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Zbl 0916.34031
Zounes, Randolph S.; Rand, Richard H.
Transition curves for the quasi-periodic Mathieu equation.
(English)
[J] SIAM J. Appl. Math. 58, No. 4, 1094-1115 (1998). ISSN 0036-1399; ISSN 1095-712X/e

The quasi-periodic Mathieu equation $$\ddot\psi+ \bigl[\delta+ \varepsilon(\cos t+\cos\omega t)\bigr]\psi=0$$ is investigated for small $\varepsilon$ and irrational $\omega$. The aim is to obtain a stability diagram in the $\delta f$-$\omega$ plane (for fixed $\varepsilon)$ for which all solutions are bounded. Numerical integration is used to produce approximations to the stability diagram both directly and through contour plots of Lyapunov exponents. Approximate analytic techniques for the transition curves between unstable and bounded solutions using (1) a regular perturbation method, and (2) the method of harmonic balance with Hill's determinants, are derived. Comparisons are made between the numerical and approximate analytic results.
[P.Smith (Keele)]
MSC 2000:
*34B30 Special ODE
34E10 Asymptotic perturbations (ODE)
34C27 Almost periodic solutions of ODE
34D08 Lyapunov exponents
34D10 Stability perturbations of ODE

Keywords: Floquet theory; Lyapunov exponents; Hill's determinants

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