Advances in Differential Equations

Almost periodicity enforced by Coulomb friction

Klaus Deimling, Georg Hetzer, and Wen Xian Shen

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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.

Article information

Adv. Differential Equations, Volume 1, Number 2 (1996), 265-281.

First available in Project Euclid: 25 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34C25: Periodic solutions
Secondary: 34A60: Differential inclusions [See also 49J21, 49K21] 70K20: Stability 70K30: Nonlinear resonances


Deimling, Klaus; Hetzer, Georg; Shen, Wen Xian. Almost periodicity enforced by Coulomb friction. Adv. Differential Equations 1 (1996), no. 2, 265--281.

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