Abstract
Third-harmonic resonance of capillary-gravity waves in two-dimensional Faraday waves due to the parametric excitation of the lower-frequency mode is examined for infinite depth. The amplitude equation incorporating both a small detuning from this internal resonance and that from the external resonance with the vertical oscillation of a container is derived using the method of reductive perturbation and also including a linear damping. This equation has mixed-wave solutions: periodic and chaotic solutions as well as stationary solutions. Moreover, we find two more hysteresis regions of stationary solutions, in addition to the hysteresis region observed for a single-mode Faraday wave. Some periodic solutions become chaotic through a series of period-doubling bifurcations.
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