  An amplitude equation for an unstable mode in a collisionless plasma is
derived from the dynamics on the unstable manifold of the equilibrium
$F_0(v)$.\\ The mode eigenvalue arises from a simple zero of the dielectric
$\epsilon_{{k}}(z)$; as the linear growth rate $\gamma$ vanishes, the
eigenvalue merges with the continuous spectrum on the imaginary axis and
disappears. The evolution of the mode amplitude $\rho(t)$ is studied using an
expansion in $\rho$. As $\glim$, the expansion coefficients diverge, but these
singularities are absorbed by rescaling the amplitude:
$\rho(t)\equiv\gamma^2\,r(\gamma t)$. This renders the theory finite and also
indicates that the electric field exhibits trapping scaling $E\sim\gamma^2$.
These singularities and scalings are independent of the specific $F_0(v)$
considered. The asymptotic dynamics of $r(\tau)$ depends on $F_0$ only through
$\exp{i\xi}$ where $d\epsilon_{{k}} /dz=|{\epsilon'_{{k}}}|\exp{-i\xi/2}$.
Similar results also hold for the electric field and distribution function.

Chaotic Dynamics;Chaotic Dynamics