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In this paper, we present a model describing the time evolution of twodimensional surface waves in gravity and infinite depth. The model of sixinteracting modes derives from the normal form of the system describing thedynamics of surface waves and is governed by a Hamiltonian system of equationsof cubic order in the amplitudes of the waves. We derive a Hamiltonian systemwith two degrees of freedom from this Hamiltonian using conserved quantities.The interactions are those of two coupled Benjamin-Feir resonances. Thetemporal evolution of the amplitude of the different modes is describedaccording to the parameters of the system. In particular, we study the energyexchange produced by the modulations of the amplitudes of the modes. Theevolution of the modes reveals a chaotic dynamics

A Hamiltonian system for interacting Benjamin–Feir resonances