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The Fast Lyapunov Indicators are functions defined on the tangent fiber ofthe phase-space of a discrete (or continuous) dynamical system, by using afinite number of iterations of the dynamics. In the last decade, they have beenlargely used in numerical computations to localize the resonances in thephase-space and, more recently, also the stable and unstable manifolds ofnormally hyperbolic invariant manifolds. In this paper, we provide an analyticdescription of the growth of tangent vectors for orbits with initial conditionswhich are close to the stable-unstable manifolds of a hyperbolic saddle pointof an area-preserving map. The representation explains why the Fast LyapunovIndicator detects the stable-unstable manifolds of all fixed points whichsatisfy a certain condition. If the condition is not satisfied, a suitablymodified Fast Lyapunov Indicator can be still used to detect thestable-unstable manifolds. The new method allows for a detection of themanifolds with a number of precision digits which increases linearly withrespect to the integration time. We illustrate the method on the criticalproblem of detection of the so-called tube manifolds of the Lyapunov orbits ofL1,L2 in the circular restricted three-body problem.

Evolution of the tangent vectors and localization of the stable and unstable manifolds of hyperbolic orbits by Fast Lyapunov Indicators