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Self-Similarity in Hyperchaotic Data

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Abstract

The brain is a giant dynamic system. Its complexity should exceed e.g. that of a dripping faucet (Rössler 1977), which is one of the work-horses of nonlinear science (Martien and Shaw 1985). There are numerical methods for estimating the embedding dimension of a signal generated by a dynamic system (Froehling et al. 1981) which have been successfully applied to many kinds of data including brain-generated ones (see Mayer-Kress 1986 for many references).

Originally published in Başar E, Bullock TH (eds) Brain dynamics. Springer, Berlin Heidelberg New York, pp 113-121 (Springer series in brain dynamics, vol 2). Cross references refer to that volume.

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References

  • Albano AM, Mees AI, de Guzman GC, Rapp P (1987) Data requirements for reliable estimation of correlation dimensions. In: Degn H, Holden AV, Olsen LF (eds) Chaos in biological systems. Plenum, New York, pp 207–220

    Google Scholar 

  • Braitenberg V (1984) Vehicles. MIT Press, Cambridge, p 65

    Google Scholar 

  • Crutchfield JP, Packard NH (1982) Symbolic dynamics of one-dimensional maps: entropies, finite precision and noise. Int J Theor Phys 21:433–466

    Article  Google Scholar 

  • Farmer D (1981) Unpublished preprint

    Google Scholar 

  • Froehling H, Crutchfield JP, Farmer D, Packard NH, Shaw R (1981) On determining the dimension of chaotic flows. Physica 3D:605–617

    Google Scholar 

  • Glass L, Mackey MC (1979) Pathological conditions resulting from instabilities in physiological control systems. Ann NY Acad Sci 316:214–235

    Article  PubMed  CAS  Google Scholar 

  • Hénon M (1976) A two-dimensional mapping with a strange attractor. Commun Math Phys 50:69–77

    Article  Google Scholar 

  • Killory H, Rossler OE, Hudson JL (1987) Higher chaos in a four-variable chemical reaction model. Phys Lett 122A:341–345

    Google Scholar 

  • Mandelbrot B (1982) The fractal geometry of nature. Freeman, San Francisco

    Google Scholar 

  • Martien P, Shaw R (1985) The dripping faucet. Phys Lett 110A:399–402

    Google Scholar 

  • Mayer-Kress G (ed) (1986) Dimensions and entropies in chaotic systems. Quantification of complex behavior. Springer, Berlin Heidelberg New York (Springer series in synergetics, vol 32)

    Google Scholar 

  • Mira C (1987) Chaotic dynamics, from the one-dimensional endomorphism to the two-dimensional diffeomorphism. World Scientific, Singapore

    Google Scholar 

  • Rössler OE (1976) Chaotic behavior in simple reaction systems. Z Naturforsch 31a:259–264

    Google Scholar 

  • Rössler OE (1977) Chemical turbulence — a synopsis. In: Haken H (ed) Synergetics. Springer, Berlin Heidelberg New York (Springer series in synergetics, vol 1)

    Google Scholar 

  • Rössler OE (1979) An equation for hyperchaos. Phys Lett 71A:155–157

    Google Scholar 

  • Rössler OE (1980) Chaos and turbulence. In: Haken H (ed) Dynamics of synergetic systems. Springer, Berlin Heidelberg New York, pp 147–153 (Springer series in synergetics)

    Chapter  Google Scholar 

  • Rössler OE (1981) An artificial cognitive-plus-motivational system. Prog Theor Biol 6:147–160

    Google Scholar 

  • Rössler OE (1983) The chaotic hierarchy. Z Naturforsch 38a:788–801

    Google Scholar 

  • Rössler OE (1987a) Anaxagoras’ idea of the infinitely exact chaos. In: Marx G (ed) Teaching nonlinear phenomena, vol 2. Chaos in education. Hungarian Publications in Physics Education, National Center for Educational Technology, Veszprém, Hungary, pp 99–113

    Google Scholar 

  • Rössler OE (1987b) Chaos in coupled optimizers. Ann NY Acad Sci 504:229–240

    Article  PubMed  Google Scholar 

  • Rössler OE, Kahlert C, Parisi J, Peinke J, Röhricht B (1986) Hyperchaos and Julia sets. Z Na-turforsch 41a:819–822

    Google Scholar 

  • Rössler OE, Klein M, Hudson JL, Wais R (1988) Self-similar basin boundary in an invertible system (folded-towel map). In: Kelso JAS, Mandell AJ, Shlesinger MF (eds) Dynamic patterns in complex systems. World Scientific, Singapore, pp 209–218

    Google Scholar 

  • Ruelle D, Takens F (1971) On the nature of turbulence. Commun Math Phys 20:167–195

    Article  Google Scholar 

  • Smale S (1967) On differentiable dynamical systems. Bull Am Math Soc 73:747–829

    Article  Google Scholar 

  • Stroop R, Meier PF (1988) Evaluation of Lyapunov exponents and scaling function from time series. J Opt Soc Am [B] 5:1037–1045

    Article  Google Scholar 

  • Todt DJ (1969) On the control of irregular behavior sequences: results of an analysis of the song of the blackbird, Turdus merula (in German) In: Marco H, Farber G (eds) Kybernetik 1968. Oldenbourg, Munich, pp 465–485

    Google Scholar 

  • Winfree AT (1980) The geometry of biological time. Springer, Berlin Heidelberg New York

    Google Scholar 

  • Wolf A, Swift JB, Swinney HL, Vastano JA (1985) Determining Lyapunov exponents from a time series. Physica 16D:285–317

    CAS  Google Scholar 

Download references

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© 1990 Springer-Verlag Berlin Heidelberg

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Rössler, O.E., Hudson, J.L. (1990). Self-Similarity in Hyperchaotic Data. In: Başar, E. (eds) Chaos in Brain Function. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-75545-3_6

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  • DOI: https://doi.org/10.1007/978-3-642-75545-3_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-52329-1

  • Online ISBN: 978-3-642-75545-3

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