Stable periodic motion of a delayed spring (Q1423964)

The paper considers the system \(\dot x(t)=v(t)\), \(\dot v(t)=-\mu v(t)+f(x(t-1))\), where \(\mu>0\) and \(f\) is an odd bounded Lipschitz continuous real function satisfying \(\xi f(\xi)<0\) if \(| \xi| \) is large. The system stands for a damped spring with a delayed position-dependent restoring force. The author proves that there exists a stable periodic orbit with strong attraction properties for sufficiently large \(\mu\) and under certain conditions on \(f\). Moreover, if \(f\) is continuously differentiable, the periodic orbit is stable and hyperbolic. The proof extends a method introduced by the author in [Discrete Contin. Dyn. Syst. 7, 259--274 (2001; Zbl 1034.34086)].