Skip to main content
SpringerLink
Log in

Unfolding Complexity in Nonlinear Dynamical Systems

  • Chapter
Measures of Complexity and Chaos

Part of the book series: NATO ASI Series ((NSSB,volume 208))

Abstract

Nonlinear dynamical systems produce complex temporal or spatial patterns by stretching and folding regions of phase-space in an iterative way. The topological and metric properties of this process can be extracted from the chaotic signal by using symbolic dynamics. The underlying “grammatical rules” are systematically detected and arranged on a logic tree. Predictions on the set of possible outcomes of the system are made and compared with the observation. The discrepancy between the two, evaluated through a generalization of the information gain, characterizes the complexity of the source. As a result of this unfolding procedure, the dynamics is described as a sequence of deterministic paths (blocks of symbols) which appear at random in time, with given transition probabilities. Fast hierarchical evaluations of invariant measures, dimensions, entropies and Lyapunov exponents are obtained from the logic tree, considering lower and lower levels (i.e., increasingly long symbol-sequences). Power spectra are accurately reproduced with a limited number of short orbits. The analysis applies to any system: dissipative or conservative, hyperbolic or not, invertible or not.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. E.N. Lorenz, J.Atmos.Sci. 20, 130 (1963).

    Article  Google Scholar 

  2. D. Ruelle and F. Takens, Comm.Math.Phys. 20, 167 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  3. B.B. Mandelbrot, “The Fractal Geometry of Nature”, Freeman, San Francisco (1982).

    MATH  Google Scholar 

  4. J.P. Eckmann and D. Ruelle, Rev.Mod.Phys. 57, 617 (1985).

    Article  MathSciNet  Google Scholar 

  5. P. Grassberger, in proc. conf. on “Chaos in Astrophysics”, Palm Coast, Florida, 1984, J. Perdang et al. editors, Reidl, Dortrecht (1985).

    Google Scholar 

  6. Chaos and Complexity”, R. Livi, S. Ruffo, S. Ciliberto and M. Buiatti Eds., World Scientific, Singapore, 1988.

    Google Scholar 

  7. Proceedings of the conference on Complex Systems, Gwatt, Switzerland, 1988.

    Google Scholar 

  8. R. Badii, “Quantitative Characterization of Complexity and Predictability”, submitted for publication.

    Google Scholar 

  9. P. Grassberger, in ref7 and Wuppertal preprint B 9, 1988;

    Google Scholar 

  10. D. Zambella and P. Grassberger, Wuppertal preprint B 11, 1988.

    Google Scholar 

  11. A.N. Kolmogorov, Probl.Inform.Transm. 1, 1 (1965);

    Google Scholar 

  12. G. Chaitin, J. Assoc.Comp.Math. 13, 547 (1966).

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Lempel and J. Ziv, IEEE Trans.Inform.Theory 22, 75 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  14. L Procaccia, S. Thomae and C. Tresser, Phys.Rev. A35, 1884 (1987).

    Article  MathSciNet  Google Scholar 

  15. M.J. Feigenbaum, M.H. Jensen and I. Procaccia, Phys.Rev.Lett. 57, 1503 (1986).

    Article  MathSciNet  Google Scholar 

  16. M.H. Jensen, L.P. Kadanoff and I. Procaccia, Phys.Rev. A36, 1409 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  17. R. Badii, Riv. Nuovo Cim. 12, N° 3, 1 (1989).

    MathSciNet  Google Scholar 

  18. M.J. Feigenbaum, J.Stat.Phys. 52, 527 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  19. D.R. Hofstadter, “Gödel, Escher, Bach: an Eternal Golden Braid”, Vintage Books, New York (1980).

    Google Scholar 

  20. P. Grassberger and H. Kantz, Phys.Lett. 113A, 235 (1985).

    Article  MathSciNet  Google Scholar 

  21. R. Badii and G. Broggi, “Hierarchies of Relations between Partial Dimensions and Local Expansion Rates in Strange Attractors”, this issue.

