Abstract. The equation $x''+a^{2}(t)x=0$ with $ a(t) :=\begin{cases} \sqrt{\frac{g}{l-\varepsilon }} &\text{ if $2kT\leq t<(2k+1)T$,}\\ \sqrt{\frac{g}{l+\varepsilon }} &\text{ if $(2k+1)T\leq t<(2k+2)T$, $(k=0,1,\dots )$,}\end{cases} $ is considered, where $g$ and $l$ denote the constant of gravity and the length of the pendulum, respectively; $\varepsilon >0$ is a parameter measuring the intensity of swinging. Concepts of solutions going away from the origin and approaching to the origin are introduced. Necessary and sufficient conditions are given in terms of $T$ and $\varepsilon $ for the existence of solutions of these types, which yield conditions for the existence of $2T$-periodic and $4T$-periodic solutions as special cases. The domain of instability, i.e., the Arnold tongues of parametric resonance are deduced from these results.

DOI: 10.14232/actasm-015-510-9

AMS Subject Classification (1991): 34D20, 70J40; 34A26, 34A37, 34C25

Keyword(s): second order linear differential equations, step function coefficients, periodic coefficients, impulsive effects, periodic solutions, parametric resonance, swinging

Received February 16, 2015, and in revised form October 14, 2015. (Registered under 10/2015.)