Journal ArticleDOI
Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory
TLDR
In this paper, a method for computing all of the Lyapunov characteristic exponents of order greater than one is presented, which is related to the increase of volumes of a dynamical system.Abstract:
Since several years Lyapunov Characteristic Exponents are of interest in the study of dynamical systems in order to characterize quantitatively their stochasticity properties, related essentially to the exponential divergence of nearby orbits. One has thus the problem of the explicit computation of such exponents, which has been solved only for the maximal of them. Here we give a method for computing all of them, based on the computation of the exponents of order greater than one, which are related to the increase of volumes. To this end a theorem is given relating the exponents of order one to those of greater order. The numerical method and some applications will be given in a forthcoming paper.read more
Citations
More filters
Journal ArticleDOI
Determining Lyapunov exponents from a time series
TL;DR: In this article, the authors present the first algorithms that allow the estimation of non-negative Lyapunov exponents from an experimental time series, which provide a qualitative and quantitative characterization of dynamical behavior.
Journal ArticleDOI
Heterogeneous beliefs and routes to chaos in a simple asset pricing model
William A. Brock,Cars Hommes +1 more
TL;DR: In this paper, the authors investigate the dynamics in a simple present discounted value asset pricing model with heterogeneous beliefs, where agents choose from a finite set of predictors of future prices of a risky asset and revise their "beliefs" in each period in a boundedly rational way, according to a fitness measure such as past realized profits.
Journal ArticleDOI
The analysis of observed chaotic data in physical systems
TL;DR: Chaotic time series data are observed routinely in experiments on physical systems and in observations in the field as mentioned in this paper, and many tools have been developed for the analysis of such data.
Journal ArticleDOI
Ensemble Forecasting at NCEP and the Breeding Method
Zoltan Toth,Eugenia Kalnay +1 more
TL;DR: In this paper, it is shown that the analysis cycle is like a breeding cycle: it acts as a nonlinear perturbation model upon the evolution of the real atmosphere, and the perturbations (i.e., the analysis error), carried forward in the first-guess forecasts, is scaled down at regular intervals by the use of observations.
Journal ArticleDOI
The dimension of chaotic attractors
TL;DR: In this paper, the authors discuss a variety of different definitions of dimension, compute their values for a typical example, and review previous work on the dimension of chaotic attractors, and conclude that dimension of the natural measure is more important than the fractal dimension.
References
More filters
Book
Mathematical Methods of Classical Mechanics
TL;DR: In this paper, Newtonian mechanics: experimental facts investigation of the equations of motion, variational principles Lagrangian mechanics on manifolds oscillations rigid bodies, differential forms symplectic manifolds canonical formalism introduction to pertubation theory.
Journal ArticleDOI
Characteristic lyapunov exponents and smooth ergodic theory
TL;DR: In this article, the authors define the ergodicity of a diffeomorphism with non-zero exponents on a set of positive measure and the Bernoullian property of geodesic flows on closed Riemannian manifolds.
Book
Lectures on Differential Geometry
TL;DR: In this article, the authors present an algebraic model of transitive differential geometry and the integrability problem for geometrical structures on manifolds, which they call integral calculus on manifold.
Journal ArticleDOI
Kolmogorov entropy and numerical experiments
TL;DR: In this paper, a numerical study of the Kolmogorov entropy for the H\'enon-Heiles model is presented, based on mathematical results of Oseledec and Piesin.