Regions of nonexistence of invariant tori for spin-orbit models Chaos 17, 043119 (2007) Stochastic embedding of dynamical systems J. Math. Phys. 48, 072703 (2007) Kepler problem with time-dependent and resonant perturbations J. Math. Phys. 48, 052701 (2007) Additional information on Chaos Control and Dynamical Systems, Caltech 107-81, Pasadena, California 91125 ͑Received 21 May 1999; accepted for publication 6 December 1999͒In this paper we apply dynamical systems techniques to the problem of heteroclinic connections and resonance transitions in the planar circular restricted three-body problem. These related phenomena have been of concern for some time in topics such as the capture of comets and asteroids and with the design of trajectories for space missions such as the Genesis Discovery Mission. The main new technical result in this paper is the numerical demonstration of the existence of a heteroclinic connection between pairs of periodic orbits: one around the libration point L 1 and the other around L 2 , with the two periodic orbits having the same energy. This result is applied to the resonance transition problem and to the explicit numerical construction of interesting orbits with prescribed itineraries. The point of view developed in this paper is that the invariant manifold structures associated to L 1 and L 2 as well as the aforementioned heteroclinic connection are fundamental tools that can aid in understanding dynamical channels throughout the solar system as well as transport between the ''interior'' and ''exterior'' Hill's regions and other resonant phenomena.

This paper concerns heteroclinic connections and resonance transitions in the planar circular restricted 3-body problem, with applications to the dynamics of comets and asteroids and the design of space missions such as the Genesis Discovery Mission and low energy Earth to Moon transfers. The existence of a heteroclinic connection between pairs of equal energy periodic orbits around two of the libration points is shown numerically. This is applied to resonance transition and the construction of orbits with prescribed itineraries. Invariant manifold structures are relevant for transport between the interior and exterior Hill's regions, and other resonant phenomena throughout the solar system.

The invariant manifold structures of the collinear libration points for the restricted three-body problem provide the framework for understanding transport phenomena from a geometrical point of view. In particular, the stable and unstable invariant manifold tubes associated with libration point orbits are the phase space conduits transporting material between primary bodies for separate three-body systems. These tubes can be used to construct new spacecraft trajectories, such as a 'Petit Grand Tour' of the moons of Jupiter. Previous work focused on the planar circular restricted three-body problem. This work extends the results to the three-dimensional case. Besides providing a full description of different kinds of libration motions in a large vicinity of these points, this paper numerically demonstrates the existence of heteroclinic connections between pairs of libration orbits, one around the libration point L 1 and the other around L 2. Since these connections are asymptotic orbits, no manoeuvre is needed to perform the transfer from one libration point orbit to the other. A knowledge of these orbits can be very useful in the design of missions such as the Genesis Discovery Mission, and may provide the backbone for other interesting orbits in the future.

We generalize the concepts of finite-time Lyapunov exponent ͑FTLE͒ and Lagrangian coherent structures to arbitrary Riemannian manifolds. The methods are illustrated for convection cells on cylinders and Möbius strips, as well as for the splitting of the Antarctic polar vortex in the spherical stratosphere and a related point vortex model. We modify the FTLE computational method and accommodate unstructured meshes of triangles and tetrahedra to fit manifolds of arbitrary shape, as well as to facilitate dynamic refinement of the FTLE mesh. © 2010 American Institute of Physics. ͓doi:10.1063/1.3278516͔Riemannian manifolds are ubiquitous in science and engineering, being the more natural mathematical setting for many dynamical systems. For instance, transport along isopycnal surfaces in the ocean and large-scale mixing in the atmosphere are processes taking place on a curved manifold, not a vector space. In this paper, we generalize the notion of finite-time Lyapunov exponent (FTLE) and Lagrangian coherent structures (LCS) to arbitrary Riemannian differentiable manifolds. We show that both notions are independent of the coordinate system but depend on the chosen metric. However, we find that LCS do not depend much on the metric. The FTLE measures separation and tends to be large and positive along LCS and very small elsewhere. For sufficiently large integration times, the steep variations of the FTLE field cannot be modified much by smooth changes in the metric and the LCS remain essentially unchanged. Approximating, or even ignoring, the manifold metric does not influence the result for large integration times, and therefore, computing the FTLE field on manifolds is robust. Aside from these conclusions, we present a general algorithm for computing the FTLE on manifolds covered with meshes of polyhedra. The algorithm requires knowledge of the mesh nodes, the image of the mesh nodes under the flow, as well as information about node neighbors (but not the full connectivity). We also used the same algorithm in Euclidian spaces where the unstructured mesh permits efficient adaptive refinement for capturing sharp LCS features. We illustrate the results and methods on several systems: convection cells in a plane, on a cylinder, and on a Möbius strip, as well as atmospheric transport resulting from the 2002 splitting of the Antarctic ozone hole in the spherical stratosphere and a related point vortex model on the sphere.