### Abstract

Original language | English |
---|---|

Article number | 023104 |

Number of pages | 5 |

Journal | Chaos |

Volume | 19 |

Issue number | 2 |

Early online date | 4 May 2009 |

DOIs | |

Publication status | Published - Jun 2009 |

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### Cite this

*Chaos*,

*19*(2), [023104]. https://doi.org/10.1063/1.3117151

**Recurrences determine the dynamics.** / Robinson, Geoffrey; Thiel, Marco.

Research output: Contribution to journal › Article

*Chaos*, vol. 19, no. 2, 023104. https://doi.org/10.1063/1.3117151

}

TY - JOUR

T1 - Recurrences determine the dynamics

AU - Robinson, Geoffrey

AU - Thiel, Marco

PY - 2009/6

Y1 - 2009/6

N2 - We show that under suitable assumptions, Poincaré recurrences of a dynamical system determine its topology in phase space. Therefore, dynamical systems with the same recurrences are dynamically equivalent. This conclusion can be drawn from a theorem proved in this paper which states that the recurrence matrix determines the topology of closed sets. The theorem states that if a set of points M is mapped onto another set N, such that two points in N are closer than some prescribed fixed distance if and only if the corresponding points in M are closer than some, in general different, prescribed fixed distance, then both sets are homeomorphic, i.e., identical up to a continuous change in the coordinate system. The theorem justifies a range of methods in nonlinear dynamics which are based on recurrence properties.

AB - We show that under suitable assumptions, Poincaré recurrences of a dynamical system determine its topology in phase space. Therefore, dynamical systems with the same recurrences are dynamically equivalent. This conclusion can be drawn from a theorem proved in this paper which states that the recurrence matrix determines the topology of closed sets. The theorem states that if a set of points M is mapped onto another set N, such that two points in N are closer than some prescribed fixed distance if and only if the corresponding points in M are closer than some, in general different, prescribed fixed distance, then both sets are homeomorphic, i.e., identical up to a continuous change in the coordinate system. The theorem justifies a range of methods in nonlinear dynamics which are based on recurrence properties.

U2 - 10.1063/1.3117151

DO - 10.1063/1.3117151

M3 - Article

VL - 19

JO - Chaos

JF - Chaos

SN - 1054-1500

IS - 2

M1 - 023104

ER -