Time-series analysis

The fractals of interest in this paper are sets of states attainable by a system whose dynamical behaviour is governed by nonlinear evolution equations. Experimentally such sets, normally associated with chaotic dynamics, can occur in chemical reaction systems, fluid flows, electronic oscillators, driven neurons: the list has been growing rapidly over the past 10-15 years. Experimental studies of these systems generally produce time series of measurements that have no obvious fractal properties. Classical signal-processing methods provide little help. However, recently, techniques have been developed that are capable of extracting geometrical information from time-series data. This paper will review some of these methods and their application to the study of the geometry of invariant sets underlying the dynamics and bifurcations of some experimental and model systems.

[1]  Lai-Sang Young,et al.  Dimension formula for random transformations , 1988 .

[2]  David S. Broomhead,et al.  Phase portraits from a time series: A singular system approach , 1987 .

[3]  G. P. King,et al.  Topological dimension and local coordinates from time series data , 1987 .

[4]  Françoise Argoul,et al.  Experimental evidence for homoclinic chaos in the Belousov-Zhabotinskii reaction , 1987 .

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

[6]  Jensen,et al.  Erratum: Fractal measures and their singularities: The characterization of strange sets , 1986, Physical review. A, General physics.

[7]  Gottfried Mayer-Kress,et al.  Dimensions and Entropies in Chaotic Systems , 1986 .

[8]  Jensen,et al.  Global universality at the onset of chaos: Results of a forced Rayleigh-Benard experiment. , 1985, Physical review letters.

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

[10]  A. Murphree.  Homocystinuria treated with pyridoxine , 1984 .

[11]  H. G. E. Hentschel,et al.  The infinite number of generalized dimensions of fractals and strange attractors , 1983 .

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

[13]  J. Yorke,et al.  Dimension of chaotic attractors , 1982 .

[14]  L. Young.  Dimension, entropy and Lyapunov exponents , 1982, Ergodic Theory and Dynamical Systems.

[15]  Josef Kittler,et al.  Pattern recognition : a statistical approach , 1982 .

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

[17]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .