We discuss the stochasticities of two-triad interactions that occur in two-degree-of-freedom autonomous Hamiltonian systems. The system we study is a two-triad test-wave system consisting of a single internal wave mode (test-wave) interacting with a spectrum of ambient internal wave modes; the ambient modes, however, do not interact among themselves except through a three-wave interaction which includes the test-wave. The present study concerns the effect of nonlinearities on the ocean internal wave field. Our numerical results using the physical parameters appropriate for the deep ocean confirm that the test-wave system is non-integrable. Moreover, there exists a certain separatrix net that fills the phase space and is covered by a thin stochastic layer for a two-triad pure resonant interaction. The stochastic web implies the existence of diffusion of the Arnold type for the minimal dimension of a non-integrable autonomous system. For the non-resonant case, the stochastic layer is formed where the separatrix from KAM theory is disrupted. However, the stochasticity does not increase monotonically with increasing energy. ©1997 American Institute of Physics.