A practical method for calculating largest Lyapunov exponents from small data sets
Physica D Nonlinear PhenomenaPublished 1 May 1993
Michael T. Rosenstein, James J. Collins, Carlo J. De Luca
Citations3,338
SJR quartileQ1
SJR score0.94
SNIP1.39
Generate an AI Snapshot to get a quick, structured summary of this paper.
Study Snapshot
ObjectiveStudy objective
MethodsResearch methodology
PopulationPopulation studied
Sample sizeSample sizes
OutcomesStudy outcomes here
ResultsStudy results comes here
LimitationsResearch study limitations comes here
A concise AI-generated summary of the paper will appear here once you click Generate AI Snapshot.
TL;DR
A new method for calculating the largest Lyapunov exponent from an experimental time series is presented that is fast, easy to implement, and robust to changes in the following quantities: embedding dimension, size of data set, reconstruction delay, and noise level.
Abstract
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 state-space 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.
Keywords
Computer ScienceEconomics, Econometrics and FinancePhysics and Astronomy
Journal of the Atmospheric SciencesDeterministic Nonperiodic Flow
19,337 Citations1963Edward N. Lorenz
Physica D Nonlinear PhenomenaDetermining Lyapunov exponents from a time series
9,368 Citations1985Alan Wolf, J. B. Swift +2 more
Physical Review LettersCharacterization of Strange Attractors
4,842 Citations1983Peter Grassberger, Itamar Procaccia
Reviews of Modern PhysicsErgodic theory of chaos and strange attractors
4,836 Citations1985Jean‐Pierre Eckmann, David Ruelle
Physical review. A, General physicsIndependent coordinates for strange attractors from mutual information
4,579 Citations1986Andrew M. Fraser, Harry L. Swinney
ScienceOscillation and Chaos in Physiological Control Systems
4,109 Citations1977Michael C. Mackey, Leon Glass
First-order nonlinear differential-delay equations describing physiological control systems displaying a broad diversity of dynamical behavior including limit cycle oscillations, with a variety of wave forms, and apparently aperiodic or "chaotic" solutions are studied.
Physical Review LettersGeometry from a Time Series
3,889 Citations1980Norman H. Packard, James P. Crutchfield +2 more
Communications in Mathematical PhysicsA two-dimensional mapping with a strange attractor
2,982 Citations1976M. Hénon
A system of three first-order differential equations, whose solutions tend toward a “strange attractor”, is investigated, and it is shown that the same properties can be observed in a simple mapping of the plane defined by:xi+1=yi+1−axi2,yi-1=bxi.
Physica D Nonlinear PhenomenaExtracting qualitative dynamics from experimental data
2,292 Citations1986D.S. Broomhead, Gregory P. King
Physical Review LettersPredicting chaotic time series
1,945 Citations1987J. Doyne Farmer, John J. Sidorowich
An error estimate is presented for this forecasting technique for chaotic data, and its effectiveness is demonstrated by applying it to several examples, including data from the Mackey-Glass delay differential equation, Rayleigh-Benard convection, and Taylor-Couette flow.
NatureNonlinear forecasting as a way of distinguishing chaos from measurement error in time series
1,943 Citations1990George Sugihara, Robert M. May
An approach is presented for making short-term predictions about the trajectories of chaotic dynamical systems, applied to data on measles, chickenpox, and marine phytoplankton populations, to show how apparent noise associated with deterministic chaos can be distinguished from sampling error and other sources of externally induced environmental noise.
Physical review. A, General physicsEstimation of the Kolmogorov entropy from a chaotic signal
1,424 Citations1983Peter Grassberger, Itamar Procaccia
Physica D Nonlinear PhenomenaNonlinear prediction of chaotic time series
1,351 Citations1989Martin Casdagli
Numerical techniques are presented for constructing nonlinear predictive models directly from time series data and scaling laws are developed which describe the data requirements for reliable predictions.
Physical review. A, General physicsKolmogorov entropy and numerical experiments
1,250 Citations1976Giancarlo Benettin, L. Galgani +1 more
Progress of Theoretical PhysicsA Numerical Approach to Ergodic Problem of Dissipative Dynamical Systems
1,041 Citations1979I. Shimada, Tomohiro Nagashima
Physical Review LettersMeasurement of the Lyapunov Spectrum from a Chaotic Time Series
982 Citations1985Masaki Sano, Yasuji Sawada
A new method is proposed to determine the spectrum of several Lyapunov exponents (including positive, zero, and even negative ones) from the observed time series of a single variable.
