Quasiperiodic Response to Parametric Excitations

Abstract

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.

Spatial and temporal resonances in a periodically forced hydrodynamic system

“…1). This system has been investigated experimentally [25] and theoretically [7,12,14]. The system is governed by a number of non-dimensional parameters.…”

“…In the present system, the Naimark-Sacker bifurcation leads to a second frequency ω s , which is analogous to the natural frequency that varies with both the forcing amplitude R a and frequency ω. It needs to be determined from the Floquet analysis, and a robust technique for this has been presented in [12].…”

Endwall effects in a periodically forced centrifugally unstable flow

“…The onset of a Naimark-Sacker bifurcation from a limit cycle to a 2-torus just described above has occurred in an axisymmetric system. It should be noted that Naimark-Sacker bifurcations detected in the stability analysis of the basic state in the → ∞ system have always been associated with the simultaneous breaking of the S and the azimuthal symmetry (Lopez and Marques, 2000b). The windows in parameter space that include Naimark-Sacker bifurcations in the inÿnite system have lower forcing frequencies than that considered here (!…”