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91
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.
Optimal Sites for Supplementary Weather Observations: Simulation with a Small Model
, 1998
"... Anticipating the opportunity to make supplementary observations at locations that can depend upon the current weather situation, the question is posed as to what strategy should be adopted to select the locations, if the greatest improvement in analyses and forecasts is to be realized. To seek a p ..."
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Cited by 140 (1 self)
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Anticipating the opportunity to make supplementary observations at locations that can depend upon the current weather situation, the question is posed as to what strategy should be adopted to select the locations, if the greatest improvement in analyses and forecasts is to be realized. To seek a preliminary answer, the authors introduce a model consisting of 40 ordinary differential equations, with the dependent variables representing values of some atmospheric quantity at 40 sites spaced equally about a latitude circle. The equations contain quadratic, linear, and constant terms representing advection, dissipation, and external forcing. Numerical integration indicates that small errors (differences between solutions) tend to double in about 2 days. Localized errors tend to spread eastward as they grow, encircling the globe after about 14 days. In the
Threshold models of interpersonal effects in consumer demand
 Journal of Economic Behavior and Organization
, 1986
"... Whether one buys may be determined in part by how many others have. When the correlation is positive we refer to 'bandwagon ' effects, and when negative, 'reverse bandwagons'. We construct demand schedules in the presence of such effects and, with simple assumptions about supply, ..."
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Cited by 57 (0 self)
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Whether one buys may be determined in part by how many others have. When the correlation is positive we refer to 'bandwagon ' effects, and when negative, 'reverse bandwagons'. We construct demand schedules in the presence of such effects and, with simple assumptions about supply, investigate the existence of and approach to equilibrium. Stable pricequantity equilibria exist, but for many plausible parameter values, equilibria are asymptotically unstable, and system trajectories consist of cycles that can move, with slight parameter changes, via successive bifurcations into what has been called 'chaotic ' dynamics, essentially indistinguishable from random noise. These conditions occur despite assumptions of perfect information, profit maximizing firms and utility maximizing individuals. 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)...
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
Diagnosis of alzheimers disease from EEG signals: Where are we standing
 Current Alzheimer Research
"... This paper reviews recent progress in the diagnosis of Alzheimer’s disease (AD) from electroencephalograms (EEG). Three major effects of AD on EEG have been observed: slowing of the EEG, reduced complexity of the EEG signals, and perturbations in EEG synchrony. In recent years, a variety of sophisti ..."
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Cited by 24 (11 self)
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This paper reviews recent progress in the diagnosis of Alzheimer’s disease (AD) from electroencephalograms (EEG). Three major effects of AD on EEG have been observed: slowing of the EEG, reduced complexity of the EEG signals, and perturbations in EEG synchrony. In recent years, a variety of sophisticated computational approaches has been proposed to detect those subtle perturbations in the EEG of AD patients. The paper first describes methods that try to detect slowing of the EEG. Next the paper deals with several measures for EEG complexity, and explains how those measures have been used to study fluctuations in EEG complexity in AD patients. Then various measures of EEG synchrony are considered in the context of AD diagnosis. Also the issue of EEG preprocessing is briefly addressed. Before one can analyze EEG, it is necessary to remove artifacts due to for example head and eye movement or interference from electronic equipment. Preprocessing of EEG has in recent years received much attention. In this paper, several stateoftheart preprocessing techniques are outlined, for example, based on blind source separation and other nonlinear filtering paradigms. In addition, the paper outlines opportunities and limitations of computational approaches for diagnosing AD based on EEG. At last, future challenges and open problems are discussed.
Transients, Metastability, and Neuronal Dynamics
, 1997
"... This paper is about neuronal dynamics and how their special complexity can be understood in terms of nonlinear dynamics. There are many aspects of neuronal interactions and connectivity that engender the complexity of brain dynamics. In this paper we consider (i) the nature of this complexity and (i ..."
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Cited by 22 (5 self)
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This paper is about neuronal dynamics and how their special complexity can be understood in terms of nonlinear dynamics. There are many aspects of neuronal interactions and connectivity that engender the complexity of brain dynamics. In this paper we consider (i) the nature of this complexity and (ii) how it depends on connections between neuronal systems (e.g., neuronal populations or cortical areas). The main conclusion is that simulated neural systems show complex behaviors, reminiscent of neuronal dynamics, when these extrinsic connections are sparse. The patterns of activity that obtain, under these conditions, show a rich form of intermittency with the recurrent and selflimiting expression of stereotyped transientlike dynamics. Despite the fact that these dynamics conform to a single (complex) attractor this metastability gives the illusion of a dynamically changing attractor manifold (i.e., a changing surface upon which the dynamics unfold). This metastability is characterized using a measure that is based on the entropy of the time series’ spectral density.
Don't Bleach Chaotic Data
, 1993
"... this paper, that observation is extended. Even when the bleaching is constrained to relatively low order (by the Akaike criterion, for instance), and even for tasks other than detecting nonlinear structure, we find that the effect of bleaching on chaotic data can be detrimental. On the other hand, b ..."
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Cited by 15 (1 self)
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this paper, that observation is extended. Even when the bleaching is constrained to relatively low order (by the Akaike criterion, for instance), and even for tasks other than detecting nonlinear structure, we find that the effect of bleaching on chaotic data can be detrimental. On the other hand, bleaching
ON DETERMINING THE DIMENSION OF CHAOTIC FLOWS
, 1981
"... We describe a method for determining the approximate fractal dimension of an attractor. Our technique fits linear subspaces of appropriate dimension to sets of points on the attractor. The deviation between points on the attractor and this local linear subspace is analyzed through standard multiline ..."
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Cited by 15 (4 self)
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We describe a method for determining the approximate fractal dimension of an attractor. Our technique fits linear subspaces of appropriate dimension to sets of points on the attractor. The deviation between points on the attractor and this local linear subspace is analyzed through standard multilinear regression techniques. We show how the local dimension of attractors underlying physical phenomena can be measured even when only a single timevarying quantity is available for analysis. These methods are applied to several dissipative dynamical systems.
Uncovering nonlinear dynamicsthe case study of sea clutter
 Proc. IEEE 90
, 2002
"... Nonlinear dynamics are basic to the characterization of many physical phenomena encountered in practice. Typically, we are given a time series of some observable(s) and the requirement is to uncover the underlying dynamics responsible for generating the time series. This problem becomes particularly ..."
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Nonlinear dynamics are basic to the characterization of many physical phenomena encountered in practice. Typically, we are given a time series of some observable(s) and the requirement is to uncover the underlying dynamics responsible for generating the time series. This problem becomes particularly challenging when the process and measurement equations of the dynamics are both nonlinear and noisy. Such a problem is exemplified by the case study of sea clutter, which refers to radar backscatter from an ocean surface. After setting the stage for this case study, the paper presents tutorial reviews of: 1) the classical models of sea clutter based on the compound K distribution and 2) the application of chaos theory to sea clutter. Experimental results are presented that cast doubts on chaos as a possible nonlinear dynamical mechanism for the generation of sea clutter. Most importantly, experimental results show that on timescales smaller than a few seconds, sea clutter is very well described as a complex autoregressive process of order four or five. On larger timescales, gravity or swell waves cause this process to be modulated in both amplitude and frequency. It is shown that the amount of frequency modulation is correlated with the nonlinearity of the clutter signal. The dynamical model is an important step forward from the classical statistical approaches, but it is in its early stages of development. Keywords—Chaos, complex autoregressive models, compound Kdistribution, modulation, nonlinear dynamics, radar, sea clutter, shorttime Fouriertransform, timeDoppler. I.