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Determining Lyapunov Exponents from a Time Series
 Physica
, 1985
"... We present the first algorithms that allow the estimation of nonnegative Lyapunov exponents from an experimental time series. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to the exponentially fast divergence or convergence of n ..."
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Cited by 495 (1 self)
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We present the first algorithms that allow the estimation of nonnegative Lyapunov exponents from an experimental time series. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to the exponentially fast divergence or convergence of nearby orbits in phase space. A system with one or more positive Lyapunov exponents is defined to be chaotic. Our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems: we monitor the longterm growth rate of small volume elements in an attractor. The method is tested on model systems with known Lyapunov spectra, and applied to data for the BelousovZhabotinskii reaction and CouetteTaylor flow. Contents 1.
Ensemble forecasting at NCEP and the breeding method
 Mon. Wea. Rev
, 1997
"... The breeding method has been used to generate perturbations for ensemble forecasting at the National Centers for Environmental Prediction (formerly known as the National Meteorological Center) since December 1992. At that time a single breeding cycle with a pair of bred forecasts was implemented. In ..."
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Cited by 196 (15 self)
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The breeding method has been used to generate perturbations for ensemble forecasting at the National Centers for Environmental Prediction (formerly known as the National Meteorological Center) since December 1992. At that time a single breeding cycle with a pair of bred forecasts was implemented. In March 1994, the ensemble was expanded to seven independent breeding cycles on the Cray C90 supercomputer, and the forecasts were extended to 16 days. This provides 17 independent global forecasts valid for two weeks every day. For efficient ensemble forecasting, the initial perturbations to the control analysis should adequately sample the space of possible analysis errors. 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. The perturbation (i.e., the analysis error), carried forward in the firstguess forecasts, is ‘‘scaled down’ ’ at regular intervals by the use of observations. Because of this, growing errors associated with the evolving state of the atmosphere develop within the analysis cycle and dominate subsequent forecast error growth. The breeding method simulates the development of growing errors in the analysis cycle. A difference field between two nonlinear forecasts is carried forward (and scaled down at regular intervals) upon the evolving atmospheric analysis fields. By construction, the bred vectors are superpositions of the leading local (timedependent)
A practical method for calculating largest Lyapunov exponents from small data sets
 PHYSICA D
, 1993
"... Detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest Lyapunov exponent. Lyapunov exponents quantify the exponential divergence of initially close statespace trajectories and estimate the amount of chaos in a system. We present a new m ..."
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Cited by 181 (0 self)
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Detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest Lyapunov exponent. Lyapunov exponents quantify the exponential divergence of initially close statespace trajectories and estimate the amount of chaos in a system. We present a new method for calculating the largest Lyapunov exponent from an experimental time series. The method follows directly from the definition of the largest Lyapunov exponent and is accurate because it takes advantage of all the available data. We show that the algorithm is fast, easy to implement, and robust to changes in the following quantities: embedding dimension, size of data set, reconstruction delay, and noise level. Furthermore, one may use the algorithm to calculate simultaneously the correlation dimension. Thus, one sequence of computations will yield an estimate of both the level of chaos and the system complexity.
Equations of motion from a data series
 Complex Systems
, 1987
"... Abstract. Temporal pattern learning, control and prediction, and chaotic data analysis share a common problem: deducing optimal equations of motion from observations of timedependent behavior. Each desires to obtain models of the physical world from limited information. We describe a method to reco ..."
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Cited by 58 (15 self)
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Abstract. Temporal pattern learning, control and prediction, and chaotic data analysis share a common problem: deducing optimal equations of motion from observations of timedependent behavior. Each desires to obtain models of the physical world from limited information. We describe a method to reconstruct the deterministic portion of the equations of motion directly from a data series. These equations of motion represent a vast reduction of a chaotic data set’s observed complexity to a compact, algorithmic specification. This approach employs an informational measure of model optimality to guide searching through the space of dynamical systems. As corollary results, we indicate how to estimate the minimum embedding dimension, extrinsic noise level, metric entropy, and Lyapunov spectrum. Numerical and experimental applications demonstrate the method’s feasibility and limitations. Extensions to estimating parametrized families of dynamical systems from bifurcation data and to spatial pattern evolution are presented. Applications to predicting chaotic data and the design of forecasting, learning, and control systems, are discussed. 1.
On the computation of Lyapunov exponents for continuous dynamical systems
 SIAM J. Numer. Anal
, 1997
"... ..."
Computation Of A Few Lyapunov Exponents For Continuous And Discrete Dynamical Systems
 Appl. Numer. Math
, 1995
"... . In this paper, an error analysis of QR based methods for computing the first few Lyapunov exponents of continuous and discrete dynamical systems is given. Algorithmic developments are discussed. Implementation details, error estimators and testing are also given. 1. INTRODUCTION In this paper, w ..."
