Study on Chaotic Behavior of the (N + 1)-Body Ring Problem

Abstract

In this paper, we study chaotic behavior of the (N + 1)-body planar ring problem. Firstly, based on the perturbation theory of integrable Hamiltonian systems, we utilize mass ratio as a disturbance parameter so that this ring problem is regarded as a perturbation of the two-body problem. Then, by applying the extended Melnikov method, we address that there are transversal homoclinic orbits in the ring problem. Afterwards, since the standard Smale–Birkhoff homoclinic theorem cannot be directly applied to the case of a degenerate saddle, we construct an invertible map f satisfying Conley–Moser condition and finally conclude that the ring problem possesses chaotic behavior of the Smale horseshoe type.