Marginal resonances and intermittent behaviour in the motion in the vicinity of a separatrix

TL;DR

This paper derives a condition for intermittent bursts in chaotic motion near a resonance separatrix, predicts a new intermittent regime in asteroidal motion, and confirms it through numerical simulations.

Contribution

It introduces a new condition for intermittency related to marginal resonances and applies it to asteroid dynamics near a mean motion resonance.

Findings

Identified a new intermittent regime in resonant asteroid motion.

Derived a condition for sporadic energy bursts in chaotic layers.

Validated the theoretical condition with numerical simulations.

Abstract

A condition upon which sporadic bursts (intermittent behaviour) of the relative energy become possible is derived for the motion in the chaotic layer around the separatrix of non-linear resonance. This is a condition for the existence of a marginal resonance, i.e. a resonance located at the border of the layer. A separatrix map in Chirikov's form [Chirikov, B. V., Phys. Reports 52, 263 (1979)] is used to describe the motion. In order to provide a straightforward comparison with numeric integrations, the separatrix map is synchronized to the surface of the section farthest from the saddle point. The condition of intermittency is applied to clear out the nature of the phenomenon of bursts of the eccentricity of chaotic asteroidal trajectories in the 3/1 mean motion commensurability with Jupiter. On the basis of the condition, a new intermittent regime of resonant asteroidal motion is predicted and then identified in numeric simulations.