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The dynamics of sliding friction is mainly governed by the frictional force.Previous studies have shown that the laboratory-scale friction is welldescribed by an empirical law stated in terms of the slip velocity and thestate variable. The state variable represents the detailed physicochemicalstate of the sliding interface. Despite some theoretical attempts to derivethis friction law, there has been no unique equation for time evolution of thestate variable. Major equations known to date have their own merits anddrawbacks. To shed light on this problem from a new aspect, here we investigatethe feasibility of periodic motion without the help of radiation damping.Assuming a patch on which the slip velocity is perturbed from the rest of thesliding interface, we prove analytically that three major evolution laws failto reproduce stable periodic motion without radiation damping. Furthermore, wepropose two new evolution equations that can produce stable periodic motionwithout radiation damping. These two equations are scrutinized from theviewpoint of experimental validity and the relevance to slow earthquakes.

Slow periodic oscillation without radiation damping: New evolution laws for rate and state friction