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Finding Chaos in Noisy Systems
, 1991
"... In the past twenty years there has been much interest in the physical and biological sciences in nonlinear dynamical systems that appear to have random, unpredictable behavior. One important parameter of a dynamic system is the dominant Lyapunov exponent (LE). When the behavior of the system is comp ..."
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Cited by 54 (1 self)
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In the past twenty years there has been much interest in the physical and biological sciences in nonlinear dynamical systems that appear to have random, unpredictable behavior. One important parameter of a dynamic system is the dominant Lyapunov exponent (LE). When the behavior of the system is compared for two similar initial conditions, this exponent is related to the rate at which the subsequent trajectories diverge. A bounded system with a positive LE is one operational definition of chaotic behavior. Most methods for determining the LE have assumed thousands of observations generated from carefully controlled physical experiments. Less attention has been given to estimating the LE for biological and economic systems that are subjected to random perturbations and observed over a limited amount of time. Using nonparametric regression techniques (Neural Networks and Thin Plate Splines) it is possible to consistently estimate the LE. The properties of these methods have been studied using simulated data and are applied to a biological time series: marten fur returns for the Hudson Bay Company (18201900). Based on a nonparametric analysis there is little evidence for lowdimensional chaos in these data. Although these methods appear to work well for systems perturbed by small amounts of noise, finding chaos in a system with a significant stochastic component may be difficult.
ChaosBased Cryptography: A Brief Overview
 Proc. of the 5th WSEAS Int. Conf. on NonLinear Analysis, NonLinear Systems and Chaos
"... Abstract—In this brief article, chaosbased cryptography is discussed from a point of view which I believe is closer to the spirit of both cryptography and chaos theory than the way the subject has been treated recently by many researchers. I hope that, although this paper raises more questions tha ..."
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Cited by 52 (2 self)
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Abstract—In this brief article, chaosbased cryptography is discussed from a point of view which I believe is closer to the spirit of both cryptography and chaos theory than the way the subject has been treated recently by many researchers. I hope that, although this paper raises more questions than provides answers, it nevertheless contains seeds for future work.
The Maintenance of Uncertainty
 in Control Systems
, 1997
"... It is important to remain uncertain, of observation, model and law. For the Fermi Summer School, Criticisms Requested email : lenny@maths.ox.ac.uk, Contents 1 ..."
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Cited by 32 (6 self)
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It is important to remain uncertain, of observation, model and law. For the Fermi Summer School, Criticisms Requested email : lenny@maths.ox.ac.uk, Contents 1
The role of positivity and connectivity in the performance of business teams: A nonlinear dynamics model
 American Behavioral Scientist
, 2004
"... Connectivity, the control parameter in a nonlinear dynamics model of team performance is mathematically linked to the ratio of positivity to negativity (P/N) in team interaction. By knowing the P/N ratio it is possible to run the nonlinear dynamics model that will portray what types of dynamics are ..."
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Cited by 30 (2 self)
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Connectivity, the control parameter in a nonlinear dynamics model of team performance is mathematically linked to the ratio of positivity to negativity (P/N) in team interaction. By knowing the P/N ratio it is possible to run the nonlinear dynamics model that will portray what types of dynamics are possible for a team. These dynamics are of three types: point attractor, limit cycle, and complexor (complex order, or “chaotic ” in the mathematical sense). Low performance teams end up in point attractor dynamics, medium perfomance teams in limit cycle dynamics, and high performance teams in complexor dynamics. Keywords: positivity; connectivity; team performance; nonlinear dynamics Positive organizational scholars have made an explicit call for the use of nonlinear models stating that their field “is especially interested in the nonlinear positive dynamics... that are frequently associated with positive organizational phenomena ” (Cameron, Dutton, & Quinn, 2003, pp. 45). This article answers this call by showing how a nonlinear dynamics model, the meta learning (ML) model, developed and validated against empirical time series data of business
The Collective Stance in Modeling Expertise in Individuals and Organizations
, 1994
"... This paper is concerned with modeling the nature of expertise and its role in society in relation to research on expert systems and enterprise models. It argues for the adoption of a collective stance in which the human species is viewed as a single organism recursively partitioned in space and time ..."
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Cited by 30 (22 self)
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This paper is concerned with modeling the nature of expertise and its role in society in relation to research on expert systems and enterprise models. It argues for the adoption of a collective stance in which the human species is viewed as a single organism recursively partitioned in space and time into suborganisms that are similar to the whole. These parts include societies, organizations, groups, individuals, roles, and neurological functions. Notions of expertise arise because the organism adapts as a whole through adaptation of its interacting parts. The phenomena of expertise correspond to those leading to distribution of tasks and functional differentiation of the parts. The mechanism is one of positive feedback from parts of the organism allocating resources for action to other parts on the basis of those latter parts past performance of similar activities. Distribution and differentiation follow if performance is rewarded, and low performers of tasks, being excluded by the f...
Perturbation theory for the approximation of Lyapunov exponents by QR methods
, 2006
"... Motivated by a recently developed backward error analysis for QR methods, we consider the error in the Lyapunov exponents of perturbed triangular systems. We consider the case of stable and distinct Lyapunov exponents as well as the case of stable but not necessarily distinct exponents. We illustra ..."
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Cited by 9 (5 self)
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Motivated by a recently developed backward error analysis for QR methods, we consider the error in the Lyapunov exponents of perturbed triangular systems. We consider the case of stable and distinct Lyapunov exponents as well as the case of stable but not necessarily distinct exponents. We illustrate our analytical results with a numerical example.
