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We revisit the problem of a triad of resonantly interacting nonlinear wavesdriven by an external force applied to the unstable mode of the triad. Theequations are Hamiltonian, and can be reduced to a dynamical system for 5 realvariables with 2 conservation laws. If the Hamiltonian, $H$, is zero we reducethis dynamical system to the motion of a particle in a one-dimensionaltime-independent potential and prove that the system is integrable. Explicitsolutions are obtained for some particular initial conditions. When explicitsolution is not possible we present a novel numerical/analytical method forapproximating the dynamics. Furthermore we show analytically that when H=0 themotion is generically bounded. That is to say the waves in the forced triad arebounded in amplitude for all times for any initial condition with the singleexception of one special choice of initial condition for which the forcing isin phase with the nonlinear oscillation of the triad. This means that theenergy in the forced triad generically remains finite for all time despite thefact that there is no dissipation in the system. We provide a detailedcharacterisation of the dependence of the period and maximum energy of thesystem on the conserved quantities and forcing intensity. When $H \neq 0$ wereduce the problem to the motion of a particle in a one-dimensionaltime-periodic potential. Poincar\'{e} sections of this system provide strongevidence that the motion remains bounded when $H \neq 0$ and is typicallyquasi-periodic although periodic orbits can certainly be found. Throughout ouranalyses, the phases of the modes in the triad play a crucial role inunderstanding the dynamics.

Externally forced triads of resonantly interacting waves: boundedness and integrability properties