  The Fast Lyapunov Indicators are functions defined on the tangent fiber of
the phase-space of a discrete (or continuous) dynamical system, by using a
finite number of iterations of the dynamics. In the last decade, they have been
largely used in numerical computations to localize the resonances in the
phase-space and, more recently, also the stable and unstable manifolds of
normally hyperbolic invariant manifolds. In this paper, we provide an analytic
description of the growth of tangent vectors for orbits with initial conditions
which are close to the stable-unstable manifolds of a hyperbolic saddle point
of an area-preserving map. The representation explains why the Fast Lyapunov
Indicator detects the stable-unstable manifolds of all fixed points which
satisfy a certain condition. If the condition is not satisfied, a suitably
modified Fast Lyapunov Indicator can be still used to detect the
stable-unstable manifolds. The new method allows for a detection of the
manifolds with a number of precision digits which increases linearly with
respect to the integration time. We illustrate the method on the critical
problem of detection of the so-called tube manifolds of the Lyapunov orbits of
L1,L2 in the circular restricted three-body problem.
