inductive inference and information geometry. We review the Maximum Relative
Entropy (MrE) formalism and the theoretical structure of the information
geometrodynamical approach to chaos (IGAC) on statistical manifolds. Special
focus is devoted to the description of the roles played by the sectional
curvature, the Jacobi field intensity and the information geometrodynamical
entropy (IGE). These quantities serve as powerful information geometric
complexity measures of information-constrained dynamics associated with
arbitrary chaotic and regular systems defined on the statistical manifold.
Finally, the application of such information geometric techniques to several
theoretical models are presented.Comment: 29 page
On the predictability of time series by metric entropy
Thesis (Master)--Izmir Institute of Technology, Mechanical Engineering, Izmir, 2006Includes bibliographical references (leaves: 48-49)Text in English; Abstract: Turkish and Englishxi, 55 leavesThe computation of the metric entropy, a measure of the loss of information along the attractor, from experimental time series is the main objective of this study. In this study, replacing the current warning systems (simple threshold based, on/off circuits), a new and promising prognosis system is tried to be achieved by the metric entropy, i.e. Kolmogorov . Sinai entropy, from chaotic time series. Additional to metric entropy, correlation dimension and time series statistical parameters were investigated.Condition monitoring of ball bearings and drill bits was achieved in the light of practical considerations of time series applications. Two different accelerated bearing run-to-failure test rigs were constructed and the prediction tests were performed.However, as a reason of shaft failure in both structures during the experiments, none of them is completed. Finally, drill bit breakage experiments were carried out. In the experiments, 10 small drill bits (1 mm ) were tested until they broke down, while vibration data were consecutively taken in equal time intervals. After the analysis, a consistent decrement in variation of metric entropy just before the breakage was observed. As a result of the experiment results, metric entropy variation could be proposed as an early warning system
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