Stochastic behavior along mean motion resonances in the restricted planar 3-body problem

Abstract

One of the most remarkable instability zones in the Solar system are Kirkwood gaps in the asteroid belt. In this paper we analyze instabilities in the famous Kirkwood gap $3:1$ in the regime of small eccentricity of Jupiter. Mathematically speaking, we study the evolution of asteroids under the influence of the Sun and Jupiter using the restricted planar elliptic 3-body problem (RPE3BP) for initial conditions near a mean motion resonance 3:1. The main result exhibits stochastic diffusing behavior of the eccentricity of the asteroid for a rich set of initial conditions. Roughly speaking, for small eccentricity $e_0$ of Jupiter, the evolution of the eccentricity of the asteroid $\mathbf{e}(t\cdot e_0^{-2})$ at the Kirkwood gap $3:1$ behaves like a diffusion process on the line, where the randomness comes from the initial conditions. Along with KAM theory, we have mixed behavior in the asteroid belt, that is coexistence of quasiperiodic (deterministic) and stochastic (diffusive) behavior. See also the companion paper arXiv:2603.19893