  In this paper we derive a quantitative dichotomy for the top Lyapunov
exponent of a class of non-dissipative SDEs on a compact manifold in the small
noise limit. Specifically, we prove that in this class, either the Lyapunov
exponent is zero for all noise strengths, or it is positive for all noise
strengths and that the decay of the exponent in the small-noise limit cannot be
faster than linear in the noise parameter. As an application, we study the top
Lyapunov exponent for the motion of a charged particle in randomly-fluctuating
magnetic fields, which also involves an interesting geometric control problem.
