Tranversality of the invariant manifolds associated to the Lyapunov family of periodic orbits near L2 in the restricted three-body problem

Abstract

The restricted three-body problem is considered for values of the Jacobi constant C near the value C 2 associated to the Euler critical point L 2. A Lyapunov family of periodic orbits near L 2, the so-called family ( c), is born for C = C 2 and exists for values of C less than C 2. These periodic orbits are hyperbolic. The corresponding invariant manifolds meet transversally along homoclinic orbits. In this paper the variation of the transversality is analyzed as a function of the Jacobi constant C and of the mass parameter μ. Asymptotical expressions of the invariant manifolds for C ≲ C 2 and μ ≳ 0 are found. Several numerical experiments provide accurate information for the manifolds and a good agreement is found with the asymptotical expressions. Symbolic dynamic techniques are used to show the existence of a large class of motions. In particular the existence of orbits passing in a random way (in a given sense) from the region near one primary to the region near the other is proved.