Hysteresis in the Forced Stuart}landau Equation: Application to Vortex Shedding from an Oscillating Cylinder

The complex Stuart}Landau equation models a prototypical Hopf bifurcation in which, when the control parameter exceeds a critical value, the null solution bifurcates into a "nite amplitude time-periodic solution. We study the response of this equation to time-harmonic forcing in the subcritical regime (i.e., before the bifurcation). We show that when a second parameter in the Stuart}Landau equation passes a critical value, a portion of the solution surface as a function of forcing frequency and amplitude becomes multivalued. For instance, at a "xed forcing amplitude, one "nds a well-de"ned range of frequencies over which two stable periodic responses may coexist, having di!erent amplitudes. We apply this result to predict the behaviour of the wake downstream of an oscillating cylinder, and compare the predictions with experimental and computational observations of such a wake. ( 2001 Academic Press