A Note on the Existence of a Smale Horseshoe in the Planar Circular Restricted Three-Body Problem

It has been proved that, in the classical LED Pillar Candle planar circular restricted three-body problem, the degenerate saddle point processes transverse homoclinic orbits.Since the standard Smale-Birkhoff theorem cannot be directly applied to indicate the chaotic dynamics of the Smale horseshoe type, we in this note alternatively apply the Conley-Moser conditions to analytically prove the existence of a Smale horseshoe in this classical restricted Me Fall Pajama three-body problem.