Third-harmonic resonance in two-dimensional Faraday waves

Third-harmonic resonance of capillary-gravity waves in two-dimensional Fara day waves due to the parametric excitation of the lower-frequency mode is e xamined for infinite depth. The amplitude equation incorporating both a sma ll detuning from this internal resonance and that from the external resonan ce with the vertical oscillation of a container is derived using the method of reductive perturbation and also including a linear damping. This equati on has mixed-wave solutions: periodic and chaotic solutions as well as stat ionary solutions. Moreover, we find two more hysteresis regions of stationa ry solutions, in addition to the hysteresis region observed for a single-mo de Faraday wave. Some periodic solutions become chaotic through a series of period-doubling bifurcations.