CENTER-MANIFOLD THEORY FOR LOW-FREQUENCY EXCITATIONS IN MAGNETIZED PLASMAS

For the dissipative trapped-ion mode, a simple one-dimensional nonlinear model equation, including the effects of instability, dissipation, and dispersion, is investigated. The center-manifold theory is applied to the situation of more than one marginally stable mode, and the dynamics in the neighborhood of the onset of instability is elucidated. Depending on the (three) relevant parameters, stable solitary waves, mixed modes, heteroclinic orbits, etc., can exist; a scenario for the nonlinear dynamical behavior is developed. The bifurcation diagrams are drawn with quantitative predictions in parameter space. An important conclusion is that the codimension-two analysis utilized can predict successive bifurcations which cannot be captured by simple analysis of one unstable mode. The analytical calculations are checked by numerical simulations.