  Many systems, arising in electrical and electronic engineering are based on
controlled phase synchronization of several periodic processes ("phase
synchronization" systems, or PSS). Typically such systems are featured by the
gradient-like behavior, i.e. the system has infinite sequence of equilibria
points, and any solution converges to one of them. This property however says
nothing about the transient behavior of the system, whose important qualitative
index is the maximal phase error. The synchronous regime of gradient-like
system may be preceded by cycle slipping, i.e. the increase of the absolute
phase error. Since the cycle slipping is considered to be undesired behavior of
PSSs, it is important to find efficient estimates for the number of slipped
cycles. In the present paper, we address the problem of cycle-slipping for
phase synchronization systems described by integro-differential Volterra
equations with a small parameter at the higher derivative. New effective
estimates for a number of slipped cycles are obtained by means of Popov's
method of "a priori integral indices". The estimates are uniform with respect
to the small parameter.
