Conclusion and Outlook

  • José María AmigóEmail author
Part of the Springer Series in Synergetics book series (SSSYN)


Ordinal (or permutation-based) analysis of dynamical systems originates from the properties of the order relations and order isomorphisms. Thereby it is assumed that the state space of the systems is equipped with a total ordering. The order relations among consecutive elements in the orbits of deterministic or random dynamical systems are then codified in the form of ordinal patterns. The ordinal patterns themselves—whether admissible or forbidden—together with other “higher level” tools based on them, like permutation entropy rates, discrete entropy, frequency or probability distributions, regularity parameters, build the main repertoire of ordinal analysis. Since the sort of properties addressed by ordinal analysis and captured by its tools are not the same as in the usual measure-theoretical and topological approaches, we proposed the term “permutation complexity” to distinguish them.


Cellular Automaton Symbolic Dynamic Ordinal Analysis Bernoulli Shift Permutation Entropy 
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.


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  1. 29.
    C. Bandt, G. Keller, and B. Pompe, Entropy of interval maps via permutations. Nonlinearity 15 (2002) 1595–1602.zbMATHCrossRefMathSciNetADSGoogle Scholar
  2. 31.
    A. Berger, Chaos and Chance, Walter de Gruyter, Berlin, 2001.zbMATHGoogle Scholar
  3. 37.
    A. Boyarsky and P. Gora, Laws of Chaos. Birkhäuser, Boston, 1997.Google Scholar
  4. 52.
    G.H. Choe, Computational Ergodic Theory. Springer Verlag, Berlin, 2005.Google Scholar
  5. 67.
    M. Denker and W.A. Woyczynski, Introductory Statistics and Random Phenomena. Birkhäuser, Boston, 1998.Google Scholar
  6. 69.
    R.L. Devaney, Chaotic Dynamical Systems (2nd edition). Westview Press, Boulder, 2003.zbMATHGoogle Scholar
  7. 72.
    J.P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors, Review of Modern Physics 57 (1985) 617–656.Google Scholar
  8. 91.
    B. Hasselbaltt and A. Katok, A First Course in Dynamics. Cambridge University Press, Cambridge, 2003.Google Scholar
  9. 105.
    O. Jenkinson and M. Pollicott, Entropy, exponents and invariant densities for hyperbolic systems: Dependence and computation. In: M. Brin, B. Hasselblatt, and Y. Pesin (Eds.), Modern Dynamical Systems and Applications. pp. 365–384 Cambridge University Press, Cambridge, 2004.Google Scholar
  10. 115.
    S. Katok, p-adic Analysis compared with real. American Mathematical Society, Providence, 2007.Google Scholar
  11. 159.
    R. Monetti, W. Bunk, T. Aschenbrenner, and F. Jamitzky, Characterizing synchronization in time series using information measures extracted from symbolic representations, Physical Review E 79 (2009) 046207.CrossRefADSGoogle Scholar
  12. 177.
    D.J. Rudolph, Fundamentals of Measurable Dynamics. Oxford University Press, Oxford, 1990.Google Scholar
  13. 202.
    P. Walters, An Introduction to Ergodic Theory. Springer Verlag, New York, 2000.Google Scholar

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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Centro de Investigacion OperativaUniversidad Miguel HernandezElcheSpain

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