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\(2:1:1\) resonance in the quasi-periodic Mathieu equation. (English) Zbl 1132.34041
The paper discusses a small \(\epsilon\) perturbation analysis of the quasi-periodic Mathieu equation
\[ {\ddot x}+(\delta+ \varepsilon\cos t+\varepsilon\cos \omega t)x=0 \] in the neighborhood of the point \(\delta=0.25\) and \(\omega=0.5\). Multiple scales including terms of \(O(\varepsilon^2)\) with three time scales are used. An asymptotic expansion for an associated instability region is obtained. Comparison with numerical integration shows good agreement for \(\varepsilon=0.1\). Then the algebraic form of the perturbation solution is used to approximate scaling factors which are conjectured to determine the size of instability regions as one goes from one resonance to another in the \(\delta- \omega\) parameter plane.

MSC:
34E13 Multiple scale methods for ordinary differential equations
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
70K28 Parametric resonances for nonlinear problems in mechanics
34C23 Bifurcation theory for ordinary differential equations
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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. · Zbl 0916.34031 · doi:10.1137/S0036139996303877
[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. · Zbl 0918.34033
[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).
[4] Zounes, R. S. and Rand, R. H., ?Global behavior of a nonlinear quasi-periodic Mathieu equation?, Nonlinear Dynamics 27, 2002, 87-105. · Zbl 1003.34041 · doi:10.1023/A:1017931712099
[5] Rand, R., Guennoun, K., and Belhaq, M., ?2:2:1 Resonance in the quasi-periodic Mathieu equation?, Nonlinear Dynamics 31, 2003, 367-374. · Zbl 1062.70596 · doi:10.1023/A:1023216817293
[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. · Zbl 1013.70018 · doi:10.1023/A:1014496917703
[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. · Zbl 1346.34030 · doi:10.1016/S0020-7462(01)00020-8
[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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