Subharmonic solutions of second order subquadratic Hamiltonian systems with potential changing sign
type: Item
instance of: scholarly article
title: Subharmonic solutions of second order subquadratic Hamiltonian systems with potential changing sign
zbMATH Open document ID: 0982.37064
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-12-14T00:00:00Z
review text: This paper considers the existence of periodic solutions of a system of equations \(-\ddot{x} = V_x (t,x)\) for \(x \in {\mathbb R}^n\). The equations are written in Hamiltonian form as \(-\dot{x} = H_y (t,x,y)\), \(\dot{y} = H_x (t,x,y)\) where \(H(t,x,y) = \frac{1}{2} |y|^2 + V(t,x)\), and where the potential \(V(t,x)\) is subquadratic and may change sign. The author considers systems where \(V = V_1 + V_2\) with both \(V_1\) and \(V_2\) 1-periodic, \(V_1\) positive at infinity, and \(V_2\) satisfying \(\int_0^1 V_{2,x} (t,x) dt = 0\), and combines both time-dependent and time-independent symplectic transformations to construct a new Hamiltonian independent of the sign-changing potential \(V_2\) so that known existence results can be used. Conditions are given for the functions \(V_1\) and \(V_2\) that guarantee the existence of \(k\)-periodic solutions \(x_k\) such that \(\text{ max }_t (|x_k(t)|+ |\dot{x}_k(t)|) \rightarrow \infty\) as \(k \rightarrow \infty\). In the case where \(x \in {\mathbb R}\), the author also gives some additional conditions on \(V\) to guarantee the existence of Aubrey-Mather sets for the system.
Mathematics Subject Classification ID: 37J45
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 1449208
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: periodic solutions
zbMATH Keywords: existence of Aubrey-Mather sets
author: Meiyue Jiang
reviewed by: David J. Saunders
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6687
OpenAlex ID: W2003651501
cites work: Q4315401
cites work: Existence and multiplicity results for homogeneous second order differential equations
cites work: Q4237799
cites work: Q4692985
cites work: Periodic solutions of ''superquadratic'' Hamiltonian systems
cites work: Subharmonics near an equilibrium for some second-order Hamiltonian systems
cites work: Existence and multiplicity results for periodic solutions of superquadratic Hamiltonian systems where the potential changes sign
cites work: Q4486834
cites work: Symplectic transformations and periodic solutions of Hamiltonian systems
cites work: On a theorem by Mather and Aubry-Mather sets for planar Hamiltonian systems
cites work: Q3142713
cites work: Solutions périodiques d'un système différentiel non linéaire du second ordre avec changement de signe. (Periodic solutions of a nonlinear second order differential system with change of sign)
cites work: Periodic solutions of a second order superquadratic system with a change of sign in the potential
cites work: The stability of the equilibrium of a conservative system
cites work: Q4315078
cites work: Mather sets for superlinear Duffing equations
cites work: Aubry-Mather sets for finite-twist maps of a cylinder and semilinear Duffing equations
cites work: Q3749777
cites work: Subharmonic solutions for subquadratic Hamiltonian systems
DOI: 10.1006/JMAA.1999.6687
description: scientific article; zbMATH DE number 1449208