Fully nonlinear phenomenology of the Berk-Breizman augmentation of the Vlasov-Maxwell system

The Berk–Breizman augmentation of the Vlasov–Maxwell system is widely used to model self-consistent resonant excitation and damping of wave fields by evolving energetic particle populations in magnetic fusion plasmas. The key model parameters are the particle annihilation rate nua, which drives bump-on-tail structure, and the linear wave damping rate gammad. A code, based on the piecewise parabolic method, is used to integrate the fully nonlinear Berk–Breizman system of equations across the whole (nua,gammad) parameter space. The results of this code show that the system's behavior can be classified into one of four types, each of which occurs in a well-defined region of parameter space: chaotic, periodic, steady state, and damped. The corresponding evolution in (x,v) phase space is also examined.