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