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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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##### References:
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