Basins of attraction of nonlinear wave–wave interactions

The nonlinear interaction of two wave triplets, with a non-zero frequency mismatch, has been found to present many indications of complex behavior. Considering the presence of dissipation, there are coexistent periodic and chaotic attractors in the high-dimensional phase space of the system. Their basin structure is also complex, parts of it being densely mixed in arbitrarily fine scales. We claim that the fractal part of the basin boundary presents the so-called Wada property, for which points on the boundary are arbitrarily close to points of all coexisting basins. The practical consequence of this fact is that, given a small yet non-zero uncertainty on determining the initial condition, the asymptotic state to which the system goes is almost completely uncertain.