  In this paper, we present a model describing the time evolution of two
dimensional surface waves in gravity and infinite depth. The model of six
interacting modes derives from the normal form of the system describing the
dynamics of surface waves and is governed by a Hamiltonian system of equations
of cubic order in the amplitudes of the waves. We derive a Hamiltonian system
with two degrees of freedom from this Hamiltonian using conserved quantities.
The interactions are those of two coupled Benjamin-Feir resonances. The
temporal evolution of the amplitude of the different modes is described
according to the parameters of the system. In particular, we study the energy
exchange produced by the modulations of the amplitudes of the modes. The
evolution of the modes reveals a chaotic dynamics.

Chaotic Dynamics