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.
ASJC Scopus subject areas
- Atomic and Molecular Physics, and Optics
- Mathematical Physics
- Condensed Matter Physics