We extend in this paper some previous results concerning the differential-algebraic index of hybrid models of electrical and electronic circuits. Specifically, we present a comprehensive index characterization which holds without passivity requirements, in contrast to previous approaches, and which applies to nonlinear circuits composed of uncoupled, one-port devices. The index conditions, which are stated in terms of the forest-structure of certain digraph minors, do not depend on the specific tree chosen in the formulation of the hybrid equations. Additionally, we show how to include memristors in hybrid circuit models; in this direction, we extend the index analysis to circuits including active memristors, which have been recently used in the design of nonlinear oscillators and chaotic circuits. We also discuss the extension of these results to circuits with controlled sources, making our framework of interest in the analysis of circuits with transistors, amplifiers and other multiterminal devices.

This paper addresses a systematic characterization of saddle-node bifurcations in nonlinear electrical and electronic circuits. Our approach is a circuit-theoretic one, meaning that the bifurcation is analyzed in terms of the devices' characteristics and the graph-theoretic properties of the digraph underlying the circuit. The analysis is based on a reformulation of independent interest of the saddle-node theorem of Sotomayor for semiexplicit index one differential-algebraic equations (DAEs), which define the natural context to set up nonlinear circuit models. The bifurcation is addressed not only for classical circuits, but also for circuits with memristors. The presence of this device systematically leads to non-isolated equilibria, and in this context the saddle-node bifurcation is shown to yield a bifurcation of manifolds of equilibria; in cases with a single memristor, this phenomenon describes the splitting of a line of equilibria into two, with different stability properties.

In this paper, we extend the topological formulae of Maxwell and Kirchhoff, characterizing the non-singularity of node-admittance and loop-impedance matrices, to mixed problems, that is, to circuits combining admittance and impedance descriptions or, in nonlinear cases, involving both voltage- and current-controlled resistors. By means of this mixed formula we analyze the index of differential-algebraic models of nonlinear uncoupled circuits in a very broad setting, namely, without assumptions on their topology, their passivity or the controlling variables for nonlinear resistors. In particular, our approach allows for a characterization of index two circuits in topologically degenerate settings, which had been so far elusive in the non-passive context. As a byproduct we address the unique solvability of mixed resistive circuits, a problem which also arises in connection to the so-called DC-solvability condition of dynamic circuits. For the sake of brevity, we discuss in less detail how to extend the analysis to problems with mixed descriptions in reactive devices.

Bifurcations without parameters describe qualitative changes in the local dynamics of nonlinear ODEs when normal hyperbolicity of a manifold of equilibria fails. Non-isolated equilibrium points are systematically exhibited by nonlinear circuits with memristors; a memristor is a nonlinear device recently introduced in circuit theory and which is expected to play a key role in electronics in the near future. In this communication we provide a graph-theoretic analysis of the transcritical bifurcation without parameters in memristive circuits, owing to the presence of a locally active memristor. The results are crucially based on the use of differential-algebraic circuit models.

Bifurcation theory plays a key role in the qualitative analysis of dynamical systems. In nonlinear circuit theory, bifurcations of equilibria describe qualitative changes in the local phase portrait near an operating point, and are important from both an analytical and a numerical point of view. This work is focused on quadratic turning points, which, in certain circumstances, yield saddle-node bifurcations. Algebraic conditions guaranteeing the existence of this kind of points are well-known in the context of explicit ordinary differential equations (ODEs). We transfer these conditions to semiexplicit differential-algebraic equations (DAEs), in order to impose them to branch-oriented models of nonlinear circuits. This way, we obtain a description of the conditions characterizing these turning points in terms of the underlying circuit digraph and the devices' characteristics.