Symbolic Dynamics in the Elliptic Isosceles Restricted Three Body Problem

TL;DR

This paper demonstrates symbolic dynamics and oscillatory motions in the elliptic isosceles restricted three body problem by constructing a Smale horseshoe, revealing complex chaotic behavior for large angular momentum values.

Contribution

It establishes the existence of symbolic dynamics in the REI3BP through a novel approach that does not rely on Melnikov theory, using transversal homoclinic connections.

Findings

Existence of symbolic dynamics via Smale horseshoe construction.

Presence of oscillatory orbits that leave and return to bounded regions.

Chaotic behavior confirmed for large angular momentum G.

Abstract

The elliptic isosceles restricted three body problem (REI3BP) models the motion of a massless body under the influence of the Newtonian gravitational force caused by two other bodies called the primaries. The primaries of masses $m_{1}=m_{2}$ move along a degenerate Keplerian elliptic collision orbit (on a line) under their gravitational attraction, whereas the third, massless particle, moves on the 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 $G$ of the massless particle along the direction of the line of the primaries is conserved. We show the existence of symbolic dynamics in the REI3BP for large $G$ by building a Smale horseshoe on a certain subset of the phase space. As a consequence we deduce that the REI3BP possesses oscillatory motions, namely orbits which leave every bounded region but return infinitely often to some fixed bounded region. The proof relies on the existence of transversal homoclinic connections associated to an invariant manifold at infinity. Since the distance between the stable and unstable manifolds of infinity is exponentially small, Melnikov theory does not apply.