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Applications of the integral equation method to delay differential equations

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Abstract

In this paper, we compare two approaches for determining the amplitude equations; namely, the integral equation method and the method of multiple scales. To describe and compare the methods, we consider three examples: the parametric resonance of a Van der Pol oscillator under state feedback control with a time delay, the primary resonance of a harmonically forced Duffing oscillator under state feedback control with a time delay, and the primary resonance together with 1:1 internal resonance of a two degree-of-freedom model. Using the integral equation method and the method of multiple scales, the amplitude equations are obtained. The stability of the periodic solution is examined by using the Floquet theorem together with the Routh–Hurwitz criterion (without time delay) and the Nyquist criterion (with time delay). By comparison with the solution obtained by the numerical integration, we find that the accuracy of the integral equation method is much better.

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Acknowledgements

This research is supported by the State Key Program of National Natural Science Foundation of China under Grant No. 11032009 and National Natural Science Foundation of China under Grant No. 11272236.

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Authors and Affiliations

  1. School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, 200092, P.R. China

    Yueli Chen & Jian Xu

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  1. Yueli Chen
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  2. Jian Xu
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Correspondence to Jian Xu.

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Chen, Y., Xu, J. Applications of the integral equation method to delay differential equations. Nonlinear Dyn 73, 2241–2260 (2013). https://doi.org/10.1007/s11071-013-0938-0

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