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Publisher DOI: 10.1016/j.proeng.2017.09.046
Title: Determining periodic orbits via nonlinear filtering and recurrence spectra in the presence of noise
Language: English
Authors: Oberst, Sebastian 
Marburg, Steffen 
Hoffmann, Norbert  
Keywords: phase space prediction;unstable periodic orbits;WIENER-KHINCHIN theorem;recurrence plot analysis
Issue Date: 2017
Publisher: Elsevier
Source: Procedia Engineering (199): 772-777 (2017)
Journal or Series Name: Procedia engineering 
Abstract (english): 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.
DOI: 10.15480/882.1824
ISSN: 1877-7058
Institute: Strukturdynamik M-14 
Type: (wissenschaftlicher) Artikel
Funded by: DFG Priority Program SPP1897 "Calm, Smooth and Smart" with grant numbers OB 444/1-1 (SO) and HO 3851/12-1 (NH)
License: CC BY-NC-ND 4.0 (Attribution-NonCommercial-NoDerivatives) CC BY-NC-ND 4.0 (Attribution-NonCommercial-NoDerivatives)
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