Characterization of the flow over a cylinder moving harmonically in the cross-flow direction

The flow over a stationary cylinder is self-excited with a specific natural frequency fN. When the cylinder is moved harmonically in the cross-flow direction, the response of the flow (in terms of the lift force) will contain two frequencies, namely, the natural frequency fN and the excitation frequency fE. When fE is close to fN, the natural flow response will be entrained by the excitation, and the response will be periodic with frequency fE, and dynamicists refer to this phenomenon as lock-in or synchronization. When fE is away from fN, the flow will be either periodic with a period that is multiple of the excitation period (i.e., period-n) and dynamicists refer to this phenomenon as secondary synchronization or quasiperiodic consisting of two incommensurate frequencies, or chaotic. We use modern methods of non-linear dynamics to characterize these responses and show that the route to chaos is torus breakdown.