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.
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