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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.
The Lyapunov Characteristic Exponents and their
 Computation, Lect. Notes Phys
, 2010
"... For want of a nail the shoe was lost. For want of a shoe the horse was lost. For want of a horse the rider was lost. For want of a rider the battle was lost. For want of a battle the kingdom was lost. And all for the want of a horseshoe nail. For Want of a Nail (proverbial rhyme) Summary. We present ..."
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Cited by 29 (2 self)
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For want of a nail the shoe was lost. For want of a shoe the horse was lost. For want of a horse the rider was lost. For want of a rider the battle was lost. For want of a battle the kingdom was lost. And all for the want of a horseshoe nail. For Want of a Nail (proverbial rhyme) Summary. We present a survey of the theory of the Lyapunov Characteristic Exponents (LCEs) for dynamical systems, as well as of the numerical techniques developed for the computation of the maximal, of few and of all of them. After some historical notes on the first attempts for the numerical evaluation of LCEs, we discuss in detail the multiplicative ergodic theorem of Oseledec [99], which provides the theoretical basis for the computation of the LCEs. Then, we analyze the algorithm for the computation of the maximal LCE, whose value has been extensively used as an indicator of chaos, and the algorithm of the so–called ‘standard method’, developed by Benettin et al. [14], for the computation of many LCEs. We also consider different discrete and continuous methods for computing the LCEs based on the QR or the singular value decomposition techniques. Although, we are mainly interested in finite–dimensional conservative systems, i. e. autonomous Hamiltonian systems and symplectic maps, we also briefly refer to the evaluation of LCEs of dissipative systems and time series. The relation of two chaos detection techniques, namely the fast Lyapunov indicator (FLI) and the generalized alignment index (GALI), to the computation of the LCEs is also discussed. 1
An evaluation of the Lyapunov characteristic exponent of chaotic continuous systems
, 2003
"... A procedure to calculate the Lyapunov characteristic exponent of the response of structural continuous systems, discretized using finite element methods, is proposed. The Lyapunov characteristic exponent can be used to characterize the asymptotic stability of the system dynamic response, and it is f ..."
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Cited by 6 (2 self)
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A procedure to calculate the Lyapunov characteristic exponent of the response of structural continuous systems, discretized using finite element methods, is proposed. The Lyapunov characteristic exponent can be used to characterize the asymptotic stability of the system dynamic response, and it is frequently employed to identify a chaotic behaviour. The proposed procedure can also be used in the stability characterization of fluid–structure interaction systems in which the focus of the analysis is on the
Quantifying Dynamical Predictability: the PseudoEnsemble Approach
, 2009
"... The ensemble technique has been widely used in numerical weather prediction and extendedrange forecasting. Current approaches to evaluate the predictability using the ensemble technique can be divided into two major groups. One is dynamical, including generating Lyapunov vectors, bred vectors, and ..."
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Cited by 5 (4 self)
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The ensemble technique has been widely used in numerical weather prediction and extendedrange forecasting. Current approaches to evaluate the predictability using the ensemble technique can be divided into two major groups. One is dynamical, including generating Lyapunov vectors, bred vectors, and singular vectors, sampling the fastest errorgrowing directions of the phase space, and examining the dependence of prediction efficiency on ensemble size. The other is statistical, including distributional analysis and quantifying prediction utility by the Shannon entropy and the relative entropy. Currently, with simple models, one could choose as many ensembles as possible, with each ensemble containing a large number of members. When the forecast models become increasingly complicated, however, one would only be able to afford a small number of ensembles, each with limited number of members, thus sacrificing estimation accuracy of the forecast errors. To uncover connections between different information theoretic approaches and between dynamical and statistical approaches, we propose an (ǫ, τ)entropy and scaledependent Lyapunov exponent—based general theoretical framework to quantify information loss in ensemble forecasting. More importantly, to tremendously expedite computations, reduce data storage, and improve forecasting accuracy, we propose a technique for constructing a large number of “pseudo” ensembles from one single solution or scalar dataset. This pseudoensemble technique appears to be applicable under rather general conditions, one important situation being that observational data are available but the exact dynamical model is unknown.
Neural Learning of Chaotic Dynamics: The Error Propagation Algorithm
 in Proc. 1998 WCCI
, 1998
"... An algorithm is introduced that trains a neural network to identify chaotic dynamics from a single measured timeseries. The algorithm has four special features: 1. The state of the system is extracted from the timeseries using delays, followed by weighted Principal Component Analysis (PCA) data red ..."
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An algorithm is introduced that trains a neural network to identify chaotic dynamics from a single measured timeseries. The algorithm has four special features: 1. The state of the system is extracted from the timeseries using delays, followed by weighted Principal Component Analysis (PCA) data reduction. 2. The prediction model consists of both a linear model and a MultiLayerPerceptron (MLP). 3. The effective prediction horizon during training is useradjustable, due to ‘error propagation’: prediction errors are partially propagated to the next time step. 4. A criterion is monitored during training to select the model that has a chaotic attractor most similar to the real system’s attractor. The algorithm is applied to laser data from the Santa Fe timeseries competition (set A). The resulting model is not only useful for shortterm predictions but it also generates timeseries with similar chaotic characteristics as the measured data.
A Chaotic Limit Cycle Paradox
"... Duffing’s equation with sinusoidal forcing produces chaos for certain combinations of the forcing amplitude and frequency. To determine the most chaotic response achievable for given bounds on the input force, an optimal control problem was investigated to maximize the largest Lyapunov exponent whi ..."
