Bounds for the period of periodic orbits of dynamical systems
type: Item
instance of: scholarly article
title: Bounds for the period of periodic orbits of dynamical systems
zbMATH Open document ID: 0616.34041
DOI: 10.1016/0022-0396(87)90132-X
author: Stavros N. Busenberg
author: Mario Martelli
published in: Journal of Differential Equations
publication date: 1987-01-01T00:00:00Z
review text: Let U be an open subset of a Banach space E and let f:U\(\to E\) be a Lipschitz map with constant \(L:\| f(x)-f(y)\| \leq L\| x-y\|.\) We consider the autonomous first order differential equation (1) \(x'=f(x)\) with the corresponding Euler difference equation (2) \(x_{i+1}=x_ i+hf(x_ i)\) and derive sharp lower bounds for the period of any periodic orbit of (2). As an application of these bounds we obtain results relating the periods of solutions of (1) to the Lipschitz constant L, as well as results on the fixed points of iterates of maps on Banach and Hilbert spaces. In one of our theorems we prove that, if T is the period of a solution of (1), then \(T\geq 4.5/L\). This improves the bound \(T\geq 4/L\) obtained by Lasota and Yorke. For (2) in Hilbert space we obtain the sharp bound hL\(\geq 2 \sin (\pi /n)\). The bound for (1) in Banach spaces has been improved to \(T\geq 6/L\) which is the best possible by the authors and \textit{D. Fisher} in Proc. Am. Math. Soc. 98, 376-378 (1986).
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 37-XX
zbMATH DE Number: 3998273
zbMATH Keywords: Lipschitz dynamical systems
zbMATH Keywords: Schauder conjecture
zbMATH Keywords: autonomous first order differential equation
zbMATH Keywords: Euler difference equation
zbMATH Keywords: Hilbert space
zbMATH Keywords: Banach spaces
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3998273