The effect of a low-frequency modulation on some codimension 2 bifurcations

Abstract

The interactions between several types of intabilities lead to bifurcations of multiple codimension. The convectioe flow of a binary mixture in a porous medium is a good example of such an interaction of two instabilities. These instabilities depend on two parameters: the Rayleigh number Ra and the separation ratio ψ. The corresponding neutral curves intersect, in the (Ra, ψ) plane, at a polycritical point. We describe, in this paper, the influence, on this bifurcation, of a small low-frequency time variation of the thermal boundary conditions. The reduced nonlinear Mathieu equation depends on a parameter ε, which characterizes the low frequency modulation. For ε= 0 the Mathieu equation possesses, in the phase plane, heteroclinic and periodic solutions. For ε ≠ 0, the system may exhibit a chaotic régime of Smale horseshoe type. The onset of the chaotic regime as well as the curve at which a saddle-node bifurcation occurs can be estimated by means of Melnikov's technique. Numerical simulations agree with the results of Melnikov's theory. This problem can be considered to be a model for other instability problems arising in other domains of the Physics, such as thermosolutal convection and convection in magnetic fields or under rotation.