Realization of tangent perturbations in discrete and continuous time conservative systems

Hassan Najafi Alishah; João Lopes Dias

doi:10.3934/dcds.2014.34.5359

Article Contents

2014, Volume 34, Issue 12: 5359-5374. Doi: 10.3934/dcds.2014.34.5359

This issue Previous Article Optimal parameter-dependent bounds for Kuramoto-Sivashinsky-type equations Next Article

Realization of tangent perturbations in discrete and continuous time conservative systems

Hassan Najafi Alishah^1, and
João Lopes Dias^2,

1.
CMAF, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edifício C6, Piso 2, 1749-016 Lisboa
2.
Departamento de Matemática ISEG, Universidade Técnica de Lisboa, Rua do Quelhas 6, 1200-781 Lisboa

Received: September 30, 2013

Revised: February 28, 2014

Published: November 2014

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Abstract

We prove that any perturbation of the symplectic part of the derivative of a Poisson diffeomorphism can be realized as the derivative of a $C^1$-close Poisson diffeomorphism. We also show that a similar property holds for the Poincaré map of a Hamiltonian on a Poisson manifold. These results are the conservative counterparts of the Franks lemma, a perturbation tool used in several contexts most notably in the theory of smooth dynamical systems.

Keywords:

Franks lemma,
symplectic and Hamiltonian $C^1$ perturbations,
Poisson Hamiltonian flowbox coordinates.

Mathematics Subject Classification: Primary: 37C05, 37J05; Secondary: 37C20.

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