Long-time stability estimates for the non-periodic pendulum equation

Abstract

In this paper we consider the non-periodic pendulum equation $\ddot{x}+G_x(t,x)=p\left(t\right)$, where $G_x(t,x)$ and $p\left(t\right)$ are not required to be periodic in t. Under natural assumptions, the existence of infinitely many bounded solutions is established, furthermore, it is shown that, for any given unbounded solution x, there is a solution $x^{\epsilon }$ which is bounded and such that the half power of energies of $x^{\epsilon }$ and x remain close on a time interval of length ${\epsilon }^{-1}$ with $\epsilon >0$ small enough. In the end, a specific $p\left(t\right)$ is constructed to illustrate the existence of the unbounded solution for the equation under this $p\left(t\right)$; moreover, the long-time closeness result also holds under this $p\left(t\right)$.