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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. ?

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Analytical study of the two first period-doubling bifurcations in a three-dimensional system is reported. The multiple scales method is first applied to construct a higher-order approximation of the periodic orbit following Hopf bifurcation. The stability analysis of this periodic orbit is then performed in terms of Floquet theory to derive the critical parameter values corresponding to the first and second period-doubling bifurcations. By introducing suitable subharmonic components in the first order of the multiple scale analysis the two critical parameter values are obtained simultaneously solving analytically the resulting system of two algebraic equations. Comparisons of analytic predictions to numerical simulations are also provided.

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