UNIVERSAL TRAPPING SCALING ON THE UNSTABLE MANIFOLD FOR A COLLISIONLESS ELECTROSTATIC MODE

An amplitude equation for an unstable mode in a collisionless plasma is derived from the dynamics on the two-dimensional unstable manifold of the equilibrium. The mode amplitude rho(t) decouples from the phase due to the spatial homogeneity of the equilibrium, and the resulting one-dimensional dynamics is analyzed using an expansion in rho. As the linear growth rate gamma vanishes, the expansion coefficients diverge; a rescaling rho(t) = gamma2r(gammat) of the mode amplitude absorbs these singularities and reveals that the mode electric field exhibits trapping scaling \E1\ is similar to gamma2 as gamma --> 0. The dynamics for r(tau) depends only on the phase e(ixi) where depsilon(k)/dz = \epsilon'k\e-ixi/2 is the derivative of the dielectric as gamma --> 0.