### 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 language | English |
---|---|

Pages (from-to) | 259-286 |

Number of pages | 28 |

Journal | Physica. D, Nonlinear Phenomena |

Volume | 154 |

Issue number | 3-4 |

Early online date | 31 May 2001 |

DOIs | |

Publication status | Published - 15 Jun 2001 |

### Keywords

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

### Cite this

*Physica. D, Nonlinear Phenomena*,

*154*(3-4), 259-286. https://doi.org/10.1016/S0167-2789(01)00242-1

**What symbolic dynamics do we get with a misplaced partition? On the validity of threshold crossings analysis of chaotic time-series.** / Bollt, Erik M. ; Stanford, Theodore; Lai, Ying-Cheng; Zyczkowski, Karol.

Research output: Contribution to journal › Article

*Physica. D, Nonlinear Phenomena*, vol. 154, no. 3-4, pp. 259-286. https://doi.org/10.1016/S0167-2789(01)00242-1

}

TY - JOUR

T1 - What symbolic dynamics do we get with a misplaced partition? On the validity of threshold crossings analysis of chaotic time-series

AU - Bollt, Erik M.

AU - Stanford, Theodore

AU - Lai, Ying-Cheng

AU - Zyczkowski, Karol

PY - 2001/6/15

Y1 - 2001/6/15

N2 - 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.

AB - 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.

KW - symbol dynamics

KW - topological entropy

KW - kneading theory

KW - devil's staircase

KW - communication

KW - construction

U2 - 10.1016/S0167-2789(01)00242-1

DO - 10.1016/S0167-2789(01)00242-1

M3 - Article

VL - 154

SP - 259

EP - 286

JO - Physica. D, Nonlinear Phenomena

JF - Physica. D, Nonlinear Phenomena

SN - 0167-2789

IS - 3-4

ER -