    Google Scholar 

  22. V.M. Alekseev and M.V. Yakobson, Phys.Rep. 75, 290 (1981).

    Article  MathSciNet  Google Scholar 

  23. G. Györgyi and P. Szépfalusy, Phys.Rev. A31, 3477 (1985).

    MathSciNet  Google Scholar 

  24. A. Renyi, “Probability Theory”, North-Holland, Amsterdam (1970).

    Google Scholar 

  25. J.D. Farmer, E. Ott and J.A. Yorke, Physica 7D, 153 (1983).

    MathSciNet  Google Scholar 

  26. C. Grebogi, E. Ott and J.A. Yorke, Phys.Rev.Lett. 48, 1507 (1982).

    Article  MathSciNet  Google Scholar 

  27. R. Badii, unpublished.

    Google Scholar 

  28. M. Hénon, Comm.Math.Phys. 50, 69 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  29. P. Cvitanovie, G. Gunaratne and I. Procaccia, Phys.Rev. A38, 1503 (1988).

    Article  MathSciNet  Google Scholar 

  30. D. Auerbach, P. Cvitanovic, J.P. Eckmann, G.H. Gunaratne and I. Procaccia, Phys.Rev.Lett. 58, 2387 (1987).

    Article  MathSciNet  Google Scholar 

  31. C. Grebogi, E. Ott and J.A. Yorke, Phys.Rev. A37, 1711 (1988).

    MathSciNet  Google Scholar 

  32. P. Grassberger, R. Badii and A. Politi, J.Stat.Phys. 51. 135 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  33. M.A. Sepúlveda and R. Badii “Symbolic Dynamical Resolution of Power Spectra”, this issue.

    Google Scholar 

  34. A. Politi, Phys.Lett. A136, 374 (1989).

    Article  MathSciNet  Google Scholar 

  35. E. Ott, W. Withers and J.A. Yorke, J.Stat.Phys. 36, 687 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  36. P. Collet and J.P. Eckmann, “Iterated Maps on the Interval as Dynamical Systems”, Birkhauser, Cambridge, MA (1980).

    Google Scholar 

  37. M.J. Feigenbaum, J.Stat.Phys. 46, 919 and 925 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  38. J.D. Farmer and J.J. Sidorowich, Phys.Rev.Lett, 59, 845 (1987);

    Article  MathSciNet  Google Scholar 

  39. J.D. Farmer and J.J. Sidorowich, in “Evolution, Learning and Cognition”, Ed. Y.C. Lee, World Scientific, Singapore (1989);

    Google Scholar 

  40. J.P. Crutchfield and B.S. McNamara, Complex Systems 1, 417 (1987);

    MathSciNet  MATH  Google Scholar 

  41. J. Cremers and A. Hübler, Z.Naturforsch. 42a, 797 (1987);

    MathSciNet  Google Scholar 

  42. M. Casdagli, Physica D, to appear (1989).

    Google Scholar 

  43. M. Sano and Y. Sawada, Phys.Rev.Lett. 55, 1082 (1985);

    Article  MathSciNet  Google Scholar 

  44. J.P. Eckmann, S. Oliffson Kamphorst, D. Ruelle and S. Ciliberto, Phys.Rev. 34A, 4971 (1986).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fakultät für Physik, Universität Konstanz, 7750, Constance, West Germany

    R. Badii

Authors
  1. R. Badii
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Physics, Bryn Mawr College, 19010-2899, Bryn Mawr, Pennsylvania, USA

    Neal B. Abraham  & Alfonso M. Albano  & 

  2. Naval Air Development Center, 18974, Warminster, Pennsylvania, USA

    Anthony Passamante

  3. Department of Physiology and Biochemistry, The Medical College of Pennsylvania, 3300 Henry Ave., 19129, Philadelphia, Pennsylvania, USA

    Paul E. Rapp

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Plenum Press, New York

About this chapter

Cite this chapter

Badii, R. (1989). Unfolding Complexity in Nonlinear Dynamical Systems. In: Abraham, N.B., Albano, A.M., Passamante, A., Rapp, P.E. (eds) Measures of Complexity and Chaos. NATO ASI Series, vol 208. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-0623-9_46

Download citation

  • DOI: https://doi.org/10.1007/978-1-4757-0623-9_46

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4757-0625-3

  • Online ISBN: 978-1-4757-0623-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation