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Determining periodic orbits via nonlinear filtering and recurrence spectra in the presence of noise
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| Veröffentlicht in: | Procedia engineering 199(2018), Seite 772-777 |
|---|---|
| Personen und Körperschaften: | , , |
| Titel: | Determining periodic orbits via nonlinear filtering and recurrence spectra in the presence of noise/ Sebastian Oberst, Steffen Marburg, Norbert Hoffmann |
| Format: | E-Book-Kapitel |
| Sprache: | Englisch |
| veröffentlicht: |
2017
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| Gesamtaufnahme: |
: Procedia engineering, 199(2018), Seite 772-777
, volume:199 |
| Schlagwörter: | |
| Quelle: | Verbunddaten SWB Lizenzfreie Online-Ressourcen |
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| 520 | |a In nonlinear dynamical systems the determination of stable and unstable periodic orbits as part of phase space prediction is problematic in particular if perturbed by noise. Fourier spectra of the time series or its autocorrelation function have shown to be of little use if the dynamic process is not strictly wide-sense stationary or if it is nonlinear. To locate unstable periodic orbits of a chaotic attractor in phase space the least stable eigenvalue can be determined by approximating locally the trajectory via linearisation. This approximation can be achieved by employing a Gaussian kernel estimator and minimising the summed up distances of the measured time series i.e. its estimated trajectory (e.g. via Levenberg-Marquardt). Noise poses a significant problem here. The application of the Wiener-Khinchin theorem to the time series in combination with recurrence plots, i.e. the Fourier transform of the recurrence times or rates, has been shown capable of detecting higher order dynamics (period-2 or period-3 orbits), which can fail using classical FouRiER-based methods. However little is known about its parameter sensitivity, e.g. with respect to the time delay, the embedding dimension or perturbations. Here we provide preliminary results on the application of the recurrence time spectrum by analysing the Hénon and the Rössler attractor. Results indicate that the combination of recurrence time spectra with a nonlinearly filtered plot of return times is able to estimate the unstable periodic orbits. Owing to the use of recurrence plot based measures the analysis is more robust against noise than the conventional Fourier transform. | ||
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| contents | In nonlinear dynamical systems the determination of stable and unstable periodic orbits as part of phase space prediction is problematic in particular if perturbed by noise. Fourier spectra of the time series or its autocorrelation function have shown to be of little use if the dynamic process is not strictly wide-sense stationary or if it is nonlinear. To locate unstable periodic orbits of a chaotic attractor in phase space the least stable eigenvalue can be determined by approximating locally the trajectory via linearisation. This approximation can be achieved by employing a Gaussian kernel estimator and minimising the summed up distances of the measured time series i.e. its estimated trajectory (e.g. via Levenberg-Marquardt). Noise poses a significant problem here. The application of the Wiener-Khinchin theorem to the time series in combination with recurrence plots, i.e. the Fourier transform of the recurrence times or rates, has been shown capable of detecting higher order dynamics (period-2 or period-3 orbits), which can fail using classical FouRiER-based methods. However little is known about its parameter sensitivity, e.g. with respect to the time delay, the embedding dimension or perturbations. Here we provide preliminary results on the application of the recurrence time spectrum by analysing the Hénon and the Rössler attractor. Results indicate that the combination of recurrence time spectra with a nonlinearly filtered plot of return times is able to estimate the unstable periodic orbits. Owing to the use of recurrence plot based measures the analysis is more robust against noise than the conventional Fourier transform. |
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| spelling | Oberst, Sebastian VerfasserIn (DE-588)1172111162 (DE-627)1040950728 (DE-576)514349042 aut, Determining periodic orbits via nonlinear filtering and recurrence spectra in the presence of noise Sebastian Oberst, Steffen Marburg, Norbert Hoffmann, 2017, Diagramme, Text txt rdacontent, Computermedien c rdamedia, Online-Ressource cr rdacarrier, In nonlinear dynamical systems the determination of stable and unstable periodic orbits as part of phase space prediction is problematic in particular if perturbed by noise. Fourier spectra of the time series or its autocorrelation function have shown to be of little use if the dynamic process is not strictly wide-sense stationary or if it is nonlinear. To locate unstable periodic orbits of a chaotic attractor in phase space the least stable eigenvalue can be determined by approximating locally the trajectory via linearisation. This approximation can be achieved by employing a Gaussian kernel estimator and minimising the summed up distances of the measured time series i.e. its estimated trajectory (e.g. via Levenberg-Marquardt). Noise poses a significant problem