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From local uncertainty to global predictions: Making predictions on fractal basins

In nonlinear systems long term dynamics is governed by the attractors present in phase space. The presence of a chaotic saddle gives rise to basins of attraction with fractal boundaries and sometimes even to Wada boundaries. These two phenomena involve extreme difficulties in the prediction of the f...

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Detalles Bibliográficos
Autores principales: Levi, Asaf, Sabuco, Juan, Small, Michael, Sanjuán, Miguel A. F.
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Public Library of Science 2018
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5905961/
https://www.ncbi.nlm.nih.gov/pubmed/29668687
http://dx.doi.org/10.1371/journal.pone.0194926
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author Levi, Asaf
Sabuco, Juan
Small, Michael
Sanjuán, Miguel A. F.
author_facet Levi, Asaf
Sabuco, Juan
Small, Michael
Sanjuán, Miguel A. F.
author_sort Levi, Asaf
collection PubMed
description In nonlinear systems long term dynamics is governed by the attractors present in phase space. The presence of a chaotic saddle gives rise to basins of attraction with fractal boundaries and sometimes even to Wada boundaries. These two phenomena involve extreme difficulties in the prediction of the future state of the system. However, we show here that it is possible to make statistical predictions even if we do not have any previous knowledge of the initial conditions or the time series of the system until it reaches its final state. In this work, we develop a general method to make statistical predictions in systems with fractal basins. In particular, we have applied this new method to the Duffing oscillator for a choice of parameters where the system possesses the Wada property. We have computed the statistical properties of the Duffing oscillator for different phase space resolutions, to obtain information about the global dynamics of the system. The key idea is that the fraction of initial conditions that evolve towards each attractor is scale free—which we illustrate numerically. We have also shown numerically how having partial information about the initial conditions of the system does not improve in general the predictions in the Wada regions.
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spelling pubmed-59059612018-05-06 From local uncertainty to global predictions: Making predictions on fractal basins Levi, Asaf Sabuco, Juan Small, Michael Sanjuán, Miguel A. F. PLoS One Research Article In nonlinear systems long term dynamics is governed by the attractors present in phase space. The presence of a chaotic saddle gives rise to basins of attraction with fractal boundaries and sometimes even to Wada boundaries. These two phenomena involve extreme difficulties in the prediction of the future state of the system. However, we show here that it is possible to make statistical predictions even if we do not have any previous knowledge of the initial conditions or the time series of the system until it reaches its final state. In this work, we develop a general method to make statistical predictions in systems with fractal basins. In particular, we have applied this new method to the Duffing oscillator for a choice of parameters where the system possesses the Wada property. We have computed the statistical properties of the Duffing oscillator for different phase space resolutions, to obtain information about the global dynamics of the system. The key idea is that the fraction of initial conditions that evolve towards each attractor is scale free—which we illustrate numerically. We have also shown numerically how having partial information about the initial conditions of the system does not improve in general the predictions in the Wada regions. Public Library of Science 2018-04-18 /pmc/articles/PMC5905961/ /pubmed/29668687 http://dx.doi.org/10.1371/journal.pone.0194926 Text en © 2018 Levi et al http://creativecommons.org/licenses/by/4.0/ This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
spellingShingle Research Article
Levi, Asaf
Sabuco, Juan
Small, Michael
Sanjuán, Miguel A. F.
From local uncertainty to global predictions: Making predictions on fractal basins
title From local uncertainty to global predictions: Making predictions on fractal basins
title_full From local uncertainty to global predictions: Making predictions on fractal basins
title_fullStr From local uncertainty to global predictions: Making predictions on fractal basins
title_full_unstemmed From local uncertainty to global predictions: Making predictions on fractal basins
title_short From local uncertainty to global predictions: Making predictions on fractal basins
title_sort from local uncertainty to global predictions: making predictions on fractal basins
topic Research Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5905961/
https://www.ncbi.nlm.nih.gov/pubmed/29668687
http://dx.doi.org/10.1371/journal.pone.0194926
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