Time-reversible Hamiltonian vector fields with symplectic symmetries
type: Item
instance of: scholarly article
title: Time-reversible Hamiltonian vector fields with symplectic symmetries
zbMATH Open document ID: 1083.37045
author: C. A. Buzzi
author: Marco A. Teixeira
published in: Journal of Dynamics and Differential Equations
publication date: 2005-03-15T00:00:00Z
review text: The main aim of this paper is to study the existence of one-parameter families of periodic orbits passing through a typical singularity for smooth time-reversible Hamiltonian systems with respect to a symplectic involution. The dynamical systems analyzed are given by vector fields presenting a \(1:-1\) resonance at the equilibrium (in two degrees of freedom). It is observed that only two distinct nontrivial symplectic involutions can arise. Moreover, these symplectic involutions force the system to have a nonpositive Hessian, and the linearized system to be a \(1:-1\) resonant. The approach used consists in using Birkhoff normal form, adapted to time-reversible systems with respect to a symplectic involution, in order to find a formal normal form which has second-order terms as a first integral. This formal normal form is used, jointly with transversality theory, restricted to the energy surfaces, to prove that the periodic solutions of the linearized system persist. The main result of the paper establishes conditions on systems of two-degrees-of-freedom for which there are two one-paremeter families of reversible periodic orbits passing through the equilibrium point. The paper is organized as follows. Reversible systems are briefly reviewed in Section 2.1. Next, some results concerning Hamiltonian systems (Section 2.2) and the anti-invariant theory (Section 2.3) are given. In Section 3.1, time-reversible Hamiltonian systems of one-degree-of-freedom are studied, classifying generic two-parameter families of reversible Hamiltonian vector fields of one-degree-of-freedom. Finally, in Section 3.2, time-reversible Hamiltonian systems with two degrees of freedom are studied, and the main result of the paper is obtained.
reviewed by: Jesús Marín-Solano
Mathematics Subject Classification ID: 37J15
Mathematics Subject Classification ID: 37J20
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 37G05
Mathematics Subject Classification ID: 37G10
zbMATH DE Number: 2145902
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: time-reversible systems
zbMATH Keywords: equilibrium point
zbMATH Keywords: normal form
zbMATH Keywords: periodic orbits
zbMATH Keywords: symplectic involution
MaRDI profile type: Publication
full work available at URL: s10884-004-4291-3
OpenAlex ID: W2005236728
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DOI: 10.1007/S10884-004-4291-3
description: scientific article; zbMATH DE number 2145902