here. The application of the Wiener-Khinchin theorem to the time series in combination with recurrence plots, i.e. the Fourier transform of the recurrence times or rates, has been shown capable of detecting higher order dynamics (period-2 or period-3 orbits), which can fail using classical FouRiER-based methods. However little is known about its parameter sensitivity, e.g. with respect to the time delay, the embedding dimension or perturbations. Here we provide preliminary results on the application of the recurrence time spectrum by analysing the Hénon and the Rössler attractor. Results indicate that the combination of recurrence time spectra with a nonlinearly filtered plot of return times is able to estimate the unstable periodic orbits. Owing to the use of recurrence plot based measures the analysis is more robust against noise than the conventional Fourier transform., phase space prediction, unstable periodic orbits, WIENER-KHINCHIN theorem, recurrence plot analysis, Marburg, Steffen 1965- VerfasserIn (DE-588)120992892 (DE-627)081016042 (DE-576)292486367 aut, Hoffmann, Norbert 1969- VerfasserIn (DE-588)113319978X (DE-627)888702930 (DE-576)489158145 aut, Enthalten in Procedia engineering Amsterdam [u.a.] : Elsevier, 2009 199(2018), Seite 772-777 Online-Ressource (DE-627)607348992 (DE-600)2509658-8 (DE-576)310099773 1877-7058 nnns, volume:199 year:2018 pages:772-777, http://nbn-resolving.de/urn:nbn:de:gbv:830-88223687 Resolving-System kostenfrei Volltext, https://doi.org/10.15480/882.1824 Resolving-System kostenfrei Volltext, http://hdl.handle.net/11420/1827 Resolving-System kostenfrei Volltext, https://doi.org/10.1016%2Fj.proeng.2017.09.046 Verlag Volltext, https://doi.org/10.15480/882.1824 LFER, LFER 2019-05-29T00:00:00Z |
| spellingShingle | Oberst, Sebastian, Marburg, Steffen, Hoffmann, Norbert, Determining periodic orbits via nonlinear filtering and recurrence spectra in the presence of noise, In nonlinear dynamical systems the determination of stable and unstable periodic orbits as part of phase space prediction is problematic in particular if perturbed by noise. Fourier spectra of the time series or its autocorrelation function have shown to be of little use if the dynamic process is not strictly wide-sense stationary or if it is nonlinear. To locate unstable periodic orbits of a chaotic attractor in phase space the least stable eigenvalue can be determined by approximating locally the trajectory via linearisation. This approximation can be achieved by employing a Gaussian kernel estimator and minimising the summed up distances of the measured time series i.e. its estimated trajectory (e.g. via Levenberg-Marquardt). Noise poses a significant problem here. The application of the Wiener-Khinchin theorem to the time series in combination with recurrence plots, i.e. the Fourier transform of the recurrence times or rates, has been shown capable of detecting higher order dynamics (period-2 or period-3 orbits), which can fail using classical FouRiER-based methods. However little is known about its parameter sensitivity, e.g. with respect to the time delay, the embedding dimension or perturbations. Here we provide preliminary results on the application of the recurrence time spectrum by analysing the Hénon and the Rössler attractor. Results indicate that the combination of recurrence time spectra with a nonlinearly filtered plot of return times is able to estimate the unstable periodic orbits. Owing to the use of recurrence plot based measures the analysis is more robust against noise than the conventional Fourier transform., phase space prediction, unstable periodic orbits, WIENER-KHINCHIN theorem, recurrence plot analysis |
| title | Determining periodic orbits via nonlinear filtering and recurrence spectra in the presence of noise |
| title_auth | Determining periodic orbits via nonlinear filtering and recurrence spectra in the presence of noise |
| title_full | Determining periodic orbits via nonlinear filtering and recurrence spectra in the presence of noise Sebastian Oberst, Steffen Marburg, Norbert Hoffmann |
| title_fullStr | Determining periodic orbits via nonlinear filtering and recurrence spectra in the presence of noise Sebastian Oberst, Steffen Marburg, Norbert Hoffmann |
| title_full_unstemmed | Determining periodic orbits via nonlinear filtering and recurrence spectra in the presence of noise Sebastian Oberst, Steffen Marburg, Norbert Hoffmann |
| title_in_hierarchy | Determining periodic orbits via nonlinear filtering and recurrence spectra in the presence of noise / Sebastian Oberst, Steffen Marburg, Norbert Hoffmann, |
| title_short | Determining periodic orbits via nonlinear filtering and recurrence spectra in the presence of noise |
| title_sort | determining periodic orbits via nonlinear filtering and recurrence spectra in the presence of noise |
| title_unstemmed | Determining periodic orbits via nonlinear filtering and recurrence spectra in the presence of noise |
| topic | phase space prediction, unstable periodic orbits, WIENER-KHINCHIN theorem, recurrence plot analysis |
| topic_facet | phase space prediction, unstable periodic orbits, WIENER-KHINCHIN theorem, recurrence plot analysis |
| url | http://nbn-resolving.de/urn:nbn:de:gbv:830-88223687, https://doi.org/10.15480/882.1824, http://hdl.handle.net/11420/1827, https://doi.org/10.1016%2Fj.proeng.2017.09.046 |
| urn | urn:nbn:de:gbv:830-88223687 |