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126
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.
Extracting macroscopic dynamics: model problems and algorithms
 NONLINEARITY
, 2004
"... In many applications, the primary objective of numerical simulation of timeevolving systems is the prediction of macroscopic, or coarsegrained, quantities. A representative example is the prediction of biomolecular conformations from molecular dynamics. In recent years a number of new algorithmic ..."
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Cited by 111 (8 self)
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In many applications, the primary objective of numerical simulation of timeevolving systems is the prediction of macroscopic, or coarsegrained, quantities. A representative example is the prediction of biomolecular conformations from molecular dynamics. In recent years a number of new algorithmic approaches have been introduced to extract effective, lowerdimensional, models for the macroscopic dynamics; the starting point is the full, detailed, evolution equations. In many cases the effective lowdimensional dynamics may be stochastic, even when the original starting point is deterministic. This review surveys a number of these new approaches to the problem of extracting effective dynamics, highlighting similarities and differences between them. The importance of model problems for the evaluation of these new approaches is stressed, and a number of model problems are described. When the macroscopic dynamics is stochastic, these model problems are either obtained through a clear separation of timescales, leading to a stochastic effect of the fast dynamics on the slow dynamics, or by considering high dimensional ordinary differential equations which, when projected onto a low dimensional subspace, exhibit stochastic behaviour through the presence of a broad frequency spectrum. Models whose stochastic microscopic behaviour leads to deterministic macroscopic dynamics are also introduced. The algorithms we overview include SVDbased methods for nonlinear problems, model reduction for linear control systems, optimal prediction techniques, asymptoticsbased mode elimination, coarse timestepping methods and transferoperator based methodologies.
Nonlinear dynamics of networks: the groupoid formalism
 Bull. Amer. Math. Soc
, 2006
"... Abstract. A formal theory of symmetries of networks of coupled dynamical systems, stated in terms of the group of permutations of the nodes that preserve the network topology, has existed for some time. Global network symmetries impose strong constraints on the corresponding dynamical systems, which ..."
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Cited by 79 (13 self)
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Abstract. A formal theory of symmetries of networks of coupled dynamical systems, stated in terms of the group of permutations of the nodes that preserve the network topology, has existed for some time. Global network symmetries impose strong constraints on the corresponding dynamical systems, which affect equilibria, periodic states, heteroclinic cycles, and even chaotic states. In particular, the symmetries of the network can lead to synchrony, phase relations, resonances, and synchronous or cycling chaos. Symmetry is a rather restrictive assumption, and a general theory of networks should be more flexible. A recent generalization of the grouptheoretic notion of symmetry replaces global symmetries by bijections between certain subsets of the directed edges of the network, the ‘input sets’. Now the symmetry group becomes a groupoid, which is an algebraic structure that resembles a group, except that the product of two elements may not be defined. The groupoid formalism makes it possible to extend grouptheoretic methods to more general networks, and in particular it leads to a complete classification of ‘robust ’ patterns of synchrony in terms of the combinatorial structure of the network. Many phenomena that would be nongeneric in an arbitrary dynamical system can become generic when constrained by a particular network topology. A network of dynamical systems is not just a dynamical system with a highdimensional phase space. It is also equipped with a canonical set of observables—the states of the individual nodes of the network. Moreover, the form of the underlying ODE is constrained by the network topology—which variables occur in which component equations, and how those equations relate to each other. The result is a rich and new range of phenomena, only a few of which are yet properly understood. Contents 1.
Generalized Redundancies for Time Series Analysis
 Physica D
, 1995
"... Extensions to various informationtheoretic quantities (such as entropy, redundancy, and mutual information) are discussed in the context of their role in nonlinear time series analysis. We also discuss "linearized" versions of these quantities and their use as benchmarks in tests for nonl ..."
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Cited by 38 (0 self)
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Extensions to various informationtheoretic quantities (such as entropy, redundancy, and mutual information) are discussed in the context of their role in nonlinear time series analysis. We also discuss "linearized" versions of these quantities and their use as benchmarks in tests for nonlinearity. Many of these quantities can be expressed in terms of the generalized correlation integral, and this expression permits us to more clearly exhibit the relationships of these quantities to each other and to other commonly used nonlinear statistics (such as the BDS and GreenSavit statistics). Further, numerical estimation of these quantities is found to be more accurate and more efficient when the the correlation integral is employed in the computation. Finally, we consider several "local" versions of these quantities, including a local KolmogorovSinai entropy, which gives an estimate of variability of the shortterm predictability. 1 Introduction In Shaw's influential (and prizewinning)...
T.: Gpubased nonlinear ray tracing
 Comput. Graph. Forum
"... In this paper, we present a mapping of nonlinear ray tracing to the GPU which avoids any data transfer back to main memory. The rendering process consists of the following parts: ray setup according to the camera parameters, ray integration, ray–object intersection, and local illumination. Bent rays ..."
