Evolution of the tangent vectors and Localization of the stable and unstable manifolds of hyperbolic orbits by Fast Laypunov Indicators

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 hyperbolic saddle points. 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 problems of detection of the so-called tube manifolds of the Lyapunov orbits of L 1 , L 2 in the planar circular restricted three-body problem; detection of the Lorenz manifold; detection of the stable manifold of a saddle equilibrium point for two strongly coupled pendula.