# Associate submersions and qualitative properties of nonlinear circuits with implicit characteristics

## TL;DR

This paper develops a framework for classifying nonlinear planar curves via submersions, applying it to electrical circuit modeling, and addressing key problems like circuit solvability and bifurcations with a novel, general approach.

## Contribution

It introduces an equivalence notion for submersions to classify nonlinear curves and applies this to model and analyze nonlinear electrical circuits systematically.

## Key findings

- Unified approach to nonlinear circuit analysis
- General expressions for characteristic polynomials at equilibria
- Application to circuits with memristors and bifurcation analysis

## Abstract

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.