Transcritical bifurcation without parameters in memristive circuits

The transcritical bifurcation without parameters (TBWP) describes a stability change along a line of equilibria, resulting from the loss of normal hyperbolicity at a given point of such a line. Memristive circuits systematically yield manifolds of non-isolated equilibria, and in this paper we address a systematic characterization of the TBWP in circuits with a single memristor. To achieve this we develop two mathematical results of independent interest; the first one is an extension of the TBWP theorem to explicit ordinary differential equations (ODEs) in arbitrary dimension; the second result drives the characterization of this phenomenon to semiexplicit differential-algebraic equations (DAEs), which provide the appropriate framework for the analysis of circuit dynamics. In the circuit context the analysis is performed in graph-theoretic terms: in this setting, our first working scenario is restricted to passive problems (exception made of the bifurcating memristor), and in a second step some results are presented for the analysis of non-passive cases. The latter context is illustrated by means of a memristive neural network model

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

Homogeneous Models of Nonlinear Circuits

This paper develops a general approach to nonlinear circuit modelling aimed at preserving the intrinsic symmetry of electrical circuits when formulating reduced models. The goal is to provide a framework accommodating such reductions in a global manner and without any loss of generality in the working assumptions; that is, we avoid global hypotheses imposing the existence of a classical circuit variable controlling each device. Classical (voltage/current but also flux/charge) models are easily obtained as particular cases of a general homogeneous model. Our approach extends the results introduced for linear circuits in a previous paper, by means of a systematic use of global parametrizations of smooth planar curves. This makes it possible to formulate reduced models in terms of homogeneous variables also in the nonlinear context: contrary to voltages and currents (and also to fluxes and charges), homogeneous variables qualify as state variables in reduced models of uncoupled circuits without any restriction in the characteristics of devices. The inherent symmetry of this formalism makes it possible to address in broad generality certain analytical problems in nonlinear circuit theory, such as the state-space problem and related issues involving impasse phenomena, as well as index analyses of differential-algebraic models. Our framework applies also to circuits with memristors, and can be extended to include controlled sources and coupling effects. Several examples illustrate the results.Comment: Updated versio