Spectral flow and bifurcation of critical points of strongly indefinite functionals. II: Bifurcation of periodic orbits of Hamiltonian systems
type: Item
instance of: scholarly article
title: Spectral flow and bifurcation of critical points of strongly indefinite functionals. II: Bifurcation of periodic orbits of Hamiltonian systems
zbMATH Open document ID: 0985.37057
author: Patrick M. Fitzpatrick
author: Jacobo Pejsachowicz
author: Lázaro Recht
published in: Journal of Differential Equations
publication date: 2000-08-07T00:00:00Z
review text: Part I, cf. J. Funct. Anal. 162, 52-95 (1999; Zbl 0915.58091). The paper deals with the bifurcation of periodic orbits for time dependent Hamiltonian systems. First the authors study the bifurcation of periodic orbits from equilibrium points under the assumption that the linearization along the trivial branch is induced by a time independent Hamiltonian. Hence the spectral flow of the path of Hessians of the associated action functionals reduces to the difference of generalized signatures of the Hessians at the ends of the path. The authors show that the index of a linear Hamiltonian system introduced by Amann and Zehnder that was used to find multiple solutions of a Hamiltonian system can also be used in order to detect bifurcation for the problems under consideration. Moreover, they present a very simple and explicit calculation of the spectral flow in the particular case when the Hessians of the Hamiltonians commute with the symplectic matrix. The authors consider the general case of bifurcation of periodic orbits (with fixed period) from a known branch of periodic solutions, too.
Mathematics Subject Classification ID: 37J20
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 37B30
Mathematics Subject Classification ID: 37C10
zbMATH DE Number: 1454724
zbMATH Keywords: spectral flow
zbMATH Keywords: Hamiltonian system
zbMATH Keywords: bifurcation
zbMATH Keywords: periodic orbit
zbMATH Keywords: generalized signature
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
full work available at URL: jdeq.1999.3723
OpenAlex ID: W1977530321
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DOI: 10.1006/JDEQ.1999.3723
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description: scientific article; zbMATH DE number 1454724