Geometric properties of nonlinear networks containing capacitor-only cutsets and/or inductor-only loops. I: Conservation laws (Q1821087)

Nonlinear networks N consisting of capacitor-only cutsets and/or inductor-only loops are considered from the point of view of differentiable manifolds. Let \(\delta_ 0\) be the sum of the number of independent capacitor-only cutsets and independent inductor-only loops. The authors establish the following: (1) sufficient conditions such that the set S of equilibrium points is a \(\delta_ 0\)-dimensional submanifold of the state space of N, (2) sufficient conditions such that N has \(\delta_ 0\) independent conservation laws and thus through each point of the state space of N, there is a codimension \(\delta_ 0\) invariant submanifold \(S^*\) of the network dynamics, and (3) sufficient conditions to guarantee S and \(S^*\) intersect transversely. The results given by the authors are not new, however, they are presented in coordinate-free form, i.e. special trees are not used in the theorems and proofs.