Temporal Modulation of a Subcritical Bifurcation to Travelling Waves

Abstract

The Hopf bifurcation to travelling and standing waves which arises in a variety of systems has drawn considerable attention during the past few years. Recently it has been shown that modulating a control parameter R in time has a strong influence on the selection properties of such a system if the modulation frequency is close to a resonance with the natural frequency of these waves (Riecke et al., 1988; Walgraef, 1988; Rehberg et al., 1988): if stable travelling waves bifurcate supercritically in the unmodulated system, the modulation lowers the threshold for standing waves below that for travelling waves. This can lead to stable standing waves in a parameter regime where travelling waves do not exist. However, in the system which has been studied experimentally the most (binary mixture convection) the unmodulated instability to travelling waves is subcritical. This naturally raises the question whether in such a system the modulation can still stabilize standing waves even though the travelling waves appear already below threshold. This is the topic of the present paper.