The presence of an invariant manifold in the dissipative nonlinear four-wave coupling

We examine the existence of an invariant manifold in the phase space structure presented by a dissipative four-wave coupling. We show that, once in such manifold, the system preserves two quantities as time evolves: a quantity related to the energy of the system and another, related to the exchange of energy between the waves. We observe that the presence of this manifold allows the system to manifest stable solutions for a specific interval of the dissipation parameters. Out of this range the system brings up only uncoupled states (no nonlinear saturated states). The dissipation parameters affect only the transversal dynamics to the manifold, changing how quickly the invariant manifold is reached. In the invariant manifold, the system exhibits a large number of coexisting periodic attractors—limit cycles (coupled states), with an interwoven structure of basins of attraction. The time series of conserved quantities and the properties of the Lyapunov spectra are used to imply the existence of a lower-dimensional invariant manifold. Since the dynamics in such invariant manifold preserves an energy-like function, we identify it as being conservative.