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In this work, we investigate regions of stability in the vicinity of 2:2:1 resonance in the quasiperiodic Mathieu equation: d2xdt2+(δ+εcost+εμcos(1+εΔ)t)x=0, using two successive perturbation methods. The parameters ε and μ are assumed to be small. The parameter ε serves for deriving the corresponding slow flow differential system and μ serves to implement a second perturbation analysis on the slow flow system near its proper resonance. This strategy allows us to obtain analytical expressions for the transition curves in the resonant quasiperiodic Mathieu equation. We compare the analytical results with those of direct numerical integration. This work has application to parametrically excited systems in which there are two periodic drivers, each with frequency close to twice the frequency of the unforced system.

This paper concerns a linear study of the convective parametric instability in the case of a Newtonian fluid confined in a Hele-Shaw cell and submitted to a vertical periodic motion. The gradient of temperature, applied to the fluid layer, is either in the same direction that gravity or in the opposite one. An asymptotic analysis shows that the Hele-Shaw approximation leads to two linear formulations depending on the order of magnitude of the Prandtl number. For these two asymptotic cases, the convective threshold is determined. It turns out that in the Hele-Shaw geometrical configuration, parametric oscillations have no influence on the criterion of stability when the Prandtl number is in the order of the unity or very superior to the unity. However, when the Prandtl number is small than unity, the parametric oscillations can affect the convective instability threshold.

The effect of a time-sinusoidal magnetic field on the onset of convection in a horizontal magnetic fluid layer heated from above and bounded by isothermal non magnetic boundaries is investigated. The analysis is restricted to static and linear laws of magnetization. A first order Galerkin method is performed to reduce the governing linear system to the Mathieu equation with damping term. Therefore, the Floquet theory is used to determine the convective threshold for the free-free and rigid-rigid cases. With an appropriate choice of the ratio of the magnetic and gravitational forces, we show the possibility to produce a competition between the harmonic and subharmonic modes at the onset of convection.

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