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Decomposing the dynamics of the Lorenz 1963 model using unstable periodic orbits: Averages, transitions, and quasi-invariant sets

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posted on 2024-02-28, 15:55 authored by CC Maiocchi, V Lucarini, A Gritsun
Unstable periodic orbits (UPOs) are a valuable tool for studying chaotic dynamical systems, as they allow one to distill their dynamical structure. We consider here the Lorenz 1963 model with the classic parameters' value. We investigate how a chaotic trajectory can be approximated using a complete set of UPOs up to symbolic dynamics' period 14. At each instant, we rank the UPOs according to their proximity to the position of the orbit in the phase space. We study this process from two different perspectives. First, we find that longer period UPOs overwhelmingly provide the best local approximation to the trajectory. Second, we construct a finite-state Markov chain by studying the scattering of the orbit between the neighborhood of the various UPOs. Each UPO and its neighborhood are taken as a possible state of the system. Through the analysis of the subdominant eigenvectors of the corresponding stochastic matrix, we provide a different interpretation of the mixing processes occurring in the system by taking advantage of the concept of quasi-invariant sets.

History

Author affiliation

College of Science & Engineering/Comp' & Math' Sciences

Version

  • AM (Accepted Manuscript)

Published in

Chaos

Volume

32

Issue

3

Pagination

033129

Publisher

AIP Publishing

issn

1054-1500

eissn

1089-7682

Copyright date

2022

Available date

2024-02-28

Spatial coverage

United States

Language

eng

Deposited by

Professor Valerio Lucarini

Deposit date

2024-02-26

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