We consider the Gyldén problem—a perturbation of the Kepler problem via an explicit function of time. For certain general classes of planar periodic perturbations, after proving a Poincaré–Melnikov-type criterion, we find a manifold of orbits in which the dynamics is given by the shift automorphism on the set of bi-infinite sequences with infinitely many symbols. We achieve the main result by computing the Melnikov integral explicitly.
This study presents a method of obtaining asymptotic approximations for motions near a Lagrange point in the planar, elliptic, restricted three-body problem by using a yon Zeipel-type method. The calculations are carded out for a second-order escape solution in the proximity of the equilateral Lagrange point, L4, where the primaries' orbital eccentricity is taken as the small parameter ¢.
Abstract. We generalize the well-known Hill's circular restricted three-body problem by assuming that the primary generates a Schwarzschild-type field of the form U = A/r + B/r 3 . The term in B influences the particle, but not the far secondary. Many concrete astronomical situations can be modelled via this problem. For the two-body problem primary-particle, a homoclinic orbit is proved to exist for a continuous range of parameters (the constants of energy and angular momentum, and the field parameter B > 0). Within the restricted three-body system, we prove that, under sufficiently small perturbations from the secondary, the homoclinic orbit persists, but its stable and unstable manifolds intersect transversely. Using a result of symbolic dynamics, this means the existence of a Smale horseshoe, hence chaotic behaviour. Moreover, we find that Hill's generalized problem (in our sense) is nonintegrable.
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