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Duffing’s equation with sinusoidal forcing produces chaos for certain combinations of the forcing amplitude and frequency. To determine the most chaotic response achievable for given bounds on the input force, an optimal control problem was investigated to maximize the largest Lyapunov exponent which, in this case, also corresponds to maximizing the KaplanYorke Lyapunov fractal dimension. The resulting bangbang optimal feedback controller yielded a bounded attractor with a positive largest Lyapunov exponent and a fractional Lyapunov dimension, indicating a chaotic strange attractor. Indeed, the largest Lyapunov exponent was approximately twice as large as that achieved with sinusoidal forcing at the same amplitude. However, the resulting attractor is just a stable limit cycle and is not chaotic or fractal at all! This contradicts the basic idea that a bounded attractor with at least one positive Lyapunov exponent must be chaotic and fractal. This paper provides details of this chaotic limit cycle paradox and the resolution of the paradox. In particular, for systems of differential equations with only piecewise differentiable righthand sides, a jump discontinuity condition must be imposed on the state perturbations in order to compute correct Lyapunov exponents.
Learning Chaotic Attractors by Neural Networks
 NEURAL COMPUTATION
, 2000
"... An algorithm is introduced that trains a neural network to identify chaotic dynamics from a single measured time series. During training, the algorithm learns to shortterm predict the time series. At the same time a criterion, developed by Diks, van Zwet, Takens, and de Goede (1996) is monitored th ..."
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Cited by 1 (0 self)
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An algorithm is introduced that trains a neural network to identify chaotic dynamics from a single measured time series. During training, the algorithm learns to shortterm predict the time series. At the same time a criterion, developed by Diks, van Zwet, Takens, and de Goede (1996) is monitored that tests the hypothesis that the reconstructed attractors of modelgenerated and measured data are the same. Training is stopped when the prediction error is low and the model passes this test. Two other features of the algorithm are (1) the way the state of the system, consisting of delays from the time series, has its dimension reduced by weighted principal component analysis data reduction, and (2) the useradjustable prediction horizon obtained by "error propagation"  partially propagating prediction errors to the next time step.The algorithm is first applied to data from an experimentaldriven chaotic pendulum, of which two of the three state variables are known. This is a comprehensive example that shows how well the Diks test can distinguish between slightly different attractors. Second, the algorithm is applied to the same problem, but now one of the two known state variables is ignored. Finally, we present a model for the laser data from the Santa Fe timeseries competition (set A). It is the first model for these data that is not only useful for shortterm predictions but also generates time series with similar chaotic characteristics as the measured data.
The Anatomy of the Chua circuit
"... The Chua circuit, known to exhibit chaotic behaviour, has been constructed and studied. The goal was to investigate the experimental underpinning of the theory. The component relationship of the resistors, capacitors, inductors and operational amplifiers have been measured and established within the ..."
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The Chua circuit, known to exhibit chaotic behaviour, has been constructed and studied. The goal was to investigate the experimental underpinning of the theory. The component relationship of the resistors, capacitors, inductors and operational amplifiers have been measured and established within their domain of validity. The system equations themselves cannot be directly validated through a traditional simulation vs. experimental study due to the chaotic property. Instead, a novel procedure is proposed where the system variables are split into groups, with each group being validated by obtaining the variables outside the group directly from experimental data. The method decisively confirms the equations and parameters values to be correct. To streamline the theoretical computations, the special piecewise linear structure of the system equations is used to obtain local analytical solutions, which are then joined together by equation solving. Using the resistance as the control parameter, the different solutions to the equations are classified and qualitatively explained. A number of experimental tests are performed to investigate the correspondence between theory and experiments; the invariance under parity inversion, the position of the Hopf bifurcation and the equilibrium voltage as a function of resistance. Finally, chaos is quantified using the Lyapunov exponent. As its definition requires a metric, such is proposed based on energy consideration. The piecewise linear structure of the equation provides a very simple formula for the
THREE CONDITIONS LYAPUNOV EXPONENTS SHOULD SATISFY
"... ABSTRACT In contrast to the unilateral claim in some papers that a positive Lyapunov exponent means chaos, it was claimed in this paper that this is just one of the three conditions that Lyapunov exponent should satisfy in a dissipative dynamical system when the chaotic motion appears. The other tw ..."
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ABSTRACT In contrast to the unilateral claim in some papers that a positive Lyapunov exponent means chaos, it was claimed in this paper that this is just one of the three conditions that Lyapunov exponent should satisfy in a dissipative dynamical system when the chaotic motion appears. The other two conditions, any continuous dynamical system without a fixed point has at least one zero exponent, and any dissipative dynamical system has at least one negative exponent and the sum of all of the 1dimensional Lyapunov exponents id negative, are also discussed. In order to verify the conclusion, a MATLAB scheme was developed for the computation of the 1dimensional and 3dimensional Lyapunov exponents of the Duffing system with square and cubic nonlinearity. KEYWORDS: Lyapunov exponent, chaos, three conditions, Duffing system INTRODUCTION Recently, chaotic motions that arise from the nonlinearity of dissipative dynamical systems have received a great concern in both physical and nonphysical fields. The most striking feature of chaos is the unpredictability of the future despite a deterministic time evolution. This unpredictability is a consequence of the inherent instability of the solutions, reflected by what is called sensitive dependence on initial conditions. The tiny deviations between the initial conditions of all the trajectories are blown up after a short time. A more careful investigation of this instability leads to two different, although related, concepts. One is the loss of information related to unpredictability, quantified by the KolmogorovSinai entropy. The other is a simple geometric one, namely, that nearby trajectories separate very fast, or more precisely, separate exponentially over time. The properly averaged exponent of this increase is the characteristics for the