Minimal period estimates on \(P\)-symmetric periodic solutions of first-order mild superquadratic Hamiltonian systems
type: Item
instance of: scholarly article
title: Minimal period estimates on \(P\)-symmetric periodic solutions of first-order mild superquadratic Hamiltonian systems
zbMATH Open document ID: 1478.37064
author: Xiao-Fei Zhang
author: Chungen Liu
published in: Frontiers of Mathematics in China
publication date: 2021-07-14T00:00:00Z
review text: A \(P\)-boundary problem of the autonomous Hamiltonian system \[ -J\dot{z}(t)=\nabla H(z(t)), \qquad z(\tau)=Pz(t), \] is considered. Here \(P\) is a \(2n\times 2n\) symplectic matrix, \(H\in C^2(\mathbb R^{2n},\mathbb R)\) with \(H(Pz)=H(z)\), and \(J\) is the standard symplectic matrix. It is assumed that \(P^k=I_{2n}\) and \(P^m\neq I_{2n}\) for smaller \(m\). A solution of the above system can be extended to a \(k\tau\)-periodic solution. A \(T\)-periodic solution is called \(P\)-symmetric if \(z(T/k)=Pz(0)\). Under assumptions similar to those in [\textit{A. Abbondandolo}, Morse theory for Hamiltonian systems. Boca Raton, FL: Chapman \& Hall/CRC (2001; Zbl 0967.37002)] the authors obtain estimates on Maslov \(P\)-index and minimal period estimates. The conditions are a bit different from the well known Ambrosetti-Rabinowitz conditions.
reviewed by: Zdzisław Dzedzej
Mathematics Subject Classification ID: 37J46
Mathematics Subject Classification ID: 37J51
Mathematics Subject Classification ID: 34B05
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 70H05
zbMATH DE Number: 7371550
zbMATH Keywords: Hamiltonian system
zbMATH Keywords: \(P\)-symmetric periodic solution
zbMATH Keywords: \(P\)-index
zbMATH Keywords: minimal period
MaRDI profile type: Publication
full work available at URL: s11464-021-0903-z
OpenAlex ID: W3135112990
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DOI: 10.1007/S11464-021-0903-Z
description: scientific article; zbMATH DE number 7371550