A forced pendulum equation without stable periodic solutions of a fixed period (Q2259322)

This paper is devoted to the investigation of the differential equation \[ \ddot{x}+\beta \sin x =f(t), \] where \(\beta>0\) is a real parameter and \(f:\mathbb{R}\to \mathbb{R}\) is a continuous and \(T\)-periodic function satisfying the condition \[ \int_0^T f(t)dt=0. \] Under above assumptions it is well known that there exist at least two \(T\)-periodic solutions. The author constructs some examples where there are exactly two \(T\)-periodic solutions and both are unstable.