Three wave resonance interactions: transition to chaos

The resonance interactions of the waves are playing very important roles in many media  including liquids, atmosphere, solids, biological systems, and plasmas. In the studies of  such interactions it is often assumed that the interacting waves are neutrally stable and  just transfer the energy, enstrophy, etc. over the wavelengths (e.g. recall the Kolmogorov  spectrum). Moreover, it is often assumed that the large number of interacting waves  causes the ergodicity in the dynamics of the ensemble of the waves. However, it was  shown by Lorenz that even low dimension systems could exhibit chaotic behavior. Some  time ago it also was demonstrated that the presence of linear source/sinks of the waves,  drastically changes the ensemble dynamics and allows the transition from a weak to the  strong turbulence regimes.  

In these studies we examine the dynamics of the interactions of just three waves  in the presence of linear source/sink terms. We show that such simple system exhibits  rich phenomena including exponential growth, limit cycles, strange attractors, and  “multi-lobe” strange attractors. Therefore, the ergodicity in the dynamics of the ensemble  of the waves could be related to these novel features of the wave interactions and not just  to “many body” interaction phenomena.