  An unexpected and somewhat surprising observation is that two
counter-cascaded systems, given the right conditions, can exhibit
multivaluedness from one of the outputs to the other. The main result presented
here is a necessary and sufficient condition for multivaluedness to be
exhibited by counter-cascaded systems using the novel notions of immanence and
its opposite, transcendence, introduced here. Subsequent corollaries provide
further characterization of multivaluedness under specific conditions.
  As an application of our theoretical results, we demonstrate how these aid in
the structural complexity reduction of complex networks.
