Damping models in the truncated derivative nonlinear Schroedinger equation

Four-dimensional flow in the phase space of three amplitudes of circularly polarized Alfven waves and one relative phase, resulting from a resonant three-wave truncation of the derivative nonlinear Schroedinger equation, has been analyzed; wave 1 is linearly unstable with growth rate Γ, and waves 2 and 3 are stable with damping γ2 and γ3, respectively. The dependence of gross dynamical features on the damping model (as characterized by the relation between damping and wave-vector ratios, γ2/γ3, k2/k3), and the polarization of the waves, is discussed; two damping models, Landau (γ∝k) and resistive (γ∝k2), are studied in depth. Very complex dynamics, such as multiple blue sky catastrophes and chaotic attractors arising from Feigenbaum sequences, and explosive bifurcations involving Intermittency-I chaos, are shown to be associated with the existence and loss of stability of certain fixed point P of the flow. Independently of the damping model, P may only exist for Γ<2(γ2+γ3)/3, as against flow contraction just requiring Γ<γ2+γ3. In the case of right-hand (RH) polarization, point P may exist for all models other than Landau damping; for the resistive model, P may exist for RH polarization only if Γ<(γ2+γ3)/2