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

TL;DR

This paper analytically describes how Fast Lyapunov Indicators can detect stable and unstable manifolds of hyperbolic orbits, improving precision in dynamical systems analysis, especially in celestial mechanics.

Contribution

It provides an analytic explanation for the growth of tangent vectors and introduces a modified Fast Lyapunov Indicator for better detection of manifolds.

Findings

Fast Lyapunov Indicators detect stable-unstable manifolds under certain conditions.

A modified indicator extends detection capabilities when conditions are not met.

Detection precision increases linearly with integration time.

Abstract

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.