Sideband Instability of Modulated Traveling Wave Convection
It has been shown theoretically (Riecke et al. 1988, Walgraef 1988), that in a system exhibiting a supercritical Hopf bifurcation a temporal modulation of the driving force with a modulation frequency ωm of about double of the Hopf frequency can stabilize standing waves (SW). An experimental verification was presented (Rehberg et al. 1988) for the electro-hydrodynamic convection of liquid crystals, where the driving ac voltage is modulated as V(t)=Vc·cos(ω·t)·[1+ε+b·cos(ωm·t)], with b being the modulation and e the reduced driving amplitude. The theoretical model is very similar to the one describing parametrically excited waves which are known to exhibit Benjamin-Feir turbulence that is characterized by a transfer of energy from the fundamental Fourier mode to the side bands (Craik 1985). When increasing e for a constant modulation amplitude b the simplified theoretical model predicts a supercritical bifurcation from SW to modulated traveling waves (TW). In the experiment SW become unstable via a different mechanism which we clarify here. The scenario includes the appearance of the sideband instability, defects and stable undulated rolls of a very short wavelength.
KeywordsHopf Bifurcation Standing Wave Travel Wave Oscillatory Convection Supercritical Bifurcation
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