Periodic solutions of spatially periodic, even Hamiltonian systems
type: Item
instance of: scholarly article
title: Periodic solutions of spatially periodic, even Hamiltonian systems
zbMATH Open document ID: 0872.58050
author: Thomas J. Bartsch
author: Zhi-Qiang Wang
published in: Journal of Differential Equations
publication date: 1997-04-27T00:00:00Z
review text: In this paper, the existence of periodic solutions of Hamiltonian systems on the cotangent bundle of the torus \(T^N\) is studied. The authors assume that the Hamiltonian function is even in spatial variables and superlinear in the tangential variables, its gradient with respect to the tangential variables has a polynomial growth, and all trivial solutions are nondegenerate. Then when there is one trivial solution possessing Conley-Zehnder index not in \([0,N]\), the existence of multiple nontrivial solutions of the system is proved. This result depends on a more general abstract critical point theorem, which is proved by measuring the change of the topologies of the level sets for the corresponding functional using the Borel cohomology method. This result is related to the earlier work Math. Z. 224, No. 1, 65-76 (1997) of the authors for multiple periodic solutions of Hamiltonian systems on tori.
reviewed by: Ding Tongren
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 1003899
zbMATH Keywords: spatially periodic
zbMATH Keywords: even
zbMATH Keywords: variational methods
zbMATH Keywords: existence
zbMATH Keywords: periodic solutions
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: Conley-Zehnder index
MaRDI profile type: Publication
OpenAlex ID: W2055699238
cites work: Periodic solutions of even Hamiltonian systems on the torus
cites work: Critical point theorems for indefinite functionals
cites work: Q3730676
cites work: On the periodic nonlinearity and the multiplicity of solutions
cites work: Infinite dimensional Morse theory and multiple solution problems
cites work: The Birkhoff-Lewis fixed point theorem and a conjecture of V.I. Arnold
cites work: Morse‐type index theory for flows and periodic solutions for Hamiltonian Equations
cites work: Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems
cites work: Periodic solutions of spatially periodic Hamiltonian systems
cites work: Q5749609
cites work: Q5630031
cites work: A generalized saddle point theorem
cites work: Maslov-type index, degenerate critical points, and asymptotically linear Hamiltonian systems
cites work: Periodic solutions of hamiltonian systems
cites work: Q3749777
cites work: On a Class of Functionals Invariant under a Z n Action
cites work: A relative category and applications to critical point theory for strongly indefinite functionals
cites work: On theorems of Borsuk-Ulam, Kakutani-Yamabe-Yujobô and Dyson. I
DOI: 10.1006/JDEQ.1996.3226
description: scientific article; zbMATH DE number 1003899