Physical review. A, General physicsLiapunov exponents from time series
960 Citations1986Jean‐Pierre Eckmann, Sylvie Oliffson Kamphorst +2 more
An algorithm for computing Liapunov exponents from an experimental time series is analyzed and a hydrodynamic experiment is investigated.
Physica D Nonlinear PhenomenaFundamental limitations for estimating dimensions and Lyapunov exponents in dynamical systems
667 Citations1992J.-P. Eckmann, David Ruelle
Physical review. A, General physicsSingular-value decomposition and the Grassberger-Procaccia algorithm
461 Citations1988A. M. Albano, J. Muench +3 more
Physical Review AComputing the Lyapunov spectrum of a dynamical system from an observed time series
347 Citations1991Reggie Brown, Paul H. Bryant +1 more
It is shown that, even with very large data sets, it is clearly advantageous to utilize local neighborhood-to-neighborhood mappings with higher-order Taylor series rather than just local linear maps.
Physics Letters AProper choice of the time delay for the analysis of chaotic time series
331 Citations1989Wolfgang Liebert, H. G. Schuster
Progress of Theoretical PhysicsPractical Methods of Measuring the Generalized Dimension and the Largest Lyapunov Exponent in High Dimensional Chaotic Systems
270 Citations1987Shin’ichi Sato, Masaki Sano +1 more
Physical Review APrediction in chaotic nonlinear systems: Methods for time series with broadband Fourier spectra
207 Citations1990Henry D. I. Abarbanel, Reggie Brown +1 more
This work considers the problem of prediction and system identification for time series having broadband power spectra that arise from the intrinsic nonlinear dynamics of the system and finds that the parameter values that minimize the least-squares criterion do not, in general, reproduce the invariants of the dynamical system.
Physics Letters AAn improved method for estimating Liapunov exponents of chaotic time series
194 Citations1990Keith Briggs
Physical Review LettersEstimating the Lyapunov-exponent spectrum from short time series of low precision
183 Citations1991Xubin Zeng, R. Eykholt +1 more
The algorithm is tested on a variety of known model systems and it is found that the algorithm can be used to obtain a reasonable Lyapunov-exponent spectrum from only 5000 data points with a precision of 10,000 points.
Physica D Nonlinear PhenomenaChaotic time series analyses of epileptic seizures
165 Citations1990Gregory W Frank, Turab Lookman +4 more
The resulting Lyapunov exponent calculation provides a clear indication of chaos in seizure events, the definitive conclusion of which cannot be made from dimension calculations alone.
NatureCalculating the rate of loss of information from chaotic time series by forecasting
149 Citations1991David J. Wales
A method for estimating from such forecasting the largest Liapunov exponent of the dynamics, which provides a measure of how chaotic the system is—that is, how rapidly information is lost from the system.
Physics Letters ACalculating the dimension of attractors from small data sets
139 Citations1986N. B. Abraham, A. M. Albano +6 more
NonlinearityThe statistical properties of dimension calculations using small data sets
136 Citations1990James B. Ramsey, H -J Yuan
Physical review. A, General physicsKolmogorov entropy of a dynamical system with an increasing number of degrees of freedom
134 Citations1979Giancarlo Benettin, Claude Froeschlé +1 more
System Dynamics ReviewEmpirical and theoretical evidence of economic chaos
134 Citations1988Ping Chen
Physics Letters AConvergence rates and data requirements for Jacobian-based estimates of Lyapunov exponents from data
113 Citations1991Stephen P. Ellner, A. Ronald Gallant +2 more
Physica D Nonlinear PhenomenaUsing higher-order correlations to define an embedding window
90 Citations1991A. M. Albano, A. Passamante +1 more
Physica D Nonlinear PhenomenaCalculation of Lyapunov exponents avoiding spurious elements
48 Citations1991R. Stoop, Jürgen Parisi
Physical review. A, General physicsMethod for calculating a Lyapunov exponent
41 Citations1984Jon Wright
Physics Letters AHierarchical training of neural networks and prediction of chaotic time series
23 Citations1991J. Deppisch, H.-U. Bauer +1 more
A new procedure for hierarchical training of multilayer perceptrons to outputs of high precision achieves a dramatic increase in accuracy, e.g. by three orders of magnitude, and can reduce training time considerably.
Calculation of Lyapunov exponents through nonlinear adaptive filters
14 Citations1991Fawad Rauf, H.M. Ahmed