Associate submersions and qualitative properties of nonlinear circuits with implicit characteristics

We introduce in this paper an equivalence notion for submersions $U \to \R$, $U$ open in $\R^2$, which makes it possible to identify a smooth planar curve with a unique class of submersions. This idea, which extends to the nonlinear setting the construction of a dual projective space, provides a systematic way to handle global implicit descriptions of smooth planar curves. We then apply this framework to model nonlinear electrical devices as {\em classes of equivalent functions}. In this setting, linearization naturally accommodates incremental resistances (and other analogous notions) in homogeneous terms. This approach, combined with a projectively-weighted version of the matrix-tree theorem, makes it possible to formulate and address in great generality several problems in nonlinear circuit theory. In particular, we tackle unique solvability problems in resistive circuits, and discuss a general expression for the characteristic polynomial of dynamic circuits at equilibria. Previously known results, which were derived in the literature under unnecessarily restrictive working assumptions, are simply obtained here by using dehomogenization. Our results are shown to apply also to circuits with memristors. We finally present a detailed, graph-theoretic study of certain stationary bifurcations in nonlinear circuits using the formalism here introduced.

💡 Research Summary

The paper introduces a novel equivalence concept for submersions—smooth maps with surjective differentials—from an open subset U ⊂ ℝ² to ℝ, and shows how this concept can be used to represent smooth planar curves globally as zero‑sets of such submersions. In the linear case a straight line through the origin can be described as the kernel of any non‑zero linear form, and all such forms are equivalent up to a non‑zero scalar; this is the familiar construction of a real projective line. The authors extend this idea to the nonlinear setting by defining “associate submersions”. Two submersions f₁:U₁→ℝ and f₂:U₂→ℝ are associates if (i) each vanishes outside the intersection of their domains, and (ii) on the common part U₁∩U₂ they differ only by multiplication with a nowhere‑zero smooth function γ. This relation is reflexive, symmetric and transitive, thus an equivalence relation on the sheaf of submersions. Consequently, a smooth planar curve C is uniquely identified with an equivalence class of associate submersions; any two submersions in the class have the same zero set C.

Having established this geometric foundation, the authors turn to electrical circuit theory. A nonlinear resistor (or any two‑terminal device) is traditionally described either by a voltage‑controlled function v = h(i) or a current‑controlled function i = g(v). Such descriptions require the existence of a global function h or g, which is often unrealistic for devices with multi‑valued or non‑monotone characteristics. By modeling the device characteristic as a global submersion f(i,v)=0 and treating f as an element of an associate class, the need for a global explicit function disappears. Linearization at a point (i₀,v₀) yields the Jacobian (∂f/∂i, ∂f/∂v), which can be interpreted as homogeneous coordinates (p:q) on a projective line, analogous to the impedance‑admittance pair (z,y) in the linear case. Thus the incremental resistance (or conductance) appears naturally as a homogeneous quantity, and the same formalism accommodates memristors (where the characteristic may involve flux or charge).

The paper’s second major contribution is a “projectively‑weighted matrix‑tree theorem”. The classical matrix‑tree theorem states that the determinant of a graph Laplacian equals the sum of the products of edge weights over all spanning trees. Here each edge e is assigned a homogeneous weight wₑ =