Modeling and synchronizing chaotic systems from time-series data.

The problem of obtaining ordinary differential equations (ODE's) from chaotic time-series data is addressed. The vector fields for the ODE's are polynomials constructed from a basis set that is orthonormal on the data. The method for constructing the model is similar to the integration of ODE's using Adams predictor-corrector integration. The method is compared to the usual Euler model and is shown to be accurate for much larger sampling intervals. In addition, the method used to construct the model is capable of determining the optimal polynomial vector field for the given data. Finally, we demonstrate that it is possible to synchronize (in the sense of Fujisaka and Yamada [Prog. Theor. Phys. 69, 32 (1983)] as well as Pecora and Carroll [Phys. Rev. Lett. 64, 821 (1990); Phys. Rev. A 44, 2374 (1991)]) the model to a time series. Synchronization is used as a nontrivial test to determine how close the model vector field is to the true vector field. Implications and possible applications of synchronization are discussed.

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

[2]  He,et al.  Analysis and synthesis of synchronous periodic and chaotic systems. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[3]  R. W. Rollins,et al.  Chaotic dynamics in a model of metal passivation , 1993 .

[4]  Gouesbet.  Reconstruction of vector fields: The case of the Lorenz system. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[5]  J. Rissanen.  A UNIVERSAL PRIOR FOR INTEGERS AND ESTIMATION BY MINIMUM DESCRIPTION LENGTH , 1983 .

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

[7]  F. Albahadily,et al.  Mixed‐mode oscillations in an electrochemical system. I. A Farey sequence which does not occur on a torus , 1989 .

[8]  J. L. Hudson,et al.  Shil'nikov chaos during copper electrodissolution , 1988 .

[9]  Breeden,et al.  Reconstructing equations of motion from experimental data with unobserved variables. , 1990, Physical review. A, Atomic, molecular, and optical physics.

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

[11]  Gouesbet.  Reconstruction of the vector fields of continuous dynamical systems from numerical scalar time series. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[12]  Fraser,et al.  Independent coordinates for strange attractors from mutual information. , 1986, Physical review. A, General physics.

[13]  H. Abarbanel,et al.  Prediction in chaotic nonlinear systems: Methods for time series with broadband Fourier spectra. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[14]  Gabriel B. Mindlin,et al.  Topological analysis of chaotic time series data from the Belousov-Zhabotinskii reaction , 1991 .

[15]  N. Rulkov,et al.  MUTUAL SYNCHRONIZATION OF CHAOTIC SELF-OSCILLATORS WITH DISSIPATIVE COUPLING , 1992 .

[16]  M. Schell,et al.  Bistability and oscillations in the electrocatalyzed oxidation of formaldehyde , 1990 .

[17]  J. L. Hudson,et al.  Quasiperiodicity and chaos during an electrochemical reaction , 1989 .

[18]  M. A. Miller.  Nonuniqueness of inverse problems in macroelectrodynamics. Spherical and toroidal sources of electromagnetic fields (review) , 1986 .

[19]  Eric J. Kostelich,et al.  Problems in estimating dynamics from data , 1992 .

[20]  Lieberman,et al.  Synchronization of regular and chaotic systems. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[21]  J. L. Hudson,et al.  Oscillations during the electrocatalytic reduction of hydrogen peroxide on a platinum electrode , 1990 .

[22]  Tambe,et al.  Driving systems with chaotic signals. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[23]  D. Broomhead,et al.  Radial Basis Functions, Multi-Variable Functional Interpolation and Adaptive Networks , 1988 .

[24]  M. Rabinovich,et al.  Stochastic synchronization of oscillation in dissipative systems , 1986 .

[25]  E. Kostelich,et al.  Characterization of an experimental strange attractor by periodic orbits. , 1989, Physical review. A, General physics.

[26]  Ioannis G. Kevrekidis,et al.  Nonlinear signal processing and system identification: applications to time series from electrochemical reactions , 1990 .

[27]  J. Rissanen,et al.  Modeling By Shortest Data Description* , 1978, Autom..

[28]  Carroll,et al.  Driving systems with chaotic signals. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[29]  L. O. Chua,et al.  The double scroll family. I: Rigorous of chaos. II: Rigorous analysis of bifurcation phenomena , 1986 .

[30]  Giona,et al.  Functional reconstruction and local prediction of chaotic time series. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[31]  Stephen M. Hammel,et al.  A noise reduction method for chaotic systems , 1990 .

[32]  Jiménez,et al.  Forecasting on chaotic time series: A local optimal linear-reconstruction method. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[33]  D. Ruelle,et al.  Ergodic theory of chaos and strange attractors , 1985 .

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

[35]  T. Sauer.  A noise reduction method for signals from nonlinear systems , 1992 .

[36]  Kowalski,et al.  Asymptotically synchronous chaotic orbits in systems of excitable elements. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[37]  Baake,et al.  Fitting ordinary differential equations to chaotic data. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[38]  Mark Schell,et al.  Mixed‐mode oscillations in an electrochemical system. II. A periodic–chaotic sequence , 1989 .

[39]  David E. Rumelhart,et al.  Predicting the Future: a Connectionist Approach , 1990, Int. J. Neural Syst..

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

[41]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[42]  H. Swinney,et al.  Observation of a strange attractor , 1983 .

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

[44]  Brown.  Calculating Lyapunov exponents for short and/or noisy data sets. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

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