2:1:1 Resonance in the Quasi-Periodic Mathieu Equation
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We present a small ε perturbation analysis of the quasi-periodic Mathieu equation
in the neighborhood of the point δ = 0.25 and ω = 0.5. We use multiple scales including terms of O(ε2) with three time scales. We obtain an asymptotic expansion for an associated instability region. Comparison with numerical integration shows good agreement for ε = 0.1. Then we use the algebraic form of the perturbation solution to approximate scaling factors which are conjectured to determine the size of instability regions as we go from one resonance to another in the δ−ω parameter plane.
$$ \ddot x + (\delta + \epsilon \cos t + \epsilon \cos \omega t) x=0 $$
Keywordsparametric excitation resonance quasi-periodic Mathieu equation
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- 1.Zounes, R. S. and Rand, R. H., ‘Transition curves in the quasi-periodic Mathieu equation’, SIAM Journal of Applied Mathematics 58, 1998, 1094–1115.Google Scholar
- 2.Rand, R., Zounes, R., and Hastings, R., ‘A quasi-periodic Mathieu equation’, Chap. 9, in Nonlinear Dynamics: The Richard Rand 50th Anniversary Volume, A. Guran (ed.), World Scientific Publications, 1997, pp. 203–221.Google Scholar
- 3.Mason, S. and Rand, R., ‘On the torus flow Y’ = A + B COS Y + C COS X and its relation to the quasi-periodic Mathieu equation’, in Proceedings of the 1999 ASME Design Engineering Technical Conferences, Las Vegas, NV, September 12–15, 1999, Paper no. DETC99/VIB-8052 (CD-ROM).Google Scholar
- 4.Zounes, R. S. and Rand, R. H., ‘Global behavior of a nonlinear quasi-periodic Mathieu equation’, Nonlinear Dynamics 27, 2002, 87–105.Google Scholar
- 5.Rand, R., Guennoun, K., and Belhaq, M., ‘2:2:1 Resonance in the quasi-periodic Mathieu equation’, Nonlinear Dynamics 31, 2003, 367–374.Google Scholar
- 6.Guennoun, K., Houssni, M., and Belhaq, M., ‘Quasi-periodic solutions and stability for a weakly damped nonlinear quasi-periodic Mathieu equation’, Nonlinear Dynamics 27, 2002, 211–236.Google Scholar
- 7.Belhaq, M., Guennoun, K., and Houssni, M., ‘Asymptotic solutions for a damped non-linear quasi-periodic Mathieu equation’, International Journal of Non-Linear Mechanics 37, 2002, 445–460.Google Scholar
- 8.Rand, R. H., Lecture Notes on Nonlinear Vibrations (version 45), Published on-line by The Internet-First University Press, Ithaca, NY, 2004, http://dspace.library.cornell.edu/handle/1813/79.
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