Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics

Abstract

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.

Controlled Lagrangians and the stabilization of mechanical systems. II. Potential shaping

“…Use can be made in this setting of heteroclinic connections. This is related to the work of [15] and [27].…”

“…Now assume that (27) holds. Then, by using the vector calculus, we derive the following solution to the PDE in (26): where is an arbitrary function.…”

“…For each , we choose any curve joining and . Then, the integration is path-independent by (27) and Stokes' Theorem (we regard as a -dependent one-form on ). In the old coordinates, (27) is expressed as (28) which is Assumption SM-5.…”

“…Then, the integration is path-independent by (27) and Stokes' Theorem (we regard as a -dependent one-form on ). In the old coordinates, (27) is expressed as (28) which is Assumption SM-5. Thus, Assumption SM-5 is a necessary and sufficient condition for the integrability of the PDE (16).…”

Optimal Control for Halo Orbit Missions

“…The unstable manifold, which winds off of the halo orbit, is used to design the return trajectory which brings the spacecraft and its precious samples back to Earth via a heteroclinic connection with L 2 . See Koon et al [1999] for the current state of the computation of homoclinic and heteroclinic orbits in this problem.…”