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Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Oct;86(4 Pt 2):046214. Epub 2012 Oct 31.

Spectral element method and the delayed feedback control of chaos.

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  • 1Mechanical Engineering and Materials Science Department, Duke University, Durham, North Carolina 27708, USA.


A spectral element approach is introduced to determine the Floquet exponents (FEs) of unstable periodic orbits (UPOs) stabilized by extended delayed feedback control (EDFC). The spectral approach does not require solving time-dependent eigenproblems that existing methods require. Instead, the spectral approach determines the stability of the delay differential equations of the system by numerical approximation. The method is capable of analyzing systems whose UPOs arise from bifurcations other than period-doubling. Results are presented for stabilizing UPOs in Duffing systems. The FEs calculated by the spectral approach are compared to published results for two examples. In both cases, the spectral method results agree well with those determined by previous methods. In addition, the spectral method was used to analyze a high-dimensional, asymmetrical system with a UPO in chaos arising from tori doubling following a Hopf bifurcation.

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