What symbolic dynamics do we get with a misplaced partition? On the validity of threshold crossings analysis of chaotic time-series

Erik M. Bollt, Theodore Stanford, Ying-Cheng Lai, Karol Zyczkowski

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Abstract

An increasingly popular method of encoding chaotic time-series from physical experiments is the so-called threshold crossings technique, where one simply replaces the real valued data with symbolic data of relative positions to an arbitrary partition at discrete times. The implication has been that this symbolic encoding describes the original dynamical system. On the other hand, the literature on generating partitions of non-hyperbolic dynamical systems has shown that a good partition is non-trivial to find. It is believed that the generating partition of non-uniformly hyperbolic dynamical system connects "primary tangencies", which are generally not simple Lines as used by a threshold crossings. Therefore, we investigate consequences of using itineraries generated by a non-generating partition. We do most of our rigorous analysis using the tent map as a benchmark example, but show numerically that our results likely generalize. In summary, we find the misrepresentation of the dynamical system by "sample-path" symbolic dynamics of an arbitrary partition can be severe, including (sometimes extremely) diminished topological entropy, and a high degree of non-uniqueness. Interestingly, we find topological entropy as a function of misplacement to be devil's staircase-like, but surprisingly non-monotone. (C) 2001 Elsevier Science B.V. All right reserved.

Original languageEnglish
Pages (from-to)259-286
Number of pages28
JournalPhysica. D, Nonlinear Phenomena
Volume154
Issue number3-4
Early online date31 May 2001
DOIs
Publication statusPublished - 15 Jun 2001

Keywords

  • symbol dynamics
  • topological entropy
  • kneading theory
  • devil's staircase
  • communication
  • construction

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