Detection of separatrices and chaotic seas based on orbit amplitudes

TL;DR

This paper introduces a new, sensitive index based on the Maximum Eccentricity Method to detect phase space structures like separatrices and chaotic regions in planetary systems, improving resolution over traditional methods.

Contribution

It develops a second-derivative index for the MEM, enhancing detection of phase space structures such as separatrices and chaotic seas in dynamical systems.

Findings

The index effectively detects minute phase space structures.

It distinguishes resonant webs and chaotic regions more accurately.

Application to N-body simulations demonstrates practical utility.

Abstract

The Maximum Eccentricity Method (MEM) is a standard tool for the analysis of planetary systems and their stability. The method amounts to estimating the maximal stretch of orbits over sampled domains of initial conditions. The present paper leverages on the MEM to introduce a sharp detector of separatrices and chaotic seas. After introducing the MEM analogue for nearly-integrable action-angle Hamiltonians, i.e., diameters, we use low-dimensional dynamical systems with multi-resonant modes and junctions, supporting chaotic motions, to recognise the drivers of the diameter metric. Once this is appreciated, we present a second-derivative based index measuring the regularity of this application. This quantity turns to be a sensitive and robust indicator to detect separatrices, resonant webs and chaotic seas. We discuss practical applications of this framework in the context of $N$-body simulations for the planetary case affected by mean-motion resonances, and demonstrate the ability of the index to distinguish minute structures of the phase space, otherwise undetected with the original MEM.