Quasiperiodic Response to Parametric Excitations

Parametric excitations are capable of stabilizing an unstable state, but they can also destabilize modes that are otherwise stable. Resonances come into play when the periodically forced base state undergoes a Hopf bifurcation from a limit cycle to a torus. A Floquet analysis of the basic state will identify such an event, but in order to characterize the resonances, one requires precise knowledge of the frequencies of the resultant quasiperiodic state. However, the Floquet analysis only returns the angle at which one of the pair of complex-conjugate eigenvalues cross the unit disk, modulo 2π. This angle is related in a non-trivial way to the new frequency resulting from the bifurcation. Here, we first present a technique to unambiguously determine the frequencies of the solutions following such a Hopf bifurcation, using only results from Floquet theory and discrete dynamical systems. We apply this technique to a periodically forced system susceptible to centrifugal instabilities and identify the conditions under which the system resonates, leading to resonance horns emerging from the Hopf bifurcation curves, and also identify locations of high codimension degenerate bifurcations resulting from space-time resonances.