A thorough investigation of the nonlinear dynamics of networks of memristor oscillators is a key step towards the design of systems based upon them, such as neuromorphic circuits and dense nonvolatile memories. A wide gamut of complex dynamic behaviors, including chaos, is observed even in a simple network of memristor oscillators, proposed here as a good candidate for the realization of oscillatory associative and dynamic memories. A detailed study of number and stability of all periodic and nonperiodic oscillations appearing in the network may not leave aside a preliminary deep understanding of the local and global behavior of the basic oscillator. Depending on two bifurcation parameters, controlling memristor nonlinearity, the oscillator exhibits different dynamic behaviors, analyzed here through application of state-of-the-art techniques from the theory of nonlinear dynamics to the oscillator model.

Since the 2008-dated discovery of memristor behavior at the nano-scale, Hewlett Packard is credited for, a large deal of efforts have been spent in the research community to derive a suitable model able to capture the nonlinear dynamics of the nano-scale structures. Despite a considerable number of models of different complexity have been proposed in the literature, there is an ongoing debate over which model should be universally adopted for the investigation of the unique opportunities memristors may offer in integrated circuit design. In order to shed some light into this passionate discussion, this paper compares some of the most noteworthy memristor models present in the literature. The strength of the Pickett?s model stands in its experiment-based development and in its ability to describe some physical mechanism at the origin of memristor dynamics. Since its parameter values depend on the excitation of the memristor and/or on the circuit employing the memristor, it may be assumed as a reference for comparison only in those scenarios for which its parameters were reported in the literature. In this work various noteworthy memristor models are fitted to the Pickett's model under one of such scenarios. This study shows how three models, Biolek's model, the Boundary Condition Memristor model and the Threshold Adaptive Memristor model, outperform the others in the replica of the dynamics observed in the Pickett's model. In the second part of this work the models are used in a couple of basic circuits to study the variance between the dynamical behaviors they give rise to. This analysis intends to make the circuit designers aware of the different behaviors which may occur in memristor-based circuits according to the memristor model under use.

The class of nonlinear dynamical systems known as memristive systems was defined by Chua and Kang back in 1976. Since then, many studies have addressed the search for physically-realisable memristive systems. In this reported work, it is proved that the class of memristive systems encloses an elementary electronic circuit comprising a full-wave rectifier with a second-order RLC filter.

A deep theoretical discussion proves that in Joglekar's and Biolek's models the memductance-flux relation of a memristor driven by a sign-varying voltage source may only exhibit single-valuedness and multi-valuedness respectively. This manuscript derives a novel boundary condition-based Model for memristor nanostructures. Unlike previous models, the proposed one allows for closed-form solutions. More importantly, subject to the nonlinear behavior under exam, this model enables a suitable tuning of boundary conditions, which may result in the detection of both single-valued and multi-valued memductance-flux relations under certain sign-varying inputs of interest. The large class of modeled dynamics include all behaviors reported in the legendary paper revealing the existence of memory-resistance at the nano scale.

The present manuscript relies on the companion paper entitled “Memristor Circuits: Flux-Charge Analysis Method,” which has introduced a comprehensive analysis method to study the nonlinear dynamics of memristor circuits in the flux-charge (φ, q)-domain. The Flux-Charge Analysis Method is based on Kirchhoff Flux and Charge Laws and constitutive relations of circuit elements in terms of incremental fluxes and incremental charges. The straightforward application of the method has previously provided a full portrait of the nonlinear dynamics and bifurcations of the simplest memristor circuit composed by a capacitor and a flux-controlled memristor. This paper aims to show that the method is effective to analyze nonlinear dynamics and bifurcations in memristor circuits with more complex dynamics including Hopf bifurcations (originating persistent oscillations) and period-doubling cascades (leading to chaotic behavior). One key feature of the method is that it makes clear how initial conditions give rise to bifurcations for an otherwise fixed set of circuit parameters. To the best of the authors' knowledge, these represent the first results that relate such bifurcations, which are referred to in the paper as Bifurcations without Parameters, with physical circuit variables as the initial conditions of dynamic circuit elements.