Almost periodicity enforced by Coulomb friction

Abstract

We describe the influence of Coulomb friction $\mu$ sgn $ \dot x$ on the behavior of the linear oscillator given by $\ddot x+x=\varphi(t)$, where $\varphi$ is continuous and almost periodic. Depending on $\varphi$, we characterize the range of $\mu>0$ such that nontrivial almost periodic motions exist. We also show that dissipation caused by Coulomb friction may be too weak to ensure uniqueness of such motions, a phenomenon which appears already in case $\varphi$ is $2k\pi$-periodic with $k>1$. Nevertheless, we get a rather complete picture of the asymptotic behavior of such a system, but also have some interesting open questions, for example concerning the shape of the almost periodic solutions.