Periodic solutions and stability for a weakly damped nonlinear Mathieu equation

A. F. El-Bassiouny, Ayman Abdel-Khalik

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

Periodic solutions are investigated for a weakly damped nonlinear Mathieu equation. The multiple scales perturbation technique is used to determine a third-order solution for a nonlinear Mathieu equation. We derive a first reduced differential amplitude-phase system having periodic components. These equations are used to determine the fixed points. The stability of stationary solutions of this reduced system is analyzed. Numerical solutions are presented, which illustrate the behavior of the steady-state response amplitude as a function of the detuning parameter. Stability analysis is carried out for each case. The effects of these (quadratic and cubic ) nonlinearities on these oscillations are specifically investigated. With this study, it has been verified that the qualitative effects of these nonlinearities are different.

Original languageEnglish
Article number015008
JournalPhysica Scripta
Volume81
Issue number1
DOIs
Publication statusPublished - 2010
Externally publishedYes

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Mathieu Equation
Mathieu function
Damped
Periodic Solution
Nonlinear Equations
nonlinearity
Nonlinearity
Multiple Scales
Perturbation Technique
Periodic Systems
Stationary Solutions
Stability Analysis
Fixed point
Numerical Solution
Oscillation
perturbation
oscillations

ASJC Scopus subject areas

  • Atomic and Molecular Physics, and Optics
  • Mathematical Physics
  • Condensed Matter Physics
  • Physics and Astronomy(all)

Cite this

Periodic solutions and stability for a weakly damped nonlinear Mathieu equation. / El-Bassiouny, A. F.; Abdel-Khalik, Ayman.

In: Physica Scripta, Vol. 81, No. 1, 015008, 2010.

Research output: Contribution to journalArticle

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