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 ρ(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 ρ. As the linear growth rate γ vanishes, the expansion coefficients diverge; a rescaling ρ(t)≡${\gamma }^2$r(γt) of the mode amplitude absorbs these singularities and reveals that the mode electric field exhibits trapping scaling |${\mathrm{E}}_1$|∼${\gamma }^2$ as γ→0. The dynamics for r(τ) depends only on the phase ${\mathrm{e}}^{\mathrm{i}\xi }$ where d${\mathrm{\epsilon }}_{\mathrm{k}}$/dz=|${\mathrm{\epsilon }}_{\mathrm{k}}^{\prime }$|${\mathrm{e}}^{\mathrm{-}\mathrm{i}\xi /2\mathrm{}}$ is the derivative of the dielectric as γ→0.