Bursting and regular dynamics in time modulated Taylor–Couette flow with two incommensurate time scales

The presence of multiple timescales in many dynamical systems leads to bursting oscillations characterized by a combination of relatively large amplitude and nearly small amplitude oscillations. In this work, we are concerned about the bursting oscillations induced by two timescale effects in a Taylor–Couette system subjected to a time-quasi-periodic forcing. We consider the case where both the inner and outer cylinders are oscillating, respectively, at the angular velocities Ω0 cos(ω1t) and Ω0 cos(ω2t) with two incommensurate frequencies ω1 and ω2, i.e., the ratio ω=ω2/ω1 is irrational. The growth rate of linear disturbances is employed to investigate the dynamics of the system through a numerical resolution of the stability equations utilizing both the Chebyshev spectral method and the Runge–Kutta numerical scheme. It turns out that at sufficiently high frequencies and arbitrary values of ω, Taylor vortices with regular oscillations are observed where two distinct and separated boundary Stokes layer flows are generated near each cylinder. By decreasing the oscillation frequencies, these oscillatory Stokes layer flows move closer together and their interaction gives rise to bursting oscillations via either synchronous or period-doubling bifurcations. In addition, it is demonstrated that such bursting behavior occurs near codimension-two bifurcation points where more than one instability modes are triggered near the primary bifurcation. However, only one instability mode is detected when dealing with regular oscillations. Moreover, it is shown that the presence of two timescales alters significantly the flow reversal of the system observed when ω=1. Although these results are obtained by the Floquet theory when irrational values of the frequency ratio are approached by rational ones, direct numerical simulations of the governing linear equations are conducted using the same numerical approach and an excellent agreement is obtained. Such agreement demonstrates the validity and robustness of Floquet theory in describing the linear dynamics of hydrodynamic systems subjected to time-quasi-periodic forcing and suggests that it can be considered as an alternative to the widely used harmonic balance analysis.