Pattern formation at the bi-critical point of the Faraday instability

We present measurements on parametrically driven surface waves (Faraday waves) performed in the vicinity of a bi-critical point in parameter space, where modes with harmonic and subharmonic time dependence interact. The primary patterns are squares in the subharmonic and hexagons in the harmonic regime. If the primary instability is harmonic we observe a hysteretic secondary transition from hexagons to squares without a perceptible variation of the fundamental wavelength. The transition is understood in terms of a set of coupled Landau equations and related to other canonical examples of phase transitions in nonlinear dissipative systems. Moreover, the subharmonic-harmonic mode competition gives rise to a variety of new superlattice states. These structures are interpreted as mediator modes involved in the transition between patterns of fourfold and sixfold rotational symmetry.