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Zbl 1003.34041
Zounes, Randolph S.; Rand, Richard H.
Global behavior of a nonlinear quasiperiodic Mathieu equation. (English)
[J] Nonlinear Dyn. 27, No.2, 87-105 (2002). ISSN 0924-090X; ISSN 1573-269X/e

Here, the interaction of subharmonic resonances in the nonlinear quasiperiodic Mathieu equation $$ \ddot x + [\delta + \varepsilon (\cos \omega_1t+\cos \omega_2t)]x + \alpha x^3=0 $$ is studied, where $\varepsilon \ll 1 $ and the coefficient of the nonlinear term $\alpha$ is positive but not necessarily small. By using Lie transform perturbation theory with elliptic functions, the authors study subharmonic resonances associated with orbits in $2m:1$ resonance with a respective driver. In particular, the authors derive analytic expressions that put conditions on the parameters $(\delta, \varepsilon, \omega_1, \omega_2)$ at which subharmonic resonance bands in a Poincaré section of action space begin. The authors obtain an overview of the $\vartheta(\varepsilon)$ global behavior of above equation as a function of $\delta$ and $\omega_2$ with $\omega_1, \alpha$ and $\varepsilon$ fixed.
[Chen Lan Sun (Beijing)]

MSC 2000:: *34D23 Global stability
70K30 Nonlinear resonances (general mechanics)

Keywords: elliptic functions; Lie transformations; resonance; quasiperiodic Mathieu equation; Poincaré section

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