A note on subharmonic instabilities

When a fluid system is subject to time-periodic forcing, it is well known that it may exhibit both harmonic and subharmonic instabilities, the classic example being Faraday oscillations. When the forcing is confined to a periodic shearing motion, however, it has been observed that the subharmonic response is absent. The underlying mathematical feature that unifies these systems is a conjugate-translation symmetry [A. C. Or, J. Fluid Mech. 335, 213 (1997)]. We show that any subharmonic solutions of periodically driven systems with conjugate-translation symmetry must have Floquet multipliers with multiplicity greater than one. The effect of this constraint is that subharmonic solutions are very difficult to locate within the system's parameter space and, more importantly, that phase locking cannot occur for such systems. (C) 1999 American Institute of Physics. [S1070-6631(99)00412-2].