Journal of Korea Water Resources Association (한국수자원학회논문집)

Korea Water Resources Association (한국수자원학회)
권호 표지

Chaos and Correlation Dimension

  • Published : 2000.05.01
Download PDF
⟨ Previous Next ⟩

Abstract

The method of delays is widely used for reconstruction chaotic attractors from experimental observations. Many studies have used a fixed delay time ${\tau}_d$ as the embedding dimension m is increased, but this is not necessarily the best choice for obtaining good convergence of the correlation dimension. Recently, some researchers have suggested that it is better to fix the delay time window ${\tau}_w$ instead. Unfortunately, ${\tau}_w$ cannot be estimated using either the autocorrelation function or the mutual information, and no standard procedure for estimating ${\tau}_w$ has yet emerged. However, a new technique, called the C-C method, can be used to estimate either ${\tau}_d\;or\;{\tau}_w$. Using this method, we show that, for small data sets, fixing ${\tau}_w$, rather than ${\tau}_d$, does indeed lead to a more rapid convergence of the correlation dimension as the embedding dimension m in increased.

Keywords

References

  1. Phys. Rev. Lett. v.45 Geometry from a time series Packard N.H.;Crutchfield J.P.;Farmer J.D.;Shaw R.S.
  2. Lecture Notes in Mathematics 898 Detecting strange attractors in turbulence in Dunamical Systems and Turbulence Takens F.;Rand D.A.(ed.);Young L.S.(ed.)
  3. Physica D v.51 State space reconstruction in the Presence of noise Casdagli M.;Eubank S.;Farmer J.D.;Gibson J.
  4. Physica D v.127 Nonlinear dynamics, delay times, and embedding windows Kim H.S.;Eyholt R.;Salas J.D.
  5. Physica D v.20 Extracting qualitative dynamics from experimental data Broomhead D.S.;King G.P.
  6. Phys. Rev. A v.38 Singular-value decomposition and the Grassberger-Procaccia algorithm Albano A.M.;Muench J.;Schwartz C.;Mees A.I.;Rapp P.E.
  7. Physica D v.73 Reconstruction expansion as a geometry-based framework for choosing proper delay times Rosenstein M.T.;Collins J.J.;De Luca C.J.
  8. Physica D v.95 no.13 State space reconstruction parameters in the analysis of chaotic time serise-the role of the time window length Kugiumtzis D.B.
  9. Phys. Rev. A v.45 Mutual information, strange attractors, and the optimal estimation of dimension Martinerie J.M.;Albano A.M.;Mees A.I.;Rapp P.E.
  10. Physica D v.85 Remark on metric analysis of reconstructed dynamics from chaotic time series Wu Z.B.
  11. Physica D v.7 Measuring the strangeness of strange attractors Grassberger P.;Procaccia I.
  12. Nonlinear Dynamics Chaos, and Instability Statistical Theory and Economic Evidence Brock W.A.;Hsieh;Lebaron B.
  13. Econ. Rev. v.15 A test for independence based on the correlation dimension Brock W.A.;Dechert W.D.;Scheinkman J.A.;LeBaron B.
  14. Sov. Phys. JETP v.50 Stochastic self-modulation of waves in nonequilibrium media Rabinovich M.I.;Fabrikant A.L.