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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 nonisolated 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 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 (an exception is made for the bifurcating memristor), and in a second step some results are presented for the analysis of nonpassive cases. The latter context is illustrated by means of a memristive neural network model. [ABSTRACT FROM AUTHOR]

In this paper we address local bifurcation properties of a family of networked dynamical systems, specifically those defined by a potential-driven flow on a (directed) graph. These network flows include linear consensus dynamics or Kuramoto models of coupled nonlinear oscillators as particular cases. As it is well-known for consensus systems, these networks exhibit a somehow unconventional dynamical feature, namely, the existence of a line of equilibria, following from a well-known property of the graph Laplacian matrix in connected networks with positive weights. Negative weights, which arise in different contexts (e.g. in consensus models in signed graphs or in Kuramoto models with antagonistic actors), may on the one hand lead to higher-dimensional manifolds of equilibria and, on the other, be responsible for bifurcation phenomena. In this direction, we prove a saddle-node bifurcation theorem for a broad family of potential-driven flows, in networks with one or more negative weights. The goal is to state the conditions in structural terms, that is, in terms of the expressions defining the flowrates and the graph-theoretic properties of the network. Not only the eigenvalue requirements but also the nonlinear transversality assumptions supporting the bifurcation motivate an analysis of independent interest concerning the rank degeneracies of nodal matrices arising in the linearized dynamics; this analysis is performed in terms of the contraction–deletion structure of spanning trees and uses several results from matrix analysis. Different examples illustrate the results; some linear problems (including signed graphs) are aimed at illustrating the analysis of nodal matrices, whereas in a nonlinear framework we apply the characterization of saddle-node bifurcations to networks with a sinusoidal (Kuramoto-like) flow. [ABSTRACT FROM AUTHOR]

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 nonisolated 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. [ABSTRACT FROM AUTHOR]

The recent discovery of a physical device behaving as a memristor has driven a lot of attention to memristive systems, which are likely to play a relevant role in electronics in the near future, especially at the nanometer scale. The derivation of explicit ODE models for these systems is important because it opens a way for the study of the dynamics of general memristive circuits, including e.g. stability aspects, oscillations, bifurcations or chaotic phenomena. We tackle this problem as a reduction of implicit ODE (differential-algebraic) models, and show how tree-based approaches can be adapted in order to accommodate memristors. Specifically, we prove that the derivation of a tree-based explicit ODE model is feasible for strictly passive memristive systems under broad coupling effects and without a priori current/voltage control assumptions on tree/cotree elements. Our framework applies in particular to topologically degenerate circuits and accommodates a wide class of controlled sources. We also discuss a quasilinear reduction of nonpassive problems, which do not admit an explicit ODE description in the presence of singularities; some related bifurcations are addressed in this context. [ABSTRACT FROM AUTHOR]

The present paper extends the singularity induced bifurcation theorem (SIBT) to higher index differential-algebraic equations (DAEs) in Hessenberg form. The SIBT arises in power system theory, and is also significant within the context of electrical circuits. This phenomenon is due to the presence of singularities in parameter-dependent problems, and it was originally proved for semiexplicit index-1 DAEs. We introduce a new proof of a matrix pencil-based version of this result, relying on the spectral features of the linearization of the underlying ordinary differential equation. This approach is then applied to higher index DAEs in Hessenberg form. For these structures, the SIBT is shown to follow from a minimal index change at the singularity. Additionally, we show that a Kronecker index change is also necessary for the existence of a singularity induced bifurcation point in semi-explicit index-1 and Hessenberg DAEs. [ABSTRACT FROM AUTHOR]