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301
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.
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.
The chaotic nature of TCP congestion control
, 2000
"... Abstract In this paper we demonstrate how TCP congestion control can show chaotic behavior. We demonstrate the major features of chaotic systems in TCPlIP networks with examples. These features include unpredictability, extreme sensitivity to initial conditions and odd periodicity. Previous work h ..."
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Cited by 138 (4 self)
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Abstract In this paper we demonstrate how TCP congestion control can show chaotic behavior. We demonstrate the major features of chaotic systems in TCPlIP networks with examples. These features include unpredictability, extreme sensitivity to initial conditions and odd periodicity. Previous work has shown the fractal nature of aggregate TCPAP traffic and one explanation to this phenomenon was that traffic can be approximated by a large number of ON/OFF sources where the random ON and/or OFF periods are of length described by a heavy tailed distribution. In this paper we show that this argument is not necessary to explain selfsimilarity, neither randomness is required. Rather, TCP itself as a deterministic process creates chaos, which generates selfsimilarity. This property is inherent in todays TCPlIP networks and it is independent of higher layer applications or protocols. The two causes: heavy tailed ONlOFF and chaotic TCP together contribute to the phenomena, called fractal nature of Internet traffic. KeywordsTCP congestion control, fractal traffic, chaotic models. I.
Estimating fractal dimension
 Journal of the Optical Society of America A
, 1990
"... Fractals arise from a variety of sources and have been observed in nature and on computer screens. One of the exceptional characteristics of fractals is that they can be described by a noninteger dimension. The geometry of fractals and the mathematics of fractal dimension have provided useful tools ..."
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Cited by 123 (4 self)
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Fractals arise from a variety of sources and have been observed in nature and on computer screens. One of the exceptional characteristics of fractals is that they can be described by a noninteger dimension. The geometry of fractals and the mathematics of fractal dimension have provided useful tools for a variety of scientific disciplines, among which is chaos. Chaotic dynamical systems exhibit trajectories in their phase space that converge to a strange attractor. The fractal dimension of this attractor counts the effective number of degrees of freedom in the dynamical system and thus quantifies its complexity. In recent years, numerical methods have been developed for estimating the dimension directly from the observed behavior of the physical system. The purpose of this paper is to survey briefly the kinds of fractals that appear in scientific research, to discuss the application of fractals to nonlinear dynamical systems, and finally to review more comprehensively the state of the art in numerical methods for estimating the fractal dimension of a strange attractor. Confusion is a word we have invented for an order which is not understood.Henry Miller, "Interlude," Tropic of Capricorn Numerical coincidence is a common path to intellectual perdition in our quest for meaning. We delight in catalogs of disparate items united by the same number, and often feel in our gut that some unity must underlie it all.
Computation at the onset of chaos
 The Santa Fe Institute, Westview
, 1988
"... Computation at levels beyond storage and transmission of information appears in physical systems at phase transitions. We investigate this phenomenon using minimal computational models of dynamical systems that undergo a transition to chaos as a function of a nonlinearity parameter. For perioddoubl ..."
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Cited by 101 (19 self)
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Computation at levels beyond storage and transmission of information appears in physical systems at phase transitions. We investigate this phenomenon using minimal computational models of dynamical systems that undergo a transition to chaos as a function of a nonlinearity parameter. For perioddoubling and bandmerging cascades, we derive expressions for the entropy, the interdependence ofmachine complexity and entropy, and the latent complexity of the transition to chaos. At the transition deterministic finite automaton models diverge in size. Although there is no regular or contextfree Chomsky grammar in this case, we give finite descriptions at the higher computational level of contextfree Lindenmayer systems. We construct a restricted indexed contextfree grammar and its associated oneway nondeterministic nested stack automaton for the cascade limit language. This analysis of a family of dynamical systems suggests a complexity theoretic description of phase transitions based on the informational diversity and computational complexity of observed data that is independent of particular system control parameters. The approach gives a much more refined picture of the architecture of critical states than is available via
State space reconstruction in the presence of noise
 Physica D: Nonlinear Phenomena
, 1991
"... SFI Working Papers contain accounts of scientific work of the author(s) and do not necessarily represent the views of the Santa Fe Institute. We accept papers intended for publication in peerreviewed journals or proceedings volumes, but not papers that have already appeared in print. Except for pap ..."
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Cited by 91 (3 self)
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SFI Working Papers contain accounts of scientific work of the author(s) and do not necessarily represent the views of the Santa Fe Institute. We accept papers intended for publication in peerreviewed journals or proceedings volumes, but not papers that have already appeared in print. Except for papers by our external faculty, papers must be based on work done at SFI, inspired by an invited visit to or collaboration at SFI, or funded by an SFI grant. ©NOTICE: This working paper is included by permission of the contributing author(s) as a means to ensure timely distribution of the scholarly and technical work on a noncommercial basis. Copyright and all rights therein are maintained by the author(s). It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author's copyright. These works may be reposted only with the explicit permission of the copyright holder. www.santafe.edu
Time Series Prediction by Using a Connectionist Network with Internal Delay Lines
 Time Series Prediction
, 1994
"... A neural network architecture, which models synapses as Finite Impulse Response (FIR) linear filters, is discussed for use in time series prediction. Analysis and methodology are detailed in the context of the Santa Fe Institute Time Series Prediction Competition. Results of the competition show tha ..."
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Cited by 78 (4 self)
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A neural network architecture, which models synapses as Finite Impulse Response (FIR) linear filters, is discussed for use in time series prediction. Analysis and methodology are detailed in the context of the Santa Fe Institute Time Series Prediction Competition. Results of the competition show that the FIR network performed remarkably well on a chaotic laser intensity time series. 1 Introduction The goal of time series prediction or forecasting can be stated succinctly as follows: given a sequence y(1); y(2); : : : y(N) up to time N , find the continuation y(N + 1); y(N + 2)::: The series may arise from the sampling of a continuous time system, and be either stochastic or deterministic in origin. The standard prediction approach involves constructing an underlying model which gives rise to the observed sequence. In the oldest and most studied method, which dates back to Yule [1], a linear autoregression (AR) is fit to the data: y(k) = T X n=1 a(n)y(k \Gamma n) + e(k) = y(k) + ...
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.