Spatial and temporal resonances in a periodically forced hydrodynamic system

The occurrence of a Naimark–Sacker bifurcation in a periodically forced spatially extended system is analyzed with Floquet theory. The basic periodic state of the flow in an annular region consists of the superposition of circular Couette flow, driven by the constant rotation of the inner cylinder, and annular Stokes flow, driven by the harmonic oscillation of the inner cylinder in the axial direction. Windows in parameter space where onset of instability is via Naimark–Sacker correspond to forcing frequencies close to the natural frequencies of the unforced system when the forcing amplitude is sufficiently large. In these windows, not only have strong temporal resonances been identified, but also competition and resonances between various spatial modes that are simultaneously excited take place. An excitation diagram mapping out these windows and loci of resonance points is produced, providing a guide for future explorations into the nonlinear regime by either experimental or computational techniques.