Non-integrability of the Armbruster-Guckenheimer-Kim quartic Hamiltonian through Morales-Ramis theory

TL;DR

This paper proves the non-integrability of a specific quartic Hamiltonian system using Morales-Ramis theory, except for known integrable cases, and illustrates the transition to chaotic behavior with Poincaré sections.

Contribution

It applies Morales-Ramis theory to establish non-integrability of the Armbruster-Guckenheimer-Kim Hamiltonian, identifying all non-integrable parameter regimes.

Findings

Non-integrability proven for most parameter values.

Identification of known integrable cases.

Visualization of chaotic dynamics via Poincaré sections.

Abstract

We show the non-integrability of the three-parameter Armburster-Guckenheimer-Kim quartic Hamiltonian using Morales-Ramis theory, with the exception of the three already known integrable cases. We use Poincar\'e sections to illustrate the breakdown of regular motion for some parameter values.