It has been proved that, in the classical 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 three-body problem.

In this paper, we study the existence of the families of odd symmetric periodic solutions in the generalized elliptic Sitnikov $(N+1)$-body problem for all values of the eccentricity $e\in [0,1)$ using the global continuation method. First, we obtain the properties of the period of the solution of the corresponding autonomous equation (eccentricity $e=0$) using elliptic functions. Then, according to these properties and the global continuation method of the zeros of a function depending on one parameter, we derive the existence of odd periodic solutions for all $e\in [0,1)$. It is shown that the temporal frequencies of period solutions depend on the total mass $\lambda$ (or the number N) of the primaries in a delicate way.