Predicting the bounds of large chaotic systems using low-dimensional manifolds.
Predicting extrema of chaotic systems in high-dimensional phase space remains a challenge. Methods, which give extrema that are valid in the long term, have thus far been restricted to models of only a few variables. Here, a method is presented which treats extrema of chaotic systems as belonging to...
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Public Library of Science (PLoS)
2017-01-01
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| Series: | PLoS ONE |
| Online Access: | https://journals.plos.org/plosone/article/file?id=10.1371/journal.pone.0179507&type=printable |
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| author | Asger M Haugaard |
| author_facet | Asger M Haugaard |
| author_sort | Asger M Haugaard |
| collection | DOAJ |
| description | Predicting extrema of chaotic systems in high-dimensional phase space remains a challenge. Methods, which give extrema that are valid in the long term, have thus far been restricted to models of only a few variables. Here, a method is presented which treats extrema of chaotic systems as belonging to discretised manifolds of low dimension (low-D) embedded in high-dimensional (high-D) phase space. As a central feature, the method exploits that strange attractor dimension is generally much smaller than parent system phase space dimension. This is important, since the computational cost associated with discretised manifolds depends exponentially on their dimension. Thus, systems that would otherwise be associated with tremendous computational challenges, can be tackled on a laptop. As a test, bounding manifolds are calculated for high-D modifications of the canonical Duffing system. Parameters can be set such that the bounding manifold displays harmonic behaviour even if the underlying system is chaotic. Thus, solving for one post-transient forcing cycle of the bounding manifold predicts the extrema of the underlying chaotic problem indefinitely. |
| format | Article |
| id | doaj-art-798ce8e5c9e44a82b7deca2fc3032dee |
| institution | DOAJ |
| issn | 1932-6203 |
| language | English |
| publishDate | 2017-01-01 |
| publisher | Public Library of Science (PLoS) |
| record_format | Article |
| series | PLoS ONE |
| spelling | doaj-art-798ce8e5c9e44a82b7deca2fc3032dee2025-08-20T03:04:39ZengPublic Library of Science (PLoS)PLoS ONE1932-62032017-01-01126e017950710.1371/journal.pone.0179507Predicting the bounds of large chaotic systems using low-dimensional manifolds.Asger M HaugaardPredicting extrema of chaotic systems in high-dimensional phase space remains a challenge. Methods, which give extrema that are valid in the long term, have thus far been restricted to models of only a few variables. Here, a method is presented which treats extrema of chaotic systems as belonging to discretised manifolds of low dimension (low-D) embedded in high-dimensional (high-D) phase space. As a central feature, the method exploits that strange attractor dimension is generally much smaller than parent system phase space dimension. This is important, since the computational cost associated with discretised manifolds depends exponentially on their dimension. Thus, systems that would otherwise be associated with tremendous computational challenges, can be tackled on a laptop. As a test, bounding manifolds are calculated for high-D modifications of the canonical Duffing system. Parameters can be set such that the bounding manifold displays harmonic behaviour even if the underlying system is chaotic. Thus, solving for one post-transient forcing cycle of the bounding manifold predicts the extrema of the underlying chaotic problem indefinitely.https://journals.plos.org/plosone/article/file?id=10.1371/journal.pone.0179507&type=printable |
| spellingShingle | Asger M Haugaard Predicting the bounds of large chaotic systems using low-dimensional manifolds. PLoS ONE |
| title | Predicting the bounds of large chaotic systems using low-dimensional manifolds. |
| title_full | Predicting the bounds of large chaotic systems using low-dimensional manifolds. |
| title_fullStr | Predicting the bounds of large chaotic systems using low-dimensional manifolds. |
| title_full_unstemmed | Predicting the bounds of large chaotic systems using low-dimensional manifolds. |
| title_short | Predicting the bounds of large chaotic systems using low-dimensional manifolds. |
| title_sort | predicting the bounds of large chaotic systems using low dimensional manifolds |
| url | https://journals.plos.org/plosone/article/file?id=10.1371/journal.pone.0179507&type=printable |
| work_keys_str_mv | AT asgermhaugaard predictingtheboundsoflargechaoticsystemsusinglowdimensionalmanifolds |