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Asymptotic solutions for a damped non-linear quasi-periodic Mathieu equation. (English) Zbl 1346.34030
Summary: Quasi-periodic (QP) solutions of a weakly damped non-linear QP Mathieu equation are investigated near a double primary parametric resonance. A double multiple scales method is applied to reduce the original QP oscillator to an autonomous system performing two successive reduction. The problem for approximating QP solutions of the original system is then transformed to the study of stationary regimes of the induced autonomous system. Explicit analytical approximations to QP oscillations are obtained and comparisons to numerical integration of the original QP oscillator are provided.

MSC:
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
37N05 Dynamical systems in classical and celestial mechanics
70K99 Nonlinear dynamics in mechanics
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