Bayes' Estimators of Generalized Entropies
, 1998
"... .<F3.733e+05> The<F3.378e+05> orderq<F3.733e+05> Tsallis<F3.378e+05> (H<F3.378e+05> q<F3.733e+05> ) and R enyi entropy<F3.378e+05> (K<F3.378e+05> q<F3.733e+05> ) receive broad applications in the statistical analysis of complex phenomena. A gene ..."
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Cited by 6 (1 self)
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.<F3.733e+05> The<F3.378e+05> orderq<F3.733e+05> Tsallis<F3.378e+05> (H<F3.378e+05> q<F3.733e+05> ) and R enyi entropy<F3.378e+05> (K<F3.378e+05> q<F3.733e+05> ) receive broad applications in the statistical analysis of complex phenomena. A generic problem arises, however, when these entropies need to be estimated from observed data. The finite size of data sets can lead to serious systematic and statistical errors in numerical estimates. In this paper, we focus upon the problem of estimating generalized entropies from finite samples and derive the Bayes estimator of the<F3.378e+05> orderq<F3.733e+05> Tsallis entropy, including the order1 (i.e. the Shannon) entropy, under the assumption of a uniform prior probability density. The Bayes estimator yields, in general, the smallest meanquadratic deviation from the true parameter as compared with any other estimator. Exploiting the functional relationship between<F3.378e+05> H<F3.378e+05> q<F3.733e+05> and<F3.378e+05> K<F3.378e+05> q<F3....
Migration and dynamical relaxation in crowded systems of giant planets
 Icarus 163, 290 – 306 (AL2003
, 2003
"... This paper explores the intermediatetime dynamics of newly formed solar systems with a focus on possible mechanisms for planetary migration. We consider two limiting corners of the available parameter space – crowded systems containing N = 10 giant planets in the outer solar system, and solar syste ..."
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This paper explores the intermediatetime dynamics of newly formed solar systems with a focus on possible mechanisms for planetary migration. We consider two limiting corners of the available parameter space – crowded systems containing N = 10 giant planets in the outer solar system, and solar systems with N = 2 planets that are tidally interacting with a circumstellar disk. Crowded planetary systems can be formed in accumulation scenarios – if the disk is metal rich and has large mass – and through gravitational instabilities. The planetary system adjusts itself toward stability by spreading out, ejecting planets, and sending bodies into the central star. For a given set of initial conditions, dynamical relaxation leads to a welldefined distribution of possible solar systems. For each class of initial conditions, we perform large numbers (hundreds to thousands) of Nbody simulations to obtain a statistical description of the possible outcomes. For N = 10 planet systems, we consider several different planetary mass distributions; we also perform secondary sets of simulations to explore chaotic behavior and longer term dynamical evolution. For systems with 10 planets initially populating the radial range 5 AU ≤ a ≤ 30 AU, these scattering processes naturally produce planetary orbits with a ∼ 1 AU and the full range of possible eccentricity (0 ≤ ǫ ≤ 1). Shorter period orbits (smaller a) are difficult to achieve. To account for the observed eccentric giant planets, we also explore a mechanism that combines dynamical scattering and tidal interactions with a circumstellar disk. This combined model naturally produces the observed range of semimajor axis a and eccentricity ǫ. We discuss the relative merits of the different migration mechanisms for producing the observed eccentric giant planets.
Bayesian Reconstruction of Chaotic Dynamical Systems
, 2000
"... We present a Bayesian approach to the problem of determining parameters of nonlinear models from time series of noisy data. Recent approaches to this problem have been statistically awed. By applying a Markov Chain Monte Carlo algorithm, specically the Gibbs sampler, we estimate the parameters of ..."
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Cited by 4 (2 self)
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We present a Bayesian approach to the problem of determining parameters of nonlinear models from time series of noisy data. Recent approaches to this problem have been statistically awed. By applying a Markov Chain Monte Carlo algorithm, specically the Gibbs sampler, we estimate the parameters of chaotic maps. A complete statistical analysis is presented, the Gibbs sampler method is described in detail, and example applications are presented. 02.70.Lq, 06.20.Dk, 05.45.Tp, 02.60.Pn Typeset using REVT E X 1 I. INTRODUCTION Many observed time series stemming from physical laboratory experiments or \real world" systems exhibit a very complex and apparently random time behavior that may be explained by an underlying chaotic process. By a chaotic process we mean a nonlinear dynamical system [14], i.e. a discrete time series of unknown (due to noise) system states x i , i = 1; : : : ; N , that are nonlinear functions of previous states x i = g(x i 1 ). Various statistical approa...
Estimating Lyapunov Exponents In Chaotic Time Series With Locally Weighted Regression
, 1994
"... Nonlinear dynamical systems often exhibit chaos, which is characterized by sensitive dependence on initial values or more precisely by a positive Lyapunov exponent. Recognizing and quantifying chaos in time series represents an important step toward understanding the nature of random behavior and re ..."
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Cited by 4 (1 self)
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Nonlinear dynamical systems often exhibit chaos, which is characterized by sensitive dependence on initial values or more precisely by a positive Lyapunov exponent. Recognizing and quantifying chaos in time series represents an important step toward understanding the nature of random behavior and revealing the extent to which shortterm forecasts may be improved. We will focus on the statistical problem of quantifying chaos and nonlinearity via Lyapunov exponents. Predicting the future or determining Lyapunov exponents requires estimation of an autoregressive function or its partial derivatives from time series. The multivariate locally weighted polynomial fit is studied for this purpose. In the nonparametric regression context, explicit asymptotic expansions for the conditional bias and conditional covariance matrix of the regression and partial derivative estimators are derived for both the local linear fit and the local quadratic fit. These results are then generalized to the time series context. The joint asymptotic normality of the estimators is established under general shortrange dependence conditions, where the asymptotic