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116
Advanced Spectral Methods for Climatic Time Series
, 2001
"... The analysis of uni or multivariate time series provides crucial information to describe, understand, and predict climatic variability. The discovery and implementation of a number of novel methods for extracting useful information from time series has recently revitalized this classical eld of ..."
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Cited by 220 (35 self)
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The analysis of uni or multivariate time series provides crucial information to describe, understand, and predict climatic variability. The discovery and implementation of a number of novel methods for extracting useful information from time series has recently revitalized this classical eld of study. Considerable progress has also been made in interpreting the information so obtained in terms of dynamical systems theory.
Interdisciplinary application of nonlinear time series methods
 Phys. Reports
, 1998
"... This paper reports on the application to field measurements of time series methods developed on the basis of the theory of deterministic chaos. The major difficulties are pointed out that arise when the data cannot be assumed to be purely deterministic and the potential that remains in this situatio ..."
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Cited by 84 (4 self)
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This paper reports on the application to field measurements of time series methods developed on the basis of the theory of deterministic chaos. The major difficulties are pointed out that arise when the data cannot be assumed to be purely deterministic and the potential that remains in this situation is discussed. For signals with weakly nonlinear structure, the presence of nonlinearity in a general sense has to be inferred statistically. The paper reviews the relevant methods and discusses the implications for deterministic modeling. Most field measurements yield nonstationary time series, which poses a severe problem for their analysis. Recent progress in the detection and understanding of nonstationarity is reported. If a clear signature of approximate determinism is found, the notions of phase space, attractors, invariant manifolds etc. provide a convenient framework for time series analysis. Although the results have to be interpreted with great care, superior performance can be achieved for typical signal processing tasks. In particular, prediction and filtering of signals are discussed, as well as the classification of system states by means of time series recordings.
The Riemann Zeros and Eigenvalue Asymptotics
 SIAM Rev
, 1999
"... Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many feat ..."
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Cited by 61 (10 self)
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Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many features of H cl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the t n have a similar structure to those of the semiclassical En ; in particular, they display randommatrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the t n can be computed accurately from formulae with quantum analogues. The RiemannSiegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian H cl = XP. Key words. spectral asymptotics, number theory AMS subject classifications. 11M26, 11M06, 35P20, 35Q40, 41A60, 81Q10, 81Q50 PII. S0036144598347497 1.
Chaosbased random number generatorspart uppercaseI: analysis [cryptography
 IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications
, 2001
"... Abstract—This paper and its companion (Part II) are devoted to the analysis of the application of a chaotic piecewiselinear onedimensional (PL1D) map as random number generator (RNG). Piecewise linearity of the map enables us to mathematically find parameter values for which a generating partition ..."
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Cited by 19 (0 self)
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Abstract—This paper and its companion (Part II) are devoted to the analysis of the application of a chaotic piecewiselinear onedimensional (PL1D) map as random number generator (RNG). Piecewise linearity of the map enables us to mathematically find parameter values for which a generating partition is Markov and the RNG behaves as a Markov information source, and then to mathematically analyze the information generation process and the RNG. In the companion paper we discuss practical aspects of our chaosbased RNGs. Index Terms—Chaos, random number generator, symbolic dynamics. I.
A universal circuit for studying and generating chaos—Part I: Routes to chaos
 IEEE Trans. Circuits Syst. I, Fundam. Theory Appl
, 1993
"... AbstractIn this introductory tutorial paper, we demonstrate the generality of Chua's oscillator in generating chaos and bifurcation phenomena by electronic laboratory experiments which illustrate the standard routes to chaos, and by giving a result which shows that Chua's oscillator can ..."
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Cited by 12 (0 self)
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AbstractIn this introductory tutorial paper, we demonstrate the generality of Chua's oscillator in generating chaos and bifurcation phenomena by electronic laboratory experiments which illustrate the standard routes to chaos, and by giving a result which shows that Chua's oscillator can generate the same qualitative behavior as any member of a 21parameter family C of continuous, oddsymmetric, piecewiselinear vector field in R3. This result is of fundamental importance because it unifies many previously published papers on chaotic circuits and systems (e.g. examples from Brockett, Sparrow, Arnkodo, Nishio, Ogorzalek, etc.) under one umbrella, thereby obviating the need to analyze these circuits and systems as separate and unrelated systems. Indeed, every bifurcation and chaotic phenomena exhibited by any member of the family C is also exhibited by this universal circuit. In a companion paper [l], we show how the generality of Chua's oscillator can be used to approximate other chaotic systems in the literature which are not necessarily piecewiselinear. I.
