On the stability of periodic solutions of the damped pendulum equation (Q1359644)

The equation \[ \ddot ambda \dot x+k \sin x=p(t) \] with positive parameters \(\lambda,k\) and \(T\)-periodic continuous function \(p(t)\) is considered. It is assumed that \(p(t)= \overline p+ \widetilde p(t)\) where \(\overline p\in \mathbb{R}\) and \(\int^T_0 \widetilde p(t)dt =0\). The main result of the paper is: For \(\widetilde p(t)\) sufficiently small in the norm it is proved the following existence and multiplicity result. For \((\lambda,k)\) belonging to the area which is determined by \[ k<\max \left\{{\lambda^2 \over 4}+ \omega^2, \omega \sqrt {\lambda^2 +\omega^2} \right\},\;\omega ={2\mu \over T}, \] there are numbers \(\mu_- =\mu_-(\lambda,k, \widetilde p)\) and \(\mu_+ =\mu_+(\lambda,k, \widetilde p)\) such that \(\mu_-< \mu_+\) and eq. (1.1) has (i) no \(T\)-periodic solution for \(\overline p\notin [\mu_-, \mu_+]\); (ii) exactly one \(T\)-periodic solution if either \(\overline p= \mu_-\) or \(\overline p= \mu_+\); (iii) exactly two \(T\)-periodic solutions if \(\overline p\in (\mu_-,\mu_+)\). The result concerning the stability of these solutions says that if a couple \((\lambda,k)\) belongs to the area which is determined by \[ k<\max \left\{{\lambda^2 \over 4} +{\omega^2 \over 4}, {\omega\over 2} \sqrt{\lambda^2 +\omega^2} \right\}, \] then (i)--(iii) remain true and the periodic solution obtained in (ii) is unstable while one solution obtained in (iii) is asymptotically stable and another unstable.