Subharmonic solutions with prescribed minimal period of a forced pendulum equation with impulses
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.
scientific article; zbMATH DE number 6991241