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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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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.
Causes and effects of chaos
 MIT Artificial Intelligence Lab
, 1990
"... Most of the recent literature on chaos and nonlinear dynamics is written either for popular science magazine readers or for advanced mathematicians. This paper gives a broad introduction to this interesting and rapidly growing field at a level that is between the two. The graphical and analytical ..."
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Cited by 4 (3 self)
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Most of the recent literature on chaos and nonlinear dynamics is written either for popular science magazine readers or for advanced mathematicians. This paper gives a broad introduction to this interesting and rapidly growing field at a level that is between the two. The graphical and analytical tools used in the literature are explained and demonstrated, the rudiments of the current theory are outlined and that theory is discussed in the context of several examples: an electronic circuit, a chemical reaction and a system of satellites in the solar system. 9112378
State Space Reconstruction: Method of Delays vs Singular Spectrum Approach
 N. Christophersen, http://citeseer.nj.nec.com/kugiumtzis97state.html
, 1997
"... Abstract The analysis of chaotic time series requires proper reconstruction of the state space from the available data in order to successfully estimate invariant properties of the embedded attractor. Using the correlation dimension, we discuss the applicability of the two most common methods of re ..."
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Cited by 2 (0 self)
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Abstract The analysis of chaotic time series requires proper reconstruction of the state space from the available data in order to successfully estimate invariant properties of the embedded attractor. Using the correlation dimension, we discuss the applicability of the two most common methods of reconstruction, the method of delays (MOD) and the Singular Spectrum Approach (SSA). Contrary to previous discussions, we found that the two methods perform equivalently in practice for noisefree data provided the parameters of the two methods are properly related. In fact, the quality of the reconstruction is in both cases determined by the choice of the time window length τw and is independent of the selected method. However, when the data are noisy, we find that SSA outperforms MOD.
Recurrent structuring of dynamical and spatial systems
 In: Colosimo, A. (Ed.). Complexity in the Living: A Modelistic Approach
, 1998
"... The complexity and nonlinearity of physiological systems typically defy comprehensive and deterministic mathematical modeling, except from a statistical perspective. Living systems are governed by numerous interacting variables (high dimensional problem) with drifting parameters (nonstationarity) in ..."
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The complexity and nonlinearity of physiological systems typically defy comprehensive and deterministic mathematical modeling, except from a statistical perspective. Living systems are governed by numerous interacting variables (high dimensional problem) with drifting parameters (nonstationarity) in the presence of noise (internal and external perturbations). Depending upon the frame of reference, many biological signals can be shown to be discontinuous alternations between deterministic trajectories and stochastic pauses (terminal dynamics). One promising approach for assessing such nondeterministic complexity is recurrence quantification analysis (RQA). As reviewed