The Existence of Odd Symmetric Periodic Solutions in the Generalized Elliptic Sitnikov (N+1)-Body Problem

Abstract

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∈[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∈[0,1). It is shown that the temporal frequencies of period solutions depend on the total mass λ (or the number N) of the primaries in a delicate way.