Chaotic behavior in the 3-center problem

We consider the motion of a particle in a plane under the gravitational action of 3 fixed centers (the 3-center planar problem). As it is well known (Vestnik Moskov. Univ. Ser. 1 Matem. Mekh 6 (1984) 65; Prikl. Matem. i Mekhan. 48 (1984) 356; Classical Planar Scattering by Coulombic Potentials, Lecture Notes in Physics, Springer, Berlin, 1992) on the non-negative level sets of the energy E there do not exist non-constant analytic first integrals, and moreover the system has chaotic trajectories. These results were proved by variational methods. Here we investigate the problem in the domain of small negative values of E. Moreover, we assume that one of the centers is far away from the other two. Then we get a two-parameter singular perturbation of an integrable dynamical system: the 2-center problem on the zero-energy level. The main problem we deal with is to prove that the Poincaré–Melnikov theory applies in the limit case E→0.