Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Attractor characterization from scaled doublet structures: simulations for small data sets

Abstract

Further developments are presented in the technique for analysing attractor behaviour from small data sets, based on the observation of scaled structures in families of slope curves of correlation integrals. Scaled doublet structures are investigated systematically for short time series obeying the Mackey and Glass delay differential equation. At an attractor correlation dimension close to 5, ranges of values of T (the length of the time sequence) and of f (the recording frequency) are described in which the scaled doublet structures are unambiguously identified and distinguished from structures that can occasionally be found with randomized time sequences. Implications for the characterization of low-dimension attractors, notably from electroencephalographic recordings, are discussed, including in particular the advantage to be gained from moderately oversampling the data.

This is a preview of subscription content, log in to check access.

References

  1. Babloyantz A, Nicolis C, Salazar M (1985) Evidence for chaotic dynamics of brain activity during the sleep cycle. Phys Lett A 111:152–156

  2. Cerf R (1993) Attractor-ruled dynamics in neurobiology: does it exist? can it be measured? In: Haken H, Mikhailov A (eds) Interdisciplinary approaches in nonlinear complex systems. Springer Series in Synergetics, Vol 62, Springer, Berlin Heidelberg New York, pp 201–214

  3. Cerf R, Ben Maati ML (1991) Trans-embedding-scaled dynamics. Phys Lett A 158:119–125

  4. Cerf R, Oumarrakchi M, Ben Maati ML, Sefrioui M (1992) Doublet-split-scaling of correlation integrals in non-linear dynamics and in neurobiology. Biol Cybern 68:115–124

  5. Farmer JD (1982) Chaotic attractors of an infinite dimensional dynamic system. Physica 4D:366–393

  6. Grassberger P, Procaccia I (1983) Characterization of strange attractors. Physica 9D:189–208

  7. Packard NH, Crutchfield JP, Farmer JD, Shaw RS (1980) Geometry from a time series. Phys Rev Lett 45:712–716

  8. Rapp PE, Zimmerman ID, Albano AM, Guzman GC de, Greenbaum NN, Bashore TR (1986) Experimental studies of chaotic neural behaviour: cellular activity and electroencephalographic signals. In: Othmer HG (ed) Non-linear oscillations in biology and chemistry. (Lecture Notes in Biomathematics, Vol 66) Springer, Berlin Heidelberg New York, pp 366–381

  9. Ruelle D (1981) Chemical kinetics and differentiable dynamic systems. In: Pacault A, Vidal C (eds) Nonlinear phenomena in chemical dynamics. (Springer Series in Synergetics, Vol 12) Springer, Berlin Heidelberg New York, pp 30–37

  10. Takens F (1981) Detecting strange attractors in turbulence. In: Rand DA, Young LS (eds) Dynamical systems and turbulence. (Lecture Notes in Mathematics, Vol 898) Springer, Berlin Heidelberg New York, pp 366–381

  11. Theiler J, Galdrikian B, Longtin A, Eubank S, Farmer JD (1992) Using surrogate data to detect nonlinearity in time series. In: Casdagli M, Eubank S (eds) Nonlinear modeling and forecasting. Addison-Wesley, New York, pp 163–182

Download references

Author information

Correspondence to R. Cerf.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Cerf, R., Ben Maati, M.L. Attractor characterization from scaled doublet structures: simulations for small data sets. Biol. Cybern. 72, 357–363 (1995). https://doi.org/10.1007/BF00202791

Download citation

Keywords

  • Differential Equation
  • Time Series
  • Time Sequence
  • Correlation Dimension
  • Attractor Behaviour