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Analytical approximations for the periodic motion of the Duffing system with delayed feedback

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Abstract

In this study, the homotopy analysis method is developed to give periodic solutions of delayed differential equations that describe time-delayed position feedback on the Duffing system. With this technique, some approximate analytical solutions of high accuracy for some possible solutions are captured, which agree well with the numerical solutions in the whole time domain. Two examples of dynamic systems are considered, focusing on the periodic motions near a Hopf bifurcation of an equilibrium point. It is found that the current technique leads to higher accurate prediction on the local dynamics of time-delayed systems near a Hopf bifurcation than the energy analysis method or the traditional method of multiple scales.

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Authors and Affiliations

  1. State Key Lab of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China

    Xiangcheng You & Hang Xu

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  1. Xiangcheng You
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  2. Hang Xu
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Correspondence to Xiangcheng You.

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You, X., Xu, H. Analytical approximations for the periodic motion of the Duffing system with delayed feedback. Numer Algor 56, 561–576 (2011). https://doi.org/10.1007/s11075-010-9404-y

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  • DOI: https://doi.org/10.1007/s11075-010-9404-y

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