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 i n which, when the control parameter exceeds a critical value, the null solu tion bifurcates into a finite amplitude time-periodic solution. We study th e response of this equation to time-harmonic forcing in the subcritical reg ime (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 solut ion surface as a function of forcing frequency and amplitude becomes multiv alued. For instance, at a fixed forcing amplitude, one finds a well-defined range of frequencies over which two stable periodic responses may coexist, having different amplitudes. We apply this result to predict the behaviour of the wake downstream of an oscillating cylinder, and compare the predict ions with experimental and computational observations of such a wake. (C) 2 001 Academic Press.