Theory and applications of fast Lyapunov indicators to model problems of celestial mechanics

In the last decades, we have seen a rapid increment in the use of finite-time chaos indicators in celestial mechanics. They have been used to analyze the complex dynamics of planetary systems, of minor planets and of space debris. In fact, theoretical studies on fundamental dynamical models have revealed that, computed on short time intervals, they allow to efficiently detect resonances, represent the phase portraits of complex dynamics, compute center-stable-unstable manifolds as well as Lagrangian coherent structures. In this paper, we focus on applications of the fast Lyapunov indicator (FLI) and review through examples why its computation is particularly powerful for those systems whose solutions may have an asymptotic behavior very different from the short-term one, as it is the case of sequences of close encounters in gravitational systems and the advection of particles in aperiodic flows. The main case study here considered is the computation of the manifold tubes and the related transit orbits in the restricted three-body problem. We also provide a new application of the FLI to a complex problem of planetary hydrodynamics, such as the detection of the stable and unstable manifolds guiding the motions of particles advected by the gas of a protoplanetary nebula.