On the number of solutions for the forced pendulum equation
type: Item
instance of: scholarly article
title: On the number of solutions for the forced pendulum equation
zbMATH Open document ID: 0686.34036
DOI: 10.1016/0022-0396(89)90096-X
published in: Journal of Differential Equations
publication date: 1989-01-01T00:00:00Z
review text: The problem considered is the problem of the number of T-periodic solutions for the forced pendulum equation: (1) ẍ\(+\lambda \dot x+k \sin x=e(t)+c\), where \(\lambda\),k,c\(\in R\), \(k>0\), \(e=e(t)\in C[0,T]\) T- periodic function with zero average value. The main result of the article is following: There exists a positive \(\lambda_ 0=\lambda_ 0(k,e)\) depending on k,e such that for every \(\lambda:| \lambda | >\lambda_ 0\) there exist two numbers \(d=d(e,\lambda)\), \(D=D(e,\lambda)\) \((d<0<D)\) such that equation (1) has 1) no T -periodic solution if \(c\not\in [d,D];\) 2) exactly one T-periodic solution if \(c=d\) or \(c=D;\) 3) exactly two T-periodic solution if \(c\in (d,D).\) The corresponding result is also obtained for the equation (1) without friction (i.e., \(\lambda =0)\) under some restrictions imposed on e(t),k,T.
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C15
zbMATH DE Number: 4124139
zbMATH Keywords: friction
zbMATH Keywords: forced pendulum equation
author: G. Tarantello
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4124139