# Dynamics in first-order mean motion resonances: analytical study of a simple model with stochastic behaviour

## TL;DR

This paper analyzes a Hamiltonian model of first-order mean motion resonance in a three-body system, revealing mechanisms of chaos and classifying long-term orbital evolution paths.

## Contribution

It introduces an analytical approach to classify the evolution of slow variables and characterizes chaos in a simplified resonant Hamiltonian model.

## Key findings

- Classification of slow variable evolution paths
- Bifurcation diagram of phase portraits
- Properties of chaos in the system

## Abstract

We examine a 2DOF Hamiltonian system, which arises in study of first-order mean motion resonance in spatial circular restricted three-body problem "star-planet-asteroid", and point out some mechanisms of chaos generation. Phase variables of the considered system are subdivided into fast and slow ones: one of the fast variables can be interpreted as resonant angle, while the slow variables are parameters characterizing the shape and orientation of the asteroid's orbit. Averaging over the fast motion is applied to obtain evolution equations which describe the long-term behavior of the slow variables. These equations allowed us to provide a comprehensive classification of the slow variables' evolution paths. The bifurcation diagram showing changes in the topological structure of the phase portraits is constructed and bifurcation values of Hamiltonian are calculated. Finally, we study properties of the chaos emerging in the system.