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Distinguished material surfaces and coherent structures in threedimensional fluid flows
 Phys. D
"... We prove analytic criteria for the existence of finitetime attracting and repelling material surfaces and lines in threedimensional unsteady flows. The longest lived such structures define coherent structures in a Lagrangian sense. Our existence criteria involve the invariants of the velocity gradi ..."
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Cited by 57 (3 self)
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We prove analytic criteria for the existence of finitetime attracting and repelling material surfaces and lines in threedimensional unsteady flows. The longest lived such structures define coherent structures in a Lagrangian sense. Our existence criteria involve the invariants of the velocity gradient tensor along fluid trajectories. An alternative approach to coherent structures is shown to lead to their characterization as local maximizers of the largest finitetime Lyapunov exponent field computed directly from particle paths. Both approaches provide effective tools for extracting distinguished Lagrangian structures from threedimensional velocity data. We illustrate the results on steady and unsteady ABCtype flows. © 2001 Elsevier Science B.V. All rights reserved.
Topological Analysis of Chaotic Dynamical Systems
 Rev. Mod. Phys
, 1997
"... onfirm the integer values. These integers can be used to determine whether or not two dynamical systems are equivalent; in particular, they can determine whether a model developed from timeseries data is an accurate representation of a physical system. Conversely, these integers can be used to p ..."
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Cited by 21 (2 self)
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onfirm the integer values. These integers can be used to determine whether or not two dynamical systems are equivalent; in particular, they can determine whether a model developed from timeseries data is an accurate representation of a physical system. Conversely, these integers can be used to provide a model for the dynamical mechanisms that generate chaotic data. In fact, the author has constructed a doubly discrete classification of strange attractors. The underlying branched manifold provides one discrete classification. Each branched manifold has an "unfolding" or perturbation in which some subset of orbits is removed. The remaining orbits are determined by a basis set of orbits that forces the presence of all remaining orbits. Branched manifolds and basis sets of orbits provide this doubly discrete classification of strange attractors. In this review the author describes the steps that have been developed to implement the topologicalanalysis procedure. In addition, th
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 deterministic ..."
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Cited by 12 (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
Nonlinear speech analysis using models for chaotic systems
 IEEE Transactions on Speech and Audio Processing
, 2005
"... Abstract—In this paper, we use concepts and methods from chaotic systems to model and analyze nonlinear dynamics in speech signals. The modeling is done not on the scalar speech signal, but on its reconstructed multidimensional attractor by embedding the scalar signal into a phase space. We have ana ..."
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Cited by 12 (3 self)
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Abstract—In this paper, we use concepts and methods from chaotic systems to model and analyze nonlinear dynamics in speech signals. The modeling is done not on the scalar speech signal, but on its reconstructed multidimensional attractor by embedding the scalar signal into a phase space. We have analyzed and compared a variety of nonlinear models for approximating the dynamics of complex systems using a small record of their observed output. These models include approximations based on global or local polynomials as well as approximations inspired from machine learning such as radial basis function networks, fuzzylogic systems and support vector machines. Our focus has been on facilitating the application of the methods of chaotic signal analysis even when only a short time series is available, like phonemes in speech utterances. This introduced an increased degree of difficulty that was dealt with by resorting to sophisticated function approximation models that are appropriate for short data sets. Using these models enabled us to compute for short time series of speech sounds useful features like Lyapunov exponents that are used to assist in the characterization of chaotic systems. Several experimental insights are reported on the possible applications of such nonlinear models and features. Index Terms—Chaos, nonlinear systems, speech analysis. I.
Reasoning About Sensor Data for Automated System Identification
 In Advances in Intelligent Data
, 1998
"... The computer program pret automatically constructs mathematical models of physical systems. A critical part of this task is automating the processing of sensor data. pret's intelligent data analyzer uses geometric reasoning to infer qualitative information from quantitative data; if critical variabl ..."
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Cited by 11 (7 self)
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The computer program pret automatically constructs mathematical models of physical systems. A critical part of this task is automating the processing of sensor data. pret's intelligent data analyzer uses geometric reasoning to infer qualitative information from quantitative data; if critical variables are either unknown or cannot be measured, it uses delaycoordinate embedding to reconstruct the internal dynamics from the external sensor measurements. Successful modeling results for two sensorequipped systems, a driven pendulum and a radiocontrolled car, demonstrate the effectiveness of these techniques.
Methods and techniques of complex systems science: An overview
, 2003
"... In this chapter, I review the main methods and techniques of complex systems science. As a ..."
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Cited by 11 (0 self)
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In this chapter, I review the main methods and techniques of complex systems science. As a
A method for modeling the intrinsic dynamics of intraindividual variability: Recovering the parameters of simulated oscillators in multi–wave panel data
 Multivariate Behavioral Research
, 2002
"... A simple method for fitting differential equations to multi–wave panel data performs remarkably well in recovering parameters from underlying continuous models with as few as three waves of data. Two techniques for fitting models of intrinsic dynamics to intraindividual variability data are examined ..."
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Cited by 10 (3 self)
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A simple method for fitting differential equations to multi–wave panel data performs remarkably well in recovering parameters from underlying continuous models with as few as three waves of data. Two techniques for fitting models of intrinsic dynamics to intraindividual variability data are examined by testing these techniques ’ behavior in recovering the parameters from data generated by two simulated systems of differential equations. Each simulated data set contains 100 “subjects ” each of whom are measured at only three points in time. A local linear approximation of the first and second derivatives of the subject’s data accurately recovers the true parameters of each simulation. A state–space embedding technique for estimating the first and second derivatives does not recover the parameters as well. An optimum sampling interval can be estimated for this model as that interval at which multiple R 2 first nears its asymptotic value.
NONLINEAR DYNAMICAL SYSTEM IDENTIFICATION FROM UNCERTAIN AND INDIRECT MEASUREMENTS
, 2002
"... We review the problem of estimating parameters and unobserved trajectory components from noisy time series measurements of continuous nonlinear dynamical systems. It is first shown that in parameter estimation techniques that do not take the measurement errors explicitly into account, like regressio ..."
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Cited by 10 (0 self)
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We review the problem of estimating parameters and unobserved trajectory components from noisy time series measurements of continuous nonlinear dynamical systems. It is first shown that in parameter estimation techniques that do not take the measurement errors explicitly into account, like regression approaches, noisy measurements can produce inaccurate parameter estimates. Another problem is that for chaotic systems the cost functions that have to be minimized to estimate states and parameters are so complex that common optimization routines may fail. We show that the inclusion of information about the timecontinuous nature of the underlying trajectories can improve parameter estimation considerably. Two approaches, which take into account both the errorsinvariables problem and the problem of complex cost functions, are described in detail: shooting approaches and recursive estimation techniques. Both are demonstrated on numerical examples.
Windowed crosscorrelation and peak picking for the analysis of variability in the association between behavioral time series
 Psychological Methods
"... Cross–correlation and most other longitudinal analyses assume that the association between two variables is stationary. Thus, a sample of occasions of measurement is expected to be representative of the association between variables regardless of the time of onset or number of occasions in the sampl ..."
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Cited by 10 (3 self)
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Cross–correlation and most other longitudinal analyses assume that the association between two variables is stationary. Thus, a sample of occasions of measurement is expected to be representative of the association between variables regardless of the time of onset or number of occasions in the sample. We propose a method to analyze the association between two variables when the assumption of stationarity may not be warranted. The method results in estimates of both the strength of peak association and the time lag when the peak association occurred for a range of starting values of elapsed time from the beginning of an experiment.