Unfolding Complexity in Nonlinear Dynamical Systems

  • R. Badii
Part of the NATO ASI Series book series (NSSB, volume 208)


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.


Periodic Orbit Lyapunov Exponent Information Gain Nonlinear Dynamical System Logic Tree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    E.N. Lorenz, J.Atmos.Sci. 20, 130 (1963).CrossRefGoogle Scholar
  2. [2]
    D. Ruelle and F. Takens, Comm.Math.Phys. 20, 167 (1971).MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    B.B. Mandelbrot, “The Fractal Geometry of Nature”, Freeman, San Francisco (1982).zbMATHGoogle Scholar
  4. [4]
    J.P. Eckmann and D. Ruelle, Rev.Mod.Phys. 57, 617 (1985).MathSciNetCrossRefGoogle Scholar
  5. [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. [6]
    Chaos and Complexity”, R. Livi, S. Ruffo, S. Ciliberto and M. Buiatti Eds., World Scientific, Singapore, 1988.Google Scholar
  7. [7]
    Proceedings of the conference on Complex Systems, Gwatt, Switzerland, 1988.Google Scholar
  8. [8]
    R. Badii, “Quantitative Characterization of Complexity and Predictability”, submitted for publication.Google Scholar
  9. [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. [10]
    A.N. Kolmogorov, Probl.Inform.Transm. 1, 1 (1965);Google Scholar
  12. G. Chaitin, J. Assoc.Comp.Math. 13, 547 (1966).MathSciNetzbMATHCrossRefGoogle Scholar
  13. [11]
    A. Lempel and J. Ziv, IEEE Trans.Inform.Theory 22, 75 (1976).MathSciNetzbMATHCrossRefGoogle Scholar
  14. [12]
    L Procaccia, S. Thomae and C. Tresser, Phys.Rev. A35, 1884 (1987).MathSciNetCrossRefGoogle Scholar
  15. [13]
    M.J. Feigenbaum, M.H. Jensen and I. Procaccia, Phys.Rev.Lett. 57, 1503 (1986).MathSciNetCrossRefGoogle Scholar
  16. [14]
    M.H. Jensen, L.P. Kadanoff and I. Procaccia, Phys.Rev. A36, 1409 (1987).MathSciNetzbMATHCrossRefGoogle Scholar
  17. [15]
    R. Badii, Riv. Nuovo Cim. 12, N° 3, 1 (1989).MathSciNetGoogle Scholar
  18. [16]
    M.J. Feigenbaum, J.Stat.Phys. 52, 527 (1988).MathSciNetzbMATHCrossRefGoogle Scholar
  19. [17]
    D.R. Hofstadter, “Gödel, Escher, Bach: an Eternal Golden Braid”, Vintage Books, New York (1980).Google Scholar
  20. [18]
    P. Grassberger and H. Kantz, Phys.Lett. 113A, 235 (1985).MathSciNetCrossRefGoogle Scholar
  21. [19]
    R. Badii and G. Broggi, “Hierarchies of Relations between Partial Dimensions and Local Expansion Rates in Strange Attractors”, this issue.Google Scholar
  22. [20]
    V.M. Alekseev and M.V. Yakobson, Phys.Rep. 75, 290 (1981).MathSciNetCrossRefGoogle Scholar
  23. [21]
    G. Györgyi and P. Szépfalusy, Phys.Rev. A31, 3477 (1985).MathSciNetGoogle Scholar
  24. [22]
    A. Renyi, “Probability Theory”, North-Holland, Amsterdam (1970).Google Scholar
  25. [23]
    J.D. Farmer, E. Ott and J.A. Yorke, Physica 7D, 153 (1983).MathSciNetGoogle Scholar
  26. [24]
    C. Grebogi, E. Ott and J.A. Yorke, Phys.Rev.Lett. 48, 1507 (1982).MathSciNetCrossRefGoogle Scholar
  27. [25]
    R. Badii, unpublished.Google Scholar
  28. [26]
    M. Hénon, Comm.Math.Phys. 50, 69 (1976).MathSciNetzbMATHCrossRefGoogle Scholar
  29. [27]
    P. Cvitanovie, G. Gunaratne and I. Procaccia, Phys.Rev. A38, 1503 (1988).MathSciNetCrossRefGoogle Scholar
  30. [28]
    D. Auerbach, P. Cvitanovic, J.P. Eckmann, G.H. Gunaratne and I. Procaccia, Phys.Rev.Lett. 58, 2387 (1987).MathSciNetCrossRefGoogle Scholar
  31. [29]
    C. Grebogi, E. Ott and J.A. Yorke, Phys.Rev. A37, 1711 (1988).MathSciNetGoogle Scholar
  32. [30]
    P. Grassberger, R. Badii and A. Politi, J.Stat.Phys. 51. 135 (1988).MathSciNetzbMATHCrossRefGoogle Scholar
  33. [31]
    M.A. Sepúlveda and R. Badii “Symbolic Dynamical Resolution of Power Spectra”, this issue.Google Scholar
  34. [32]
    A. Politi, Phys.Lett. A136, 374 (1989).MathSciNetCrossRefGoogle Scholar
  35. [33]
    E. Ott, W. Withers and J.A. Yorke, J.Stat.Phys. 36, 687 (1984).MathSciNetzbMATHCrossRefGoogle Scholar
  36. [34]
    P. Collet and J.P. Eckmann, “Iterated Maps on the Interval as Dynamical Systems”, Birkhauser, Cambridge, MA (1980).Google Scholar
  37. [35]
    M.J. Feigenbaum, J.Stat.Phys. 46, 919 and 925 (1987).MathSciNetzbMATHCrossRefGoogle Scholar
  38. [36]
    J.D. Farmer and J.J. Sidorowich, Phys.Rev.Lett, 59, 845 (1987);MathSciNetCrossRefGoogle 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);MathSciNetzbMATHGoogle Scholar
  41. J. Cremers and A. Hübler, Z.Naturforsch. 42a, 797 (1987);MathSciNetGoogle Scholar
  42. M. Casdagli, Physica D, to appear (1989).Google Scholar
  43. [37]
    M. Sano and Y. Sawada, Phys.Rev.Lett. 55, 1082 (1985);MathSciNetCrossRefGoogle Scholar
  44. J.P. Eckmann, S. Oliffson Kamphorst, D. Ruelle and S. Ciliberto, Phys.Rev. 34A, 4971 (1986).MathSciNetCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1989

Authors and Affiliations

  • R. Badii
    • 1
  1. 1.Fakultät für PhysikUniversität KonstanzConstanceWest Germany

Personalised recommendations