First order devices, hybrid memristors, and the frontiers of nonlinear circuit theory

Arxiv – First order devices, hybrid memristors, and the frontiers of nonlinear circuit theory

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 C1-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.

* nonlinear circuit theory is concerned with the study of constrained ordinary
differential equations involving time and four m-dimensional variables charge, flux, current and voltage.

* voltage-controlled) memcapacitor

* current-controlled) meminductor

* hybrid memristors, since their memristance and memductance involve both the charge q and the flux ‘. These novel circuit elements, together with their fully nonlinear variants, are introduced here for mathematical completeness, but they also provide a natural framework to accommodate actual physical devices in which different memory effects (namely, memristive, memcapacitive and/or meminductive features) coexist

* the list of devices with differential order one will include

• capacitors, inductors, q-memristors and ‘-memristors, all of them with state order one;

• voltage-controlled memcapacitors, current-controlled meminductors, and hybrid memristors, with state order two.

We have presented in this paper a comprehensive taxonomy of a variety of devices which have arisen in nonlinear circuit theory in the last few years. This taxonomy is organized around the notions of the differential and the state order of a device. An exhaustive list of possibly nonlinear devices with differential order one has been discussed: besides capacitors and inductors, these include q- and ‘-memristors, memcapacitors, meminductors, and also the hybrid memristors here introduced. Hybrid memristors display a characteristic relating all four fundamental circuit variables, and account for devices in which memory effects of resistive, capacitive and inductive nature coexist. All these devices are discussed using fully nonlinear characteristics, and particularize to Chua’s memristors and to the memcapacitors and meminductors of Di Ventra et al., respectively, when the corresponding constitutive relations are linear in certain variables. A detailed analysis of the differential-algebraic index of circuits including all possible types of first order devices is also included.

Many aspects remain open and define lines for future investigation. These include numerical issues, modelling aspects involving e.g. branch-oriented systems and hybrid analysis, or dynamical properties related to the nature of these circuits’ operating points, their stability, bifurcations, as well as the eventual existence and characterization of periodic solutions, oscillations or chaotic effects. Higher order devices and mem-systems are also in the scope of future research.

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