Applications of the integral equation method to delay differential equations

作者：Chen Yueli; Xu Jian^*

来源：Nonlinear Dynamics, 2013, 73(4): 2241-2260.

DOI：10.1007/s11071-013-0938-0

• Integral equation method
• Method of multiple scales
• Parametric resonance
• Primary resonance
• Internal resonance

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.

• 出版日期2013-9
• 单位同济大学