Computation of a separatrix map and a normally hyperbolic invariant lamination for the RP3BP

TL;DR

This paper proves the existence of a normally hyperbolic invariant lamination in the Sun-Jupiter-Asteroid model at the Kirkwood gap, analyzing its induced dynamics and laying groundwork for stochastic diffusion results.

Contribution

It constructs a NHIL for the RP3BP at the Kirkwood gap and characterizes its dynamics as a partially hyperbolic skew-shift, with some conditions verified numerically.

Findings

Existence of a NHIL at the Kirkwood gap in the RP3BP.

The induced dynamics is a partially hyperbolic skew-shift.

Numerical verification of key non-degeneracy conditions.

Abstract

In this paper we discuss the existence of a normally hyperbolic invariant lamination (NHIL) at the Kirkwood gap $3:1$ for the Restricted Planar Elliptic 3 Body Problem. This problem models the Sun-Jupiter-Asteroid dynamics. We also show that the induced dynamics on the NHIL is a partially hyperbolic skew-shift which is of the form \[ f:(\omega,I,\theta)\to (\sigma \omega, I+e_0 A_\omega(I)\cos(\theta+\psi_\omega)+\mathcal{O}(e^2_0), \theta+\Omega_\omega(I)+\mathcal{O}(e_0)),\] where $I\in [a,b], \theta\in \mathbb T, \omega\in\Sigma=\{0,1\}^\mathbb Z$, the space of sequences of $0,1$'s, $\sigma:\Sigma \to \Sigma$ is the shift in this space, $\Omega_\omega$ is the shear, $A_\omega$ is an amplitude, and $e_0$ is the eccentricity of Jupiter, which is taken as a small parameter. In the companion paper arXiv:2603.19894, relying on these skew-shift, we show the existence of stochastic diffusing behavior for Asteroids belonging to the Kirkwood gap provided the eccentricity of Jupiter is $e_0$ small enough. Key ingredients to construct the NHIL are the separatrix map associated to homoclinic channels to a normally hyperbolic invariant cylinder and an isolating block construction. Some of the necessary non-degeneracy conditions are verified numerically.