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Nonlinear Dynamics

, Volume 40, Issue 2, pp 195–203 | Cite as

2:1:1 Resonance in the Quasi-Periodic Mathieu Equation

  • Richard Rand
  • Tina Morrison
Article

Abstract

We present a small ε perturbation analysis of the quasi-periodic Mathieu equation
$$ \ddot x + (\delta + \epsilon \cos t + \epsilon \cos \omega t) x=0 $$
in the neighborhood of the point δ = 0.25 and ω = 0.5. We use multiple scales including terms of O2) with three time scales. We obtain an asymptotic expansion for an associated instability region. Comparison with numerical integration shows good agreement for ε = 0.1. Then we use the algebraic form of the perturbation solution to approximate scaling factors which are conjectured to determine the size of instability regions as we go from one resonance to another in the δ−ω parameter plane.

Keywords

parametric excitation resonance quasi-periodic Mathieu equation 

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References

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Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of Theoretical & Applied MechanicsCornell UniversityIthacaU.S.A.

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