| Abstract: |
| The Restricted $3$-Body Problem models the motion of a body of negligible mass under the gravitational influence of two massive bodies, called the primaries. If the primaries perform circular motions and the massless body is coplanar with them, one has the so called Restricted Planar Circular $3$-Body Problem (RPC$3$BP). In synodic coordinates, can be modeled by a two degrees of freedom Hamiltonian system with five critical points, $L_1,..,L_5$, called the Lagrange points.
The Lagrange point $L_3$ is a saddle-center critical point, which is collinear with the primaries and beyond the largest one. When the ratio between the masses of the primaries $\mu$ is small, the modulus of the hyperbolic eigenvalues are weaker, by a factor of order $\sqrt\mu$, than the elliptic ones. Due to the rapidly rotating dynamics, the $1$-dimensional unstable and stable manifold of $L_3$ are exponentially close to each other with respect to $\sqrt\mu$. In previous works we provided an asymptotic formula for the distance between these invariant manifolds for small ratios of the mass parameter. This result relies on a Stokes constant which, using computer assisted proofs, we will prove that is non zero. In this paper, we study different chaotic and homoclinic phenomena occurring in a neighborhood of $L_3$ and its invariant manifolds.
The first result concerns the existence of $2$-round homoclinic connections to $L_3$, i.e. homoclinic orbits that approach the critical point $2$-times. More concretely, we prove the existence of $2$-round homoclinic orbits for a specific sequence of mass ratio parameters. The second result studies the family of Lyapunov periodic orbits of $L_3$ with Hamiltonian energy level exponentially close to that of $L_3$. In particular, we show that there exists a set of periodic orbits whose unstable and stable manifolds intersect transversally. By the Smale-Birkhoff homoclinic theorem, this implies the existence of chaotic motions (Smale horseshoe) exponentially close to $L_3$ and its invariant manifolds. In addition, we also show the existence of a generic unfolding of a quadratic homoclinic tangency and, as a consequence, the existence of Newhouse domains for the RPC3BP at coorbital motions. |
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