Recurrences determine the dynamics

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Abstract

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.
Original languageEnglish
Article number023104
Number of pages5
JournalChaos
Volume19
Issue number2
Early online date4 May 2009
DOIs
Publication statusPublished - Jun 2009

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Recurrence
Dynamical systems
theorems
Topology
dynamical systems
topology
Dynamical system
Theorem
Homeomorphic
Closed set
Justify
Set of points
Nonlinear Dynamics
Phase Space
matrices
If and only if
Range of data

Cite this

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

In: Chaos, Vol. 19, No. 2, 023104, 06.2009.

Research output: Contribution to journalArticle

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