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zbMATH Open
Article . 2023
Data sources: zbMATH Open
https://dx.doi.org/10.48550/ar...
Article . 2023
License: CC BY NC ND
Data sources: Datacite
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Ordinal Poincaré sections: Reconstructing the first return map from an ordinal segmentation of time series

Ordinal Poincaré sections: reconstructing the first return map from an ordinal segmentation of time series
Authors: Zahra Shahriari; Shannon D. Algar; David M. Walker; Michael Small;

Ordinal Poincaré sections: Reconstructing the first return map from an ordinal segmentation of time series

Abstract

We propose a robust algorithm for constructing first return maps of dynamical systems from time series without the need for embedding. A first return map is typically constructed using a convenient heuristic (maxima or zero-crossings of the time series, for example) or a computationally nuanced geometric approach (explicitly constructing a Poincaré section from a hyper-surface normal to the flow and then interpolating to determine intersections with trajectories). Our method is based on ordinal partitions of the time series, and the first return map is constructed from successive intersections with specific ordinal sequences. We can obtain distinct first return maps for each ordinal sequence in general. We define entropy-based measures to guide our selection of the ordinal sequence for a “good” first return map and show that this method can robustly be applied to time series from classical chaotic systems to extract the underlying first return map dynamics. The results are shown for several well-known dynamical systems (Lorenz, Rössler, and Mackey–Glass in chaotic regimes).

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Keywords

Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.), FOS: Mathematics, Time series analysis of dynamical systems, Dynamical Systems (math.DS), Mathematics - Dynamical Systems

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
5
Top 10%
Average
Top 10%
Green