Corrigendum to “Oscillatory motions in restricted N-body problems” [J. Differential Equations 265 (2018) 779–803]

We consider the planar restricted $N$-body problem where the $N-1$ primaries are assumed to be in a central configuration whereas the infinitesimal particle escapes to infinity in a parabolic orbit. We prove the existence of transversal intersections between the stable and unstable manifolds of the parabolic orbits at infinity which guarantee the existence of a Smale's horseshoe. This implies the occurrence of chaotic motions but also of oscillatory motions, that is, orbits for which the massless particle leaves every bounded region but it returns infinitely often to some fixed bounded region. Our achievement is based in an adequate scaling of the variables which allows us to write the Hamiltonian function as the Hamiltonian of the Kepler problem multiplied by a small quantity plus higher-order terms that depend on the chosen configuration. We compute the Melnikov function related to the first two non-null perturbative terms and characterize the cases where it has simple zeroes. Concretely, for some combinations of the configuration parameters, i.e. mass values and positions of the primaries, the Melnikov function vanishes, otherwise it has simple zeroes and the transversality condition is satisfied. The theory is illustrated for various cases of restricted $N$-body problems, including the circular restricted three-body problem. No restrictions on the mass parameters are assumed.