Subharmonic solutions in the restricted three-body problem (Q1355015)

The restricted three-body problem is considered where the primaries \(P_1\) and \(P_2\) with masses \(\mu\) and \(1-\mu\) move in circular orbits around their center of mass with period \(2\pi\) and a third particle with zero mass, \(P_3\), moves in the plane of the primaries. Solutions of this problem are considered which are close to a circular solution of Kepler problem in the case \(\mu=0\). The main result is stated in theorem: Let \((x_0(s,\delta),y_0(s,\delta))\) be the periodic solution of the unperturbed system with initial condition \(x_0= a+\delta\), \(y_0=0\) and period \(T={m\over n} 2\pi\). If \(n>1\) then, for \(\delta\) small, there bifurcate from this solution exactly \(4m\) periodic solutions of the perturbed system with period \(m\cdot 2\pi\), for small values of the parameter \(\mu\). The proof of this theorem is based on the Melnicov method which is discussed in detail.