Linearly unstable dissipative systems with quadratic nonlinearity occurring in plasma physics, optics, fluid mechanics, etc. are often modeled by a general set of three-wave mode-coupled ordinary differential equations for complex variables. Bounded attractors of the set approximate nonlinearly saturated turbulent states of real physical systems. Exact criteria for boundedness of the attractors are found. Fundamentally different kinds of asymptotic behavior of the wave triad are classified in the parameter space and quantitatively assessed.

• Received 9 June 1999

DOI:https://doi.org/10.1103/PhysRevLett.84.622