On the parametrically excited pendulum equation with a step function coefficient

•Elementary method is given to study non-linear Meissner type equations.•Periodic excitation models of swinging are not realistic.•Optimal swinging can be described by a self-oscillating system.•Oscillation Theorem is generalized to the excited non-linear pendulum equation.

An elementary geometric method is established to study non-linear second order differential equations with step function coefficientx″+a2(t)g(x)=0,a(t)≔akiftk−1≤t0ak>0, tk↗∞tk↗∞ as k→∞k→∞. The equation is rewritten into a discrete dynamical system on the plane. The method is applied to the excited pendulum equation when g(x)=sinx. Starting from the usual periodic model, the problem of parametric resonance (problem of swinging) is investigated. It will be pointed out that the realistic model of swinging is not a periodically excited system, instead swing is a self-oscillating system. Finally, the classical Oscillation Theorem is extended to the non-linear periodic pendulum equation ψ″+a2(t)sinψ=0,a(t)={gl−εif2kT≤t<(2k+1)T,gl+εif(2k+1)T≤t<(2k+2)T(k∈Z+),where g and l   denote the constant of gravity and the length of the pendulum, respectively; ε>0ε>0 is a parameter measuring the intensity of swinging