On stabilizability of the upper equilibrium of the asymmetrically excited inverted pendulum

Using purely elementary methods, necessary and sufficient conditions are given for the existence of $T$-periodic and $2T$-periodic solutions around the upper equilibrium of the mathematical pendulum when the suspension point is vibrating vertically with asymmetric high frequency. The equation of the motion is of the form \begin{equation*} \ddot{\theta}-\dfrac{1}{l} \left(g+a(t)\right) \theta=0, \end{equation*} where \begin{equation*} a(t) :=\begin{cases} A_h, &\hbox{if } kT\leq t<kT+T_h,\\ -A_e, &\hbox{if } kT+T_h\leq t<(kT+T_h)+T_e,\\ \end{cases}\qquad (k=0,1,\dots); \end{equation*} $A_h, A_e, T_h, T_e$ are positive constants ($T_h+T_e=T$); $g$ and $l$ denote the acceleration of gravity and the length of the pendulum, respectively. An extended Oscillation Theorem is given. The exact stability regions for the upper equilibrium are presented