Stability, singular perturbations, and the vector Liénard equation

The rapid rate at which solutions of the vector equation
\epsilon\ddot{x} + R(x)\dot{x} + g(x) = 0, (0 < \epsilon \ll 1)
approach the subspace
\dot{x} = -R^{-1}(x)g(x)
is obtained and the subsequent slow decay to zero is estimated.