Basic Elements and Problems of Probability Theory
, 1999
"... After a brief review of ontic and epistemic descriptions, and of subjective, logical and statistical interpretations of probability, we summarize the traditional axiomatization of calculus of probability in terms of Boolean algebras and its settheoretical realization in terms of Kolmogorov probabil ..."
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Cited by 12 (0 self)
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After a brief review of ontic and epistemic descriptions, and of subjective, logical and statistical interpretations of probability, we summarize the traditional axiomatization of calculus of probability in terms of Boolean algebras and its settheoretical realization in terms of Kolmogorov probability spaces. Since the axioms of mathematical probability theory say nothing about the conceptual meaning of “randomness” one considers probability as property of the generating conditions of a process so that one can relate randomness with predictability (or retrodictability). In the measuretheoretical codification of stochastic processes genuine chance processes can be defined rigorously as socalled regular processes which do not allow a longterm prediction. We stress that stochastic processes are equivalence classes of individual point functions so that they do not refer to individual processes but only to an ensemble of statistically equivalent individual processes. Less popular but conceptually more important than statistical descriptions are individual descriptions which refer to individual chaotic processes. First, we review the individual description based on the generalized harmonic analysis by Norbert Wiener. It allows the definition of individual purely chaotic processes which can be interpreted as trajectories of regular statistical stochastic processes. Another individual description refers to algorithmic procedures which connect the intrinsic randomness of a finite sequence with the complexity of the shortest program necessary to produce the sequence. Finally, we ask why there can be laws of chance. We argue that random events fulfill the laws of chance if and only if they can be reduced to (possibly hidden) deterministic events. This mathematical result may elucidate the fact that not all nonpredictable events can be grasped by the methods of mathematical probability theory.
Fluctuation spectroscopy
 Chaos, Solitons, and Fractals, special issue on Complexity
, 1993
"... We review the thermodynamics of estimating the statistical fluctuations of an observed process. Since any statistical analysis involves a choice of model class  either explicitly or implicitly  we demonstrate the benefits of a careful choice. For each of three classes a particular model is recon ..."
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Cited by 12 (1 self)
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We review the thermodynamics of estimating the statistical fluctuations of an observed process. Since any statistical analysis involves a choice of model class  either explicitly or implicitly  we demonstrate the benefits of a careful choice. For each of three classes a particular model is reconstructed from data streams generated by four sample processes. Then each estimated model's thermodynamic structure is used to estimate the typical behavior and the magnitude of deviations for the observed system. These are then compared to the known fluctuation properties. The type of analysis advocated here, which uses estimated model class information, recovers the correct statistical structure of these processes from simulated data. The current alternative  direct estimation of the Renyi entropy from time series histograms  uses neither prior nor reconstructed knowledge of the model class. And, in most cases, it fails to recover the process's statistical structure from finite data  unpredictability is overestimated.
Boolean delay equations: A simple way of looking at complex system
 Physica D
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Chaotic time series Part I: Estimation of some invariant properties in state space
 Modeling, Identification and Control, 15(4):205  224
, 1995
"... Certain deterministic nonlinear systems may show chaotic behaviour. Time series derived from such systems seem stochastic when analyzed with linear techniques. However, uncovering the deterministic structure is important because it allows constructing more realistic and better models and thus impro ..."
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Cited by 10 (5 self)
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Certain deterministic nonlinear systems may show chaotic behaviour. Time series derived from such systems seem stochastic when analyzed with linear techniques. However, uncovering the deterministic structure is important because it allows constructing more realistic and better models and thus improved predictive capabilities. This paper provides a review of two main key features of chaotic systems, the dimensions of their strange attractors and the Lyapunov exponents. The emphasis is on state space reconstruction techniques that are used to estimate these properties, given scalar observations. Data generated from equations known to display chaotic behaviour are used for illustration. A compilation of applications to real data from widely different fields is given. If chaos is found to be present, one may proceed to build nonlinear models, which is the topic of the second paper in this series.