Chaos and stability in a two-parameter family of convex billiard tables

We study, by numerical simulations and semi-rigorous arguments, a two-parameter family of convex, two-dimensional billiard tables, generalizing the one-parameter class of oval billiards of Benettin--Strelcyn [Phys. Rev. A 17, 773 (1978)]. We observe interesting dynamical phenomena when the billiard tables are continuously deformed from the integrable circular billiard to different versions of completely-chaotic stadia. In particular, we conjecture that a new class of ergodic billiard tables is obtained in certain regions of the two-dimensional parameter space, when the billiards are close to skewed stadia. We provide heuristic arguments supporting this conjecture, and give numerical confirmation using the powerful method of Lyapunov-weighted dynamics.Comment: 19 pages, 13 figures. Submitted for publication. Supplementary video available at http://sistemas.fciencias.unam.mx/~dsanders

Time-frequency analysis of the restricted three-body problem: transport and resonance transitions

A method of time-frequency analysis based on wavelets is applied to the problem of transport between different regions of the solar system, using the model of the circular restricted three-body problem in both the planar and the spatial versions of the problem.. The method is based on the extraction of instantaneous frequencies from the wavelet transform of numerical solutions. Time-varying frequencies provide a good diagnostic tool to discern chaotic trajectories from regular ones, and we can identify resonance islands that greatly affect the dynamics. Good accuracy in the calculation of time-varying frequencies allows us to determine resonance trappings of chaotic trajectories and resonance transitions. We show the relation between resonance transitions and transport in different regions of the phase space

Fast numerical computation of Lissajous and quasi-halo libration point trajectories

In this paper we present a methodology for the automatic generation of quasi–periodic libration point trajectories (Lissajous and quasi–halo) of the Spatial, Circular Restricted Three–Body Problem. This methodology is based on the computation of a mesh of orbits which, using interpolation strategies, gives an accurate quantitative representation of the full set of libration point orbits. This representation, when combined with the one obtained using Poincar´e maps, provides a useful tool for the design of missions to libration points fulfilling specific requirements. The same methodology applies to stable and unstable manifolds as well. This paper extends and improves results presented in [10].Postprint (published version