The stability of the periodic solutions of second order Hamiltonian systems
type: Item
instance of: scholarly article
title: The stability of the periodic solutions of second order Hamiltonian systems
zbMATH Open document ID: 0960.37029
DOI: 10.1142/S025295990000025X
published in: Chinese Annals of Mathematics. Series B
publication date: 2000-06-21T00:00:00Z
review text: This paper is devoted to the stability of the periodic solutions of the following second order Hamiltonian systems \[ \ddot x+ V_x'(t,x)= 0, \quad x\in \mathbb{R}^n, \tag{1} \] where \(n\) is a positive integer, \(V: \mathbb{R}\times \mathbb{R}^n\to \mathbb{R}\) is a function which is \(\tau\)-periodic in the variable \(t\). \(V_x'\) denotes its gradient with respect to \(x\). Under some natural conditions on \(H\) the author analyzes two different cases: superquadratic and subquadratic cases. He proves that in the subquadratic case, there exist infinitely many geometrically distinct elliptic periodic solutions, and in the superquadratic case, there exist infinitely many geometrically distinct periodic solutions with at most one instability direction if they are half period non-degenerated, otherwise they are elliptic.
Mathematics Subject Classification ID: 37J45
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 1462264
zbMATH Keywords: stability
zbMATH Keywords: periodic solutions
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: subquadratic case
zbMATH Keywords: superquadratic case
author: Chungen Liu
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
full work available at URL: s025295990000025x
OpenAlex ID: W2117647271
description: scientific article; zbMATH DE number 1462264