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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 version

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.

This paper presents a general framework for linear circuit analysis based on elementary aspects of projective geometry. We use a flexible approach in which no a priori assignment of an electrical nature to the circuit branches is necessary. Such an assignment is eventually done just by setting certain model parameters, in a way which avoids the need for a distinction between voltage and current sources and, additionally, makes it possible to get rid of voltage- or current-control assumptions on the impedances. This paves the way for a completely general $m$-dimensional reduction of any circuit defined by $m$ two-terminal, uncoupled linear elements, contrary to most classical methods which at one step or another impose certain restrictions on the allowed devices. The reduction has the form $$\begin{pmatrix} AP \\ BQ \end{pmatrix} u = \begin{pmatrix} AQ \\ -BP \end{pmatrix} \bar{s}.$$ Here, $A$ and $B$ capture the graph topology, whereas $P$, $Q$, $\bar{s}$ comprise homogeneous descriptions of all the circuit elements; the unknown $u$ is an $m$-dimensional vector of (say) ``seed'' variables from which currents and voltages are obtained as $i=Pu -Q\bar{s}$, $v=Qu + P\bar{s}$. Computational implementations are straightforward. These models allow for a general characterization of non-degenerate configurations in terms of the multihomogeneous Kirchhoff polynomial, and in this direction we present some results of independent interest involving the matrix-tree theorem. Our approach can be easily combined with classical methods by using homogeneous descriptions only for certain branches, yielding partially homogeneous models. We also indicate how to accommodate controlled sources and coupled devices in the homogeneous framework. Several examples illustrate the results., Comment: Updated version

A standard approach to reduce the complexity of very large networks is to group together sets of nodes into clusters according to some criterion which reflects certain structural properties of the network. Beyond the well-known modularity measures defining communities, there are criteria based on the existence of similar or identical connection patterns of a node or sets of nodes to the remainder of the network. A key notion in this context is that of structurally equivalent or twin nodes, displaying exactly the same connection pattern to the remainder of the network. The first goal of this paper is to extend this idea to subgraphs of arbitrary order of a given network, by means of the notions of T-twin and F-twin subgraphs. This is motivated by the need to provide a systematic approach to the analysis of core-semiperiphery-periphery (CSP) structures, a notion which somehow lacks a formal treatment in the literature. The goal is to provide an analytical framework accommodating and extending the idea that the unique (ideal) core-periphery (CP) structure is a 2-partitioned K2. We provide a formal definition of CSP structures in terms of core eccentricities and periphery degrees, with semiperiphery vertices acting as intermediaries. The T-twin and F-twin notions then make it possible to reduce the large number of resulting structures by identifying isomorphic substructures which share the connection pattern to the remainder of the graph, paving the way for the decomposition and enumeration of CSP structures. We compute the resulting CSP structures up to order six. We illustrate the scope of our results by analyzing a subnetwork of the network of 1994 metal manufactures trade. Our approach can be further applied in complex network theory and seems to have many potential extensions.

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.

This paper addresses analytical aspects of deterministic, continuous-time dynamical systems defined on networks. The goal is to model and analyze certain phenomena which must be framed beyond the context of networked dynamical systems, understood as a set of interdependent dynamical systems defined on the nodes of a (possibly evolving) graph. In order to advocate for a more flexible approach to the study of network dynamics, we tackle some qualitative problems which do not fall in this working scenario. First, we address a stability problem on a network of heterogeneous agents, some of which are of dynamic nature while others just impose restrictions on the system behavior. Our second context assumes that the network is clustered, and we address a two-level stability problem involving the dynamics of both individual agents and groups. The aforementioned problems exhibit lines of non-isolated equilibria, and the analysis implicitly assumes a positiveness condition on the edge weights; the removal of this restriction complicates matters, and our third problem concerns a graph-theoretic characterization of the equilibrium set in the dynamics of certain networks with positive and negative weights, the results applying in particular to signed graphs. Our approach combines graph theory and dynamical systems theory, but also uses specific tools coming from linear algebra, including e.g. Gersgorin discs or Maxwell-type determinantal expansions. The results are of application in social, economic, and flow networks, among others, and are also aimed at motivating further research.

Several devices exhibiting memory effects have shown up in nonlinear circuit theory in recent years. Among others, these circuit elements include Chua's memristors, as well as memcapacitors and meminductors. These and other related devices seem to be beyond the, say, classical scope of circuit theory, which is formulated in terms of resistors, capacitors, inductors, and voltage and current sources. We explore in this paper the potential extent of nonlinear circuit theory by classifying such mem-devices in terms of the variables involved in their constitutive relations and the notions of the differential- and the state-order of a device. Within this framework, the frontier of first order circuit theory is defined by so-called hybrid memristors, which are proposed here to accommodate a characteristic relating all four fundamental circuit variables. Devices with differential order two and mem-systems are discussed in less detail. We allow for fully nonlinear characteristics in all circuit elements, arriving at a rather exhaustive taxonomy of C^1-devices. Additionally, we extend the notion of a topologically degenerate configuration to circuits with memcapacitors, meminductors and all types of memristors, and characterize the differential-algebraic index of nodal models of such circuits., Comment: Published in 2013. Journal reference included as a footnote in the first page

The recent design of a nanoscale device with a memristive characteristic has had a great impact in nonlinear circuit theory. Such a device, whose existence was predicted by Leon Chua in 1971, is governed by a charge-dependent voltage-current relation of the form $v=M(q)i$. In this paper we show that allowing for a fully nonlinear characteristic $v=\eta(q, i)$ in memristive devices provides a general framework for modeling and analyzing a very broad family of electrical and electronic circuits; Chua's memristors are particular instances in which $\eta(q,i)$ is linear in $i$. We examine several dynamical features of circuits with fully nonlinear memristors, accommodating not only charge-controlled but also flux-controlled ones, with a characteristic of the form $i=\zeta(\varphi, v)$. Our results apply in particular to Chua's memristive circuits; certain properties of these can be seen as a consequence of the special form of the elastance and reluctance matrices displayed by Chua's memristors., Comment: 19 pages