Global behavior of a nonlinear quasiperiodic Mathieu equation
type: Item
instance of: scholarly article
title: Global behavior of a nonlinear quasiperiodic Mathieu equation
zbMATH Open document ID: 1003.34041
DOI: 10.1023/A:1017931712099
author: Randolph S. Zounes
author: Richard Rand
published in: Nonlinear Dynamics
publication date: 2002-05-30T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 34D23
Mathematics Subject Classification ID: 70K30
zbMATH DE Number: 1747905
zbMATH Keywords: elliptic functions
zbMATH Keywords: Lie transformations
zbMATH Keywords: resonance
zbMATH Keywords: quasiperiodic Mathieu equation
zbMATH Keywords: Poincaré section
describes a project that uses: DSTool
reviewed by: Lansun Chen
MaRDI profile type: Publication
full work available at URL: a:1017931712099
OpenAlex ID: W2151252324
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description: scientific article; zbMATH DE number 1747905