Subharmonic solutions with prescribed minimal period of a forced pendulum equation with impulses
type: Item
instance of: scholarly article
title: Subharmonic solutions with prescribed minimal period of a forced pendulum equation with impulses
zbMATH Open document ID: 1407.34055
published in: Acta Applicandae Mathematicae
publication date: 2018-12-10T00:00:00Z
review text: Motivated by the problem of periodic solutions for the classical forced pendulum equation \[ x''(t)+\left(\frac{g}{l}\right)\sin x(t)=f(t),\qquad t\in{\mathbb R},\tag{1.1} \] \textit{J. Xie} and \textit{Z. Luo} [Appl. Math. Lett. 52, 169--175 (2016; Zbl 1380.34065)] studied the subharmonic solutions with prescribed minimal period for the impulsive forced pendulum equation of the form \[ \begin{cases} x''(t)+A\sin x(t)=f(t), &t\neq t_j,\\ \Delta x'(t)=I_j(x(t))+d_j, &t=t_j. \end{cases} \] In this paper, the author is concerned with a forced pendulum equation with impulses \[ x''(t)+A(t)\sin x(t)=e(t), \qquad t\neq t_j,\tag{1.2} \] \[ \Delta x'(t)=I_j(x(t))+d_j, \qquad t=t_j,\tag{1.3} \] where \(A(t)>0\) and the external force term \(e(t)\) are \(T\)-periodic functions and \(j\in{\mathbb Z}\). We note that the constants \(g\) and \(l\) in Eq. (1.1) denote the \textit{gravity constant} and \textit{length of the pendulum}, respectively. Whereas, the potential \(g/l\) in Eq. (1.1) is replaced by the potential function \(A(t)\) in Eq. (1.2) which implies that the length of the pendulum is variable in the impulsive system. That is why the results obtained generalize and improve those given in [\textit{G. Hamel}, Math. Ann. 86, 1--13 (1922; JFM 48.0519.03)] and [Xie and Luo, loc. cit.]. The main instrument used to establish the results on the subharmonic solutions with prescribed minimal period is the theorem of the least action principle due to \textit{J. Mawhin} and \textit{M. Willem} [Critical point theory and Hamiltonian systems. New York etc.: Springer-Verlag (1989; Zbl 0676.58017)]. Three examples are given, and the corresponding numerical subharmonic solutions for the examples are obtained by applying Matlab software.
reviewed by: A. Özbekler
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34A37
Mathematics Subject Classification ID: 58E50
Mathematics Subject Classification ID: 70K40
Mathematics Subject Classification ID: 70K42
zbMATH DE Number: 6991241
zbMATH Keywords: subharmonic solution
zbMATH Keywords: forced pendulum equation
zbMATH Keywords: impulses
zbMATH Keywords: prescribed minimal period
zbMATH Keywords: least action principle
author: Fanchao Kong
describes a project that uses: Matlab
MaRDI profile type: Publication
full work available at URL: s10440-018-0177-y
OpenAlex ID: W2795666397
cites work: An \(n\)-dimensional pendulum-like equation via topological methods
cites work: The existence of subharmonic solutions with prescribed minimal period for forced pendulum equations with impulses
cites work: Subharmonic oscillations of forced pendulum-type equations
cites work: Twist Character of the Least Amplitude Periodic Solution of the Forced Pendulum
cites work: Subharmonic solutions with prescribed minimal period for some second-order impulsive differential equations
cites work: Critical point theory and Hamiltonian systems
cites work: Q5703734
cites work: Subharmonic oscillations for forced pendulum type equations
cites work: Subharmonic solutions with prescribed minimal period of an impulsive forced pendulum equation
cites work: The minimal period problem for the classical forced pendulum equation
Wikidata QID: Q130034676
DOI: 10.1007/S10440-018-0177-Y
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description: scientific article; zbMATH DE number 6991241