Dynamics and Chaos in the Elliptic Isosceles Restricted Three-Body Problem with Collision

Abstract

The elliptic isosceles restricted three-body problem (EIR3BP) with collision is defined as follows: two point masses \(m_1= m_2\) move along a degenerate elliptic collision orbit under their gravitational attraction, then describe the motion of a third massless particle moving on a plane perpendicular to their line of motion and passing through the center of mass of the primaries. By symmetry, the component of the angular momentum of the massless particle along the direction of the line of the primaries is conserved. We fixed it to a non-zero value in order to avoid total collision, and perform the reduction to one and a half degrees of freedom. We prove that the flow defined by the EIR3BP is complete and if a solution escapes to infinity when time \(t \rightarrow \pm \infty \), then it is parabolic or hyperbolic. A description of the parabolic orbits is given and they are asymptotic to a degenerate periodic orbit at infinity. We verify that the unstable and stable manifolds \(P^{u,s}\) of this periodic orbit at infinity are differentiable (in fact, \(C^{\infty }\)) at the origin and analytic outside. For sufficiently large angular momentum, we prove that \(P^{u}\) and \(P^{s}\) intersect a surface of section \(\Sigma \) in simple closed curves \(\gamma ^{u,s}\) having two points of intersection and we show that \(\gamma ^{u}\) and \(\gamma ^{s}\) have a transversal intersection at these points. We prove that there exists a subset of \(\Sigma \) where the Poincare map is topologically conjugate to a Bernoulli shift, in particular this shows the existence of a very complicated dynamic (chaotic dynamic) in the EIR3BP.