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...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
Public Library of Science (PLoS)
2018-01-01
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| Series: | PLoS ONE |
| Online Access: | http://europepmc.org/articles/PMC5905961?pdf=render |
| _version_ | 1818864353370177536 |
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| author | Asaf Levi Juan Sabuco Michael Small Miguel A F Sanjuán |
| author_facet | Asaf Levi Juan Sabuco Michael Small Miguel A F Sanjuán |
| author_sort | Asaf Levi |
| collection | DOAJ |
| 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. |
| first_indexed | 2024-12-19T10:30:18Z |
| format | Article |
| id | doaj.art-4b6af9b2b3be482a87c8ea78707f6bc4 |
| institution | Directory Open Access Journal |
| issn | 1932-6203 |
| language | English |
| last_indexed | 2024-12-19T10:30:18Z |
| publishDate | 2018-01-01 |
| publisher | Public Library of Science (PLoS) |
| record_format | Article |
| series | PLoS ONE |
| spelling | doaj.art-4b6af9b2b3be482a87c8ea78707f6bc42022-12-21T20:25:47ZengPublic Library of Science (PLoS)PLoS ONE1932-62032018-01-01134e019492610.1371/journal.pone.0194926From local uncertainty to global predictions: Making predictions on fractal basins.Asaf LeviJuan SabucoMichael SmallMiguel A F SanjuánIn 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.http://europepmc.org/articles/PMC5905961?pdf=render |
| spellingShingle | Asaf Levi Juan Sabuco Michael Small Miguel A F Sanjuán From local uncertainty to global predictions: Making predictions on fractal basins. PLoS ONE |
| 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 |
| url | http://europepmc.org/articles/PMC5905961?pdf=render |
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