In order to avoid non-convergence and other numerical problems associated with the simulation of extremely large memristive networks, suitable models of the individual memristors need to be developed. A circuit containing thousands of memristors for finding the shortest path in a complicated maze is a typical example of a numerically challenging simulation. A benchmark circuit for testing the applications of various complexities is used for the transient analysis. It is shown that various commonly used models, from simple behavioral to Pickett models, require special modifications for minimizing the simulation time and improving the convergence issue.

The paper introduces a set of models of memristive devices for a reliable, accurate and fast analysis of large networks in the SPICE (Simulation Program with Integrated Circuit Emphasis) environment. The modeling starts from the recently introduced TEAM (ThrEshold Adaptive Memristor Model) and VTEAM (Voltage ThrEshold Adaptive Memristor Model). A number of improvements are made towards the stick effect elimination and other numerical refinements to make the analysis of large networks fast and accurate. A set of models are proposed that utilize the synergy of several techniques such as window asymmetrization, integration with saturation, state equation preprocessing, scaling, and smoothing. The performance of models is tested in Cadence PSPICE 17.2 and particularly in HSPICE v2017, the latter on a large-scale CNN (Cellular Nonlinear Network) for detecting edges of binary images. The simulations manifest the usability of developed models for fast and reliable operation in networks containing more than one million nodes.

A special subset of two-terminal elements providing pinched hysteresis loops in the voltage-current plane with the lobe area increasing with the frequency is analysed. These devices are identified as non-memristive systems and the sufficient condition for their hysteresis loop to be pinched at the origin is derived. It turns out that the analysed behaviour can be observed only for just one concrete initial state of the device. This knowledge is conclusive for understanding why such devices cannot be regarded as memristors.

The classical fingerprint of the ideal memristor with differentiable constitutive relation has recently been extended with a piece of knowledge that the area of the pinched hysteresis loop of a memristor driven by sinusoidal signal with fixed amplitude and increasing frequency disappears with an integer power of the frequency. This fingerprint is extended to the memcapacitor and meminductor, and it is also shown that the way the hysteresis disappears can be determined directly from the behaviour of the parameter versus state map (PSM) of the element in the neighbourhood of the initial operating point.

The duality of nonlinear systems built from higher‐order two‐terminal Chua’s elements and independent voltage and current sources is analyzed. Two different approaches are now being generalized for circuits with higher‐order elements: the classical duality principle, hitherto restricted to circuits built from R‐C‐L elements, and Chua’s duality of memristive circuits. The so‐called storeyed structure of fundamental elements is used as an integrating platform of both approaches. It is shown that the combination of associated flip‐type and shift‐type transformations of the circuit elements can generate dual networks with interesting features. The regularities of the duality can be used for modeling, hardware emulation, or synthesis of systems built from elements that are not commonly available, such as memristors, via classical dual elements.

The necessary and sufficient conditions of the validity of Hamilton’s variational principle for circuits consisting of (α,β) elements from Chua’s periodical table are derived. It is shown that the principle holds if and only if all the circuit elements lie on the so-called Σ-diagonal with a constant sum of the indices α and β. In this case, the Lagrangian is the sum of the state functions of the elements of the L or ⁺R types minus the sum of the state functions of the elements of the C or ⁻R types. The equations of motion generated by this Lagrangian are always of even-order. If all the elements are linear, the equations of motion contain only even-order derivatives of the independent variable. Conclusions are illustrated on an example of the synthesis of the Pais–Uhlenbeck oscillator via the elements from Chua’s table.