Timing of transients: quantifying reaching times and transient behavior in complex systems
In dynamical systems, one may ask how long it takes for a trajectory to reach the attractor, i.e. how long it spends in the transient phase. Although for a single trajectory the mathematically precise answer may be infinity, it still makes sense to compare different trajectories and quantify which o...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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IOP Publishing
2017-01-01
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| Series: | New Journal of Physics |
| Subjects: | |
| Online Access: | https://doi.org/10.1088/1367-2630/aa7b61 |
| _version_ | 1797750635932680192 |
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| author | Tim Kittel Jobst Heitzig Kevin Webster Jürgen Kurths |
| author_facet | Tim Kittel Jobst Heitzig Kevin Webster Jürgen Kurths |
| author_sort | Tim Kittel |
| collection | DOAJ |
| description | In dynamical systems, one may ask how long it takes for a trajectory to reach the attractor, i.e. how long it spends in the transient phase. Although for a single trajectory the mathematically precise answer may be infinity, it still makes sense to compare different trajectories and quantify which of them approaches the attractor earlier. In this article, we categorize several problems of quantifying such transient times. To treat them, we propose two metrics, area under distance curve and regularized reaching time, that capture two complementary aspects of transient dynamics. The first, area under distance curve, is the distance of the trajectory to the attractor integrated over time. It measures which trajectories are ‘reluctant’, i.e. stay distant from the attractor for long, or ‘eager’ to approach it right away. Regularized reaching time, on the other hand, quantifies the additional time (positive or negative) that a trajectory starting at a chosen initial condition needs to approach the attractor as compared to some reference trajectory. A positive or negative value means that it approaches the attractor by this much ‘earlier’ or ‘later’ than the reference, respectively. We demonstrated their substantial potential for application with multiple paradigmatic examples uncovering new features. |
| first_indexed | 2024-03-12T16:35:42Z |
| format | Article |
| id | doaj.art-9bd4c4db6bd64c8bbb6bd5940896796c |
| institution | Directory Open Access Journal |
| issn | 1367-2630 |
| language | English |
| last_indexed | 2024-03-12T16:35:42Z |
| publishDate | 2017-01-01 |
| publisher | IOP Publishing |
| record_format | Article |
| series | New Journal of Physics |
| spelling | doaj.art-9bd4c4db6bd64c8bbb6bd5940896796c2023-08-08T14:53:02ZengIOP PublishingNew Journal of Physics1367-26302017-01-0119808300510.1088/1367-2630/aa7b61Timing of transients: quantifying reaching times and transient behavior in complex systemsTim Kittel0Jobst Heitzig1Kevin Webster2Jürgen Kurths3Potsdam Institute for Climate Impact Research , Telegrafenberg A31—(PO) Box 60 12 03, D-14412 Potsdam, Germany; Institut für Physik , Humboldt-Universität zu Berlin, Newtonstraße 15, D-12489 Berlin, GermanyPotsdam Institute for Climate Impact Research , Telegrafenberg A31—(PO) Box 60 12 03, D-14412 Potsdam, GermanyPotsdam Institute for Climate Impact Research , Telegrafenberg A31—(PO) Box 60 12 03, D-14412 Potsdam, GermanyPotsdam Institute for Climate Impact Research , Telegrafenberg A31—(PO) Box 60 12 03, D-14412 Potsdam, Germany; Institut für Physik , Humboldt-Universität zu Berlin, Newtonstraße 15, D-12489 Berlin, Germany; Institute for Complex Systems and Mathematical Biology, University of Aberdeen , Aberdeen AB24 3UE, United KingdomIn dynamical systems, one may ask how long it takes for a trajectory to reach the attractor, i.e. how long it spends in the transient phase. Although for a single trajectory the mathematically precise answer may be infinity, it still makes sense to compare different trajectories and quantify which of them approaches the attractor earlier. In this article, we categorize several problems of quantifying such transient times. To treat them, we propose two metrics, area under distance curve and regularized reaching time, that capture two complementary aspects of transient dynamics. The first, area under distance curve, is the distance of the trajectory to the attractor integrated over time. It measures which trajectories are ‘reluctant’, i.e. stay distant from the attractor for long, or ‘eager’ to approach it right away. Regularized reaching time, on the other hand, quantifies the additional time (positive or negative) that a trajectory starting at a chosen initial condition needs to approach the attractor as compared to some reference trajectory. A positive or negative value means that it approaches the attractor by this much ‘earlier’ or ‘later’ than the reference, respectively. We demonstrated their substantial potential for application with multiple paradigmatic examples uncovering new features.https://doi.org/10.1088/1367-2630/aa7b61early-warning signalscomplex systemsnonlinear dynamicsordinary differential equationsstability against shocks05.45.-a |
| spellingShingle | Tim Kittel Jobst Heitzig Kevin Webster Jürgen Kurths Timing of transients: quantifying reaching times and transient behavior in complex systems New Journal of Physics early-warning signals complex systems nonlinear dynamics ordinary differential equations stability against shocks 05.45.-a |
| title | Timing of transients: quantifying reaching times and transient behavior in complex systems |
| title_full | Timing of transients: quantifying reaching times and transient behavior in complex systems |
| title_fullStr | Timing of transients: quantifying reaching times and transient behavior in complex systems |
| title_full_unstemmed | Timing of transients: quantifying reaching times and transient behavior in complex systems |
| title_short | Timing of transients: quantifying reaching times and transient behavior in complex systems |
| title_sort | timing of transients quantifying reaching times and transient behavior in complex systems |
| topic | early-warning signals complex systems nonlinear dynamics ordinary differential equations stability against shocks 05.45.-a |
| url | https://doi.org/10.1088/1367-2630/aa7b61 |
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