8 NONLINEAR TIME-SERIES ANALYSIS

This tutorial review presents an overview of the achievements and some present research activities in the field of state space based methods for non­ linear time-series analysis. In particular, questions of state space reconstruction, of modelling and prediction, of filtering and noise reduction, of detecting non­ linearities in time series, and applications using chaotic synchronization are ad­ dressed. Furthermore, a new approach for modeling data from spatia-temporal systems is presented.

[1]  R. Gencay,et al.  An algorithm for the n Lyapunov exponents of an n -dimensional unknown dynamical system , 1992 .

[2]  G. Benettin,et al.  Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory , 1980 .

[3]  Nikolai F. Rulkov,et al.  Modeling and synchronizing chaotic systems from experimental data , 1994 .

[4]  H. Schuster,et al.  Proper choice of the time delay for the analysis of chaotic time series , 1989 .

[5]  G. P. King,et al.  Extracting qualitative dynamics from experimental data , 1986 .

[6]  Alfonso M Albano,et al.  Phase-randomized surrogates can produce spurious identifications of non-random structure , 1994 .

[7]  Floris Takens,et al.  DETECTING NONLINEARITIES IN STATIONARY TIME SERIES , 1993 .

[8]  M. Rosenstein,et al.  Reconstruction expansion as a geometry-based framework for choosing proper delay times , 1994 .

[9]  D. T. Kaplan,et al.  Exceptional events as evidence for determinism , 1994 .

[10]  Pierre Bergé,et al.  Order within chaos : towards a deterministic approach to turbulence , 1984 .

[11]  Brown,et al.  Computing the Lyapunov spectrum of a dynamical system from an observed time series. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[12]  T. Sauer,et al.  Correlation dimension of attractors through interspike intervals , 1997 .

[13]  Rice,et al.  Method of false nearest neighbors: A cautionary note. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[14]  Kevin Judd,et al.  Reconstructing noisy dynamical systems by triangulations , 1997 .

[15]  Ulrich Parlitz,et al.  Identification of True and Spurious Lyapunov Exponents from Time Series , 1992 .

[16]  J. Fell,et al.  Resonance-like phenomena in Lyapunov calculations from data reconstructed by the time-delay method , 1994 .

[17]  Michael Rosenblum,et al.  Time series analysis for system identification and diagnostics , 1991 .

[18]  F. Takens.  Detecting strange attractors in turbulence , 1981 .

[19]  James Theiler,et al.  Using surrogate data to detect nonlinearity in time series , 1991 .

[20]  Henry D. I. Abarbanel,et al.  Analysis of Observed Chaotic Data , 1995 .

[21]  H. Kantz.  A robust method to estimate the maximal Lyapunov exponent of a time series , 1994 .

[22]  Ulrich Parlitz,et al.  Methods of chaos physics and their application to acoustics , 1988 .

[23]  P. Grassberger,et al.  On noise reduction methods for chaotic data. , 1993, Chaos.

[24]  R. Savit,et al.  Time series and dependent variables , 1991 .

[25]  L. Tsimring,et al.  The analysis of observed chaotic data in physical systems , 1993 .

[26]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

[27]  James Theiler,et al.  Testing for nonlinearity in time series: the method of surrogate data , 1992 .

[28]  Pfister,et al.  Optimal delay time and embedding dimension for delay-time coordinates by analysis of the global static and local dynamical behavior of strange attractors. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[29]  James B. Kadtke,et al.  GLOBAL DYNAMICAL EQUATIONS AND LYAPUNOV EXPONENTS FROM NOISY CHAOTIC TIME SERIES , 1993 .

[30]  H. Abarbanel,et al.  Lyapunov exponents from observed time series. , 1990, Physical review letters.

[31]  Markus Eiswirth,et al.  Computation of Lyapunov spectra: effect of interactive noise and application to a chemical oscillator , 1993 .

[32]  Stephen A. Billings,et al.  Identification of models for chaotic systems from noisy data: implications for performance and nonlinear filtering , 1995 .

[33]  Roger A. Pielke,et al.  EXTRACTING LYAPUNOV EXPONENTS FROM SHORT TIME SERIES OF LOW PRECISION , 1992 .

[34]  A. Fowler,et al.  A correlation function for choosing time delays in phase portrait reconstructions , 1993 .

[35]  A. Mees,et al.  On selecting models for nonlinear time series , 1995 .

[36]  Andrew M. Fraser,et al.  Information and entropy in strange attractors , 1989, IEEE Trans. Inf. Theory.