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Cited by 33 (9 self)
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. In this paper, an error analysis of QR based methods for computing the first few Lyapunov exponents of continuous and discrete dynamical systems is given. Algorithmic developments are discussed. Implementation details, error estimators and testing are also given. 1. INTRODUCTION In this paper, we consider the computation of a few Lyapunov exponents for continuous and discrete finite dimensional dynamical systems. In [DRV2], we considered approximating all of the exponents of continuous dynamical systems, and gave error analysis and convergence results, as well as algorithmic details, for the class of QR based methods. Here we consider the case in which only the first few exponents are desired. As it turns out, the extension to this case is not automatic, and new theoretical and algorithmic aspects need to be addressed. We still focus on continuous and discrete QR methods, give convergence results, algorithmic details and testing. For continuous dynamical systems, a proper implement...
The Lyapunov Characteristic Exponents and their
 Computation, Lect. Notes Phys
, 2010
"... For want of a nail the shoe was lost. For want of a shoe the horse was lost. For want of a horse the rider was lost. For want of a rider the battle was lost. For want of a battle the kingdom was lost. And all for the want of a horseshoe nail. For Want of a Nail (proverbial rhyme) Summary. We present ..."
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Cited by 29 (2 self)
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For want of a nail the shoe was lost. For want of a shoe the horse was lost. For want of a horse the rider was lost. For want of a rider the battle was lost. For want of a battle the kingdom was lost. And all for the want of a horseshoe nail. For Want of a Nail (proverbial rhyme) Summary. We present a survey of the theory of the Lyapunov Characteristic Exponents (LCEs) for dynamical systems, as well as of the numerical techniques developed for the computation of the maximal, of few and of all of them. After some historical notes on the first attempts for the numerical evaluation of LCEs, we discuss in detail the multiplicative ergodic theorem of Oseledec [99], which provides the theoretical basis for the computation of the LCEs. Then, we analyze the algorithm for the computation of the maximal LCE, whose value has been extensively used as an indicator of chaos, and the algorithm of the so–called ‘standard method’, developed by Benettin et al. [14], for the computation of many LCEs. We also consider different discrete and continuous methods for computing the LCEs based on the QR or the singular value decomposition techniques. Although, we are mainly interested in finite–dimensional conservative systems, i. e. autonomous Hamiltonian systems and symplectic maps, we also briefly refer to the evaluation of LCEs of dissipative systems and time series. The relation of two chaos detection techniques, namely the fast Lyapunov indicator (FLI) and the generalized alignment index (GALI), to the computation of the LCEs is also discussed. 1
Symbolic dynamics of onedimensional maps: Entropies, finite precision, and noise
, 1982
"... In the study of nonlinear physical systems, one encounters apparently random or chaotic behavior, although the systems may be completely deterministic. Applying techniques from symbolic dynamics to maps of the interval, we compute two measures of chaotic behavior commonly employed in dynamical syste ..."
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Cited by 17 (7 self)
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In the study of nonlinear physical systems, one encounters apparently random or chaotic behavior, although the systems may be completely deterministic. Applying techniques from symbolic dynamics to maps of the interval, we compute two measures of chaotic behavior commonly employed in dynamical systems theory: the topological and metric entropies. For the quadratic logistic equation, we find that the metric entropy converges very slowly in comparison to maps which are strictly hyperbolic. The effects of finite precision arithmetic and external noise on chaotic behavior are characterized with the symbolic dynamics entropies. Finally, we discuss the relationship of these measures of chaos to algorithmic complexity, and use algorithmic information theory as a framework to discuss the construction of models for chaotic dynamics.
An Efficient QR Based Method for the Computation of Lyapunov Exponents
, 1997
"... An efficient and numerically stable method to determine all the Lyapunov characteristic exponents of a dynamical system is developed. Numerical experiments are presented highlighting some aspects of convergence, accuracy and efficiency in the computation of the Lyapunov characteristic exponents. ..."
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Cited by 12 (2 self)
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An efficient and numerically stable method to determine all the Lyapunov characteristic exponents of a dynamical system is developed. Numerical experiments are presented highlighting some aspects of convergence, accuracy and efficiency in the computation of the Lyapunov characteristic exponents.
Lyapunov Exponents From Random Fibonacci Sequences To The Lorenz Equations
 Department of Computer Science, Cornell University
, 1998
"... this paper (Mathematical Reviews:29 #648) with the words "This is a profound memoir." 9 will show in Chapter 3, there are simple algorithms for bounding the Lyapunov exponents in this setting. The advanced state of the theory for random matrix products is a peculiar situation because dete ..."
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Cited by 11 (1 self)
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this paper (Mathematical Reviews:29 #648) with the words "This is a profound memoir." 9 will show in Chapter 3, there are simple algorithms for bounding the Lyapunov exponents in this setting. The advanced state of the theory for random matrix products is a peculiar situation because deterministic matrix products that govern sensitive dependence on initial conditions are barely understood; it is as if the strong law of large numbers were well understood without a satisfactory theory of convergence of infinite series. The elements of the theory of random matrix products are carefully explained in the beautiful monograph by Bougerol [16]. The basic result about Lyapunov exponents, lim