Periodic solutions of dynamical systems with bounded potential
type: Item
instance of: scholarly article
title: Periodic solutions of dynamical systems with bounded potential
zbMATH Open document ID: 0646.34049
DOI: 10.1016/0022-0396(87)90134-3
author: Vittorio Coti Zelati
published in: Journal of Differential Equations
publication date: 1987-01-01T00:00:00Z
review text: This paper deals with the existence of T-periodic solutions for systems of ordinary differential equations of the form \(-\ddot x=\nabla_ xV(t,x),\) where \(V\in C^ 2(R\times R^ N,R)\), \(V(t+T,x)=V(t,x)\) for any \((t,x)\in {\mathbb{R}}\times {\mathbb{R}}^ N\), V is bounded and \(\nabla_ xV(t,x)\to 0\) as \(| x| \to +\infty\). The author uses Morse theory to prove the existence of periodic solutions for the given equation. The use of Morse theory is done through defining on a suitable function space E a functional f whose critical points are the T-periodic solutions of the equation. Then he shows that the Palais-Smale condition holds on \(\{x\in E:f(x)\geq c+\epsilon \}\) and estimates the homology groups \(H_ q(\{x\in E;f(x)\leq c+\epsilon \})\).
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 37-XX
zbMATH DE Number: 4054099
zbMATH Keywords: Morse theory
zbMATH Keywords: Palais-Smale condition
zbMATH Keywords: estimates
zbMATH Keywords: homology groups
reviewed by: Ubiratan D'Ambrosio
MaRDI profile type: Publication
full work available at URL: 0022-0396(87)90134-3
OpenAlex ID: W1988083867
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description: scientific article; zbMATH DE number 4054099