The existence of a Smale horseshoe in a planar circular restricted four-body problem

In this paper we study the existence of a Smale horseshoe in a planar circular restricted four-body problem. For this planar four-body system there exists a transversal homoclinic orbit, but the fixed point is a degenerate saddle, so that the standard Smale–Birkhoff homoclinic theorem cannot be directly applied. We therefore apply the Conley–Moser conditions to prove the existence of a Smale horseshoe. Specifically, we first use the transversal structure of stable and unstable manifolds to make a linear transformation and then introduce a nonlinear Poincaré map $P$ by considering the truncated flow near the degenerate saddle; based on this Poincaré map $P$ , we define an invertible map $f$ , which is a composite function; by carefully checking the satisfiability of the Conley–Moser conditions for $f$ we finally prove that $f$ is a Smale horseshoe map, which implies that our restricted four-body problem has the chaotic dynamics of the Smale horseshoe type.