Study on Chaotic Behavior of the Restricted Four-Body Problem with an Equilateral Triangle Configuration

In this paper, the chaotic behavior of a planar restricted four-body problem with an equilateral triangle configuration is analytically studied. Firstly, according to the perturbation method of Melnikov, the planar restricted four-body problem is regarded as a perturbation of the two-body model. Then, we show that the Melnikov integral function has a simple zero, arriving at the existence of transversal homoclinic orbits. Afterwards, since the standard Smale-Birkhoff homoclinic theorem cannot be directly applied to the case of a degenerate saddle, we alternatively construct an invertible map f and check that f is a Smale horseshoe map, showing that our restricted four-body problem possesses chaotic behavior of the Smale horseshoe type.