Differential topology and nonlinear circuits

Several fundamental problems and conjectures in nonlinear circuit theory posed in the sixti??s have been satisfactorily resolved only recently. Virtually every circuit-and system-theoretic technique have been tried during the intervening years with little success. The breakthrough came only after modern tools from differential topology and algebraic geometry have become available and applied in a non-trivial way. The purpose of this lecture is to sample some of these results from several recent papers [1-6], with special emphasis on the roles played by several differential-geometric concepts and tools, such as differentiable manifolds, transversality, degree theory, Hodge decomposition, etc.