[37]  Fedor Mitschke,et al.  Estimation of Lyapunov exponents from time series: the stochastic case , 1993 .

[38]  K. Pawelzik,et al.  Optimal Embeddings of Chaotic Attractors from Topological Considerations , 1991 .

[39]  Ioannis G. Kevrekidis,et al.  DISCRETE- vs. CONTINUOUS-TIME NONLINEAR SIGNAL PROCESSING OF Cu ELECTRODISSOLUTION DATA , 1992 .

[40]  Zoran Aleksic,et al.  Estimating the embedding dimension , 1991 .

[41]  Gerd Pfister,et al.  Optimal Reconstruction of Strange Attractors from Purely Geometrical Arguments , 1990 .

[42]  D. Ruelle,et al.  Fundamental limitations for estimating dimensions and Lyapunov exponents in dynamical systems , 1992 .

[43]  Mark R. Muldoon,et al.  Linear Filters and Non‐Linear Systems , 1992 .

[44]  Ulrich Parlitz,et al.  Experimental Nonlinear Physics , 1997 .

[45]  Carroll,et al.  Synchronous chaos in coupled oscillator systems. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[46]  Antanas Cenys,et al.  Estimation of the number of degrees of freedom from chaotic time series , 1988 .

[47]  Antonio Politi,et al.  Hausdorff Dimension and Uniformity Factor of Strange Attractors , 1984 .

[48]  Eckmann,et al.  Liapunov exponents from time series. , 1986, Physical review. A, General physics.

[49]  Peter Grassberger,et al.  Generalizations of the Hausdorff dimension of fractal measures , 1985 .

[50]  Schreiber,et al.  Noise reduction in chaotic time-series data: A survey of common methods. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[51]  M. Rosenstein,et al.  A practical method for calculating largest Lyapunov exponents from small data sets , 1993 .

[52]  Francis C. Moon,et al.  Chaotic and fractal dynamics , 1992 .

[53]  E. Ott.  Chaos in Dynamical Systems: Contents , 1993 .

[54]  H. Fujisaka,et al.  Stability Theory of Synchronized Motion in Coupled-Oscillator Systems , 1983 .

[55]  J. D. Farmer,et al.  State space reconstruction in the presence of noise" Physica D , 1991 .

[56]  Ivan Dvořák,et al.  Singular-value decomposition in attractor reconstruction: pitfalls and precautions , 1992 .

[57]  L. Cao.  Practical method for determining the minimum embedding dimension of a scalar time series , 1997 .

[58]  George G. Szpiro.  Forecasting chaotic time series with genetic algorithms , 1997 .

[59]  Schreiber,et al.  Improved Surrogate Data for Nonlinearity Tests. , 1996, Physical review letters.

[60]  Mike E. Davies,et al.  Linear Recursive Filters and Nonlinear Dynamics , 1996 .

[61]  G. Broggi,et al.  Evaluation of dimensions and entropies of chaotic systems , 1988 .

[62]  Ulrich Parlitz,et al.  Comparison of Different Methods for Computing Lyapunov Exponents , 1990 .

[63]  Parlitz,et al.  Predicting low-dimensional spatiotemporal dynamics using discrete wavelet transforms. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[64]  Parlitz,et al.  Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems. , 1996, Physical review letters.

[65]  Passamante,et al.  Recognizing determinism in a time series. , 1993, Physical review letters.

[66]  R. Eykholt,et al.  Estimating the Lyapunov-exponent spectrum from short time series of low precision. , 1991, Physical review letters.

[67]  Martin Casdagli,et al.  Nonlinear prediction of chaotic time series , 1989 .

[68]  Bulsara,et al.  Array enhanced stochastic resonance and spatiotemporal synchronization. , 1995, Physical review letters.

[69]  Brown,et al.  Modeling and synchronizing chaotic systems from time-series data. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[70]  W. Singer.  Synchronization of cortical activity and its putative role in information processing and learning. , 1993, Annual review of physiology.

[71]  Farmer,et al.  Predicting chaotic time series. , 1987, Physical review letters.

[72]  Sauer,et al.  Reconstruction of dynamical systems from interspike intervals. , 1994, Physical review letters.

[73]  Gao,et al.  Direct dynamical test for deterministic chaos and optimal embedding of a chaotic time series. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[74]  James P. Crutchfield,et al.  Geometry from a Time Series , 1980 .

[75]  Thomas Schreiber,et al.  Constrained Randomization of Time Series Data , 1998, chao-dyn/9909042.

