  The onset of generalized synchronization of chaos in directionally-coupled
systems corresponds to the formation of a continuous mapping which enables one
to persistently define the state of the response system from the trajectory of
the drive system. The recently developed theory of generalized synchronization
of chaos deals only with the case where this synchronization mapping is a
single-valued function. In this paper, we explore generalized synchronization
in a regime where the synchronization mapping can become a multi-valued
function. Specifically, we study the properties of the multi-valued mapping
which occurs between the drive and response systems when the systems are
synchronized with a frequency ratio other than one-to-one, and address the
issues of the existence and continuity of such mappings. The basic theoretical
framework underlying the considered synchronization regimes is then developed.
