Chaos in the 1:2:3 Hamiltonian normal form
type: Item
instance of: scholarly article
title: Chaos in the 1:2:3 Hamiltonian normal form
zbMATH Open document ID: 0706.58036
DOI: 10.1016/0167-2789(90)90154-H
author: Igor Hoveijn
author: Ferdinand Verhulst
published in: Physica D
publication date: 1990-01-01T00:00:00Z
review text: The normal form of the Hamiltonian 1:2:3 resonance to degree 3 contains seven families of periodic solutions of which one can be complex unstable. Associated with this complex unstable solution is an invariant manifold N on which the dynamics can be characterized completely; one of the ingredients of N is a set of homoclinic orbits. In the normal form to degree 4 the set of homoclinic orbits break up, for certain parameter values, into one homoclinic orbit. This enables the authors to apply Shilnikov-Devaney theory to prove, at this stage numerically, the existence of a horseshoe map in the system with the implication of non- integrability and chaos in the normal form.
Mathematics Subject Classification ID: 37D45
Mathematics Subject Classification ID: 37G05
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 37J99
zbMATH DE Number: 4159691
zbMATH Keywords: Hamiltonian system
zbMATH Keywords: normal form
zbMATH Keywords: periodic solutions
zbMATH Keywords: chaos
reviewed by: Klaus Brod
MaRDI profile type: Publication
cites work: Asymptotic Integrability and Periodic Solutions of a Hamiltonian System in $1:2:2$-Resonance
cites work: Q4174811
cites work: Q4040102
cites work: Non-integrability of the 1:1:2–resonance
cites work: Averaging methods in nonlinear dynamical systems
cites work: Q4027661
full work available at URL: 0167-2789(90)90154-h
OpenAlex ID: W1998397519
description: scientific article; zbMATH DE number 4159691