[76]  Schreiber,et al.  Signal separation by nonlinear projections: The fetal electrocardiogram. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[77]  Lauterborn,et al.  Liapunov exponents from a time series of acoustic chaos. , 1989, Physical review. A, General physics.

[78]  Joachim Holzfuss,et al.  Approach to error-estimation in the application of dimension algorithms , 1986 .

[79]  I. Stewart,et al.  Bubbling of attractors and synchronisation of chaotic oscillators , 1994 .

[80]  J. Yorke,et al.  HOW MANY DELAY COORDINATES DO YOU NEED , 1993 .

[81]  L. Sirovich.  Chaotic dynamics of coherent structures , 1989 .

[82]  A. Gallant,et al.  Convergence rates and data requirements for Jacobian-based estimates of Lyapunov exponents from data , 1991 .

[83]  Leonard A. Smith.  Identification and prediction of low dimensional dynamics , 1992 .

[84]  H. Abarbanel,et al.  LYAPUNOV EXPONENTS IN CHAOTIC SYSTEMS: THEIR IMPORTANCE AND THEIR EVALUATION USING OBSERVED DATA , 1991 .

[85]  U. Parlitz,et al.  Lyapunov exponents from time series , 1991 .

[86]  Ulrich Parlitz.  Lyapunov exponents from Chua's Circuit , 1993, J. Circuits Syst. Comput..

[87]  Parlitz,et al.  Encoding messages using chaotic synchronization. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[88]  James Theiler,et al.  On the evidence for low-dimensional chaos in an epileptic electroencephalogram , 1995 .

[89]  Theiler,et al.  Spurious dimension from correlation algorithms applied to limited time-series data. , 1986, Physical review. A, General physics.

[90]  Parlitz,et al.  Synchronization-based parameter estimation from time series. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[91]  Thomas Schreiber,et al.  EFFICIENT NEIGHBOR SEARCHING IN NONLINEAR TIME SERIES ANALYSIS , 1995 .

[92]  G. Benettin,et al.  Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 2: Numerical application , 1980 .

[93]  P. F. Meier,et al.  Evaluation of Lyapunov exponents and scaling functions from time series , 1988 .

[94]  Ljupco Kocarev,et al.  General approach for chaotic synchronization with applications to communication. , 1995, Physical review letters.

[95]  Yasuji Sawada,et al.  Practical Methods of Measuring the Generalized Dimension and the Largest Lyapunov Exponent in High Dimensional Chaotic Systems , 1987 .

[96]  J. Kurths,et al.  An attractor in a solar time series , 1987 .

[97]  Mees,et al.  Mutual information, strange attractors, and the optimal estimation of dimension. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[98]  Holger Kantz,et al.  Effective deterministic models for chaotic dynamics perturbed by noise , 1997 .

[99]  James Theiler,et al.  Estimating fractal dimension , 1990 .

[100]  Kurt Wiesenfeld,et al.  Disorder-enhanced synchronization , 1995 .

[101]  D. Broomhead,et al.  Takens embedding theorems for forced and stochastic systems , 1997 .

[102]  Martin Casdagli,et al.  An analytic approach to practical state space reconstruction , 1992 .

[103]  Carroll,et al.  Synchronization in chaotic systems. , 1990, Physical review letters.

[104]  P. Grassberger,et al.  NONLINEAR TIME SEQUENCE ANALYSIS , 1991 .

[105]  Peter W. Milonni,et al.  Dimensions and entropies in chaotic systems: Quantification of complex behavior , 1986 .

[106]  Schreiber,et al.  Nonlinear noise reduction: A case study on experimental data. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[107]  Sawada,et al.  Measurement of the Lyapunov spectrum from a chaotic time series. , 1985, Physical review letters.

[108]  Ljupco Kocarev,et al.  Subharmonic Entrainment of Unstable Period Orbits and Generalized Synchronization , 1997 .

[109]  Leonard A. Smith,et al.  Distinguishing between low-dimensional dynamics and randomness in measured time series , 1992 .

[110]  H. Abarbanel,et al.  Determining embedding dimension for phase-space reconstruction using a geometrical construction. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[111]  Cawley,et al.  Smoothness implies determinism: A method to detect it in time series. , 1994, Physical review letters.

[112]  K. Briggs.  An improved method for estimating Liapunov exponents of chaotic time series , 1990 .

[113]  André Longtin,et al.  Interspike interval attractors from chaotically driven neuron models , 1997 .

[114]  H. Kantz,et al.  Nonlinear time series analysis , 1997 .

[115]  Michael Peter Kennedy.  Chaos in the Colpitts oscillator , 1994 .