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Cited by 30 (2 self)
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In this paper, we present a mapping of nonlinear ray tracing to the GPU which avoids any data transfer back to main memory. The rendering process consists of the following parts: ray setup according to the camera parameters, ray integration, ray–object intersection, and local illumination. Bent rays are approximated by polygonal lines that are represented by textures. Ray integration is based on an iterative numerical solution of ordinary differential equations whose initial values are determined during ray setup. To improve the rendering performance, we propose acceleration techniques such as early ray termination and adaptive ray integration. Finally, we discuss a variety of applications that range from the visualization of dynamical systems to the general relativistic visualization in astrophysics and the rendering of the continuous refraction in media with varying density. Categories and Subject Descriptors (according to ACM CCS): I.3.3 [Computer Graphics]: Picture/Image Generation I.3.7 [Computer Graphics]: ThreeDimensional Graphics and Realism
Detecting Nonlinearity in Data with Long Coherence Times
, 1992
"... this article, we will describe (yet) another source of difficulty that arises in the analysis of time series data. The particular problem of detecting nonlinear structure  either by comparison of the data to linear surrogate data, or by comparing linear and nonlinear predictors  is seen to be ..."
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Cited by 25 (2 self)
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this article, we will describe (yet) another source of difficulty that arises in the analysis of time series data. The particular problem of detecting nonlinear structure  either by comparison of the data to linear surrogate data, or by comparing linear and nonlinear predictors  is seen to be complicated when the data exhibits long coherence times. In this section we define some terms and discuss linear modeling of time series. Section 2 describes the method of surrogate data, and compares two approaches to generating surrogate data. We find that both have difficulties trying to mimic data with long coherence time. We illustrate these problems with real and computergenerated time series in Section 3, including the time series E.dat from the the SFI competition. In the last section, we discuss what it is about the analysis or the data that is problematic.
State space reconstruction parameters in the analysis of chaotic time series  the role of the time window length
 Physica D, 95:13
, 1996
"... dimension The most common state space reconstruction method in the analysis of chaotic time series is the Method of Delays (MOD). Many techniques have been suggested to estimate the parameters of MOD, i.e. the time delay τ and the embedding dimension m. We discuss the applicability of these techniqu ..."
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Cited by 25 (3 self)
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dimension The most common state space reconstruction method in the analysis of chaotic time series is the Method of Delays (MOD). Many techniques have been suggested to estimate the parameters of MOD, i.e. the time delay τ and the embedding dimension m. We discuss the applicability of these techniques with a critical view as to their validity, and point out the necessity of determining the overall time window length, τw, for successful embedding. Emphasis is put on the relation between τw and the dynamics of the underlying chaotic system, and we suggest to set τw ≥ τp, the mean orbital period; τp is approximated from the oscillations of the time series. The procedure is assessed using the correlation dimension for both synthetic and real data. For clean synthetic data, values of τw larger than τp always give good results given enough data and thus τp can be considered as a lower limit (τw ≥ τp). For noisy synthetic data and real data, an upper limit is reached for τw which approaches τp for increasing noise amplitude. 1
Coarsegrained embedding of time series: Random walks Gaussian random processes, and deterministic chaos
 Physica D
, 1993
"... A new method for studying time series is described based on a statistic indicating the degree to which trajectories passing through a small region of an embedding space are parallel. The method is particularly suited to time series with a significant correlation time. Analytic results are presented ..."
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Cited by 18 (2 self)
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A new method for studying time series is described based on a statistic indicating the degree to which trajectories passing through a small region of an embedding space are parallel. The method is particularly suited to time series with a significant correlation time. Analytic results are presented for Brownian motion and Gaussian random processes. These are generally different from the results for chaotic systems, allowing a test for deterministic dynamics in a time series. A variety of examples are presented of the application of the method to lowand highdimensional systems.
Taming Chaotic Circuits
, 1992
"... Control algorithms that exploit chaotic behavior and its precursors can vastly improve the performance of many practical and useful systems. Phaselocked loops, for example, are normally designed using linearization. This approximation hides the global dynamics that lead to lock and capture range li ..."
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Cited by 17 (3 self)
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Control algorithms that exploit chaotic behavior and its precursors can vastly improve the performance of many practical and useful systems. Phaselocked loops, for example, are normally designed using linearization. This approximation hides the global dynamics that lead to lock and capture range limits. Design techniques that are equipped to exploit the real nonlinear and chaotic nature of the device can loosen these limitations. The program Perfect Moment is built around a collection of such techniques. Given a differential equation, a control parameter, and two statespace points, the program explores the system's behavior, automatically choosing interesting and useful parameter values and constructing statespace portraits at each one. It then chooses a set of trajectory segments from those portraits, uses them to construct a composite path between the objectives, and finally causes the system to follow that path by switching the parameter value at the segment junctions. Rules embo...