Semi-Explicit Composition Methods in Memcapacitor Circuit Simulation

Semi-Explicit Composition Methods in Memcapacitor Circuit Simulation

Denis N. Butusov (Saint-Petersburg Electrotechnical University “LETI”, Saint Petersburg, Russia), Valerii Y. Ostrovskii (Saint-Petersburg Electrotechnical University “LETI”, Saint Petersburg, Russia), Artur I. Karimov (Saint-Petersburg Electrotechnical University “LETI”, Saint Petersburg, Russia) and Valery S. Andreev (Saint-Petersburg Electrotechnical University “LETI”, Saint Petersburg, Russia)
DOI: 10.4018/IJERTCS.2019040103

Abstract

Composition algorithms make up a prospective class of methods for solving ordinary differential equations. Their main advantage is an ability to retain some properties of the simulated continuous systems, e.g. phase space volume. Meanwhile, computational costs of composition solvers for non-Hamiltonian systems are high because the implicit midpoint rule should be used as a basic method. This also complicates the development of embedded applications based on the numerical solution of ODEs, such as hardware chaos generators. In this article, a new semi-explicit composition methods are proposed. The stability regions for different composition algorithms were plotted and a memcapacitor circuit was studied as a test problem. Computational experiments reveal the superior properties of semi-explicit composition algorithms as a hardware-targeted ODE solvers. The obtained results imply that the development of semi-explicit composition algorithms is a step towards construction a new generation of simulation software for nonlinear dynamical systems and embedded chaos generators.
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Introduction

Today, a role of dynamical chaos is rising in such fields as data encryption (Kocarev et al., 1998), pseudo-random number generation for security application (Falcioni, 2005), chaos-based communications (Vaidyanathan, & Volos, 2016), etc. One of the most challenging problems in chaos-based systems design is the implementation of chaos generators in real-time embedded systems (Pande & Zambreno, 2013). This is especially crucial when the requirements to the system imply low power consumption, compactification and other hardware limitations, while the security and performance must be enough, for instance, in RFID devices (Khan, & Moessner, 2011) and e-Health applications (Moosavi et al., 2016). The key point for solving this problem is to choose an appropriate hardware architecture of the target device and a proper mathematical basis, allowing to avoid or reduce such negative effects of computer simulation as precision loss, slipping into quasi-chaotic regime (Persohn et al., 2012), bad scaling, numerical instability and others.

One of prospective solutions for hardware chaotic generators are memristive elements – nonlinear devices that change their main characteristics upon electrical stimuli in a nonvolatile manner. Their theory originates from the prediction of the fourth basic passive element by L. Chua (1971), who called this element a memristor and placed it on a par with resistors, capacitors and inductors. Later, in 1976, the mathematical description of the memristor was extended to a wider class of memristive systems (Chua & Kang, 1976, pp. 209-223), within the framework of which the modern terms of other passive two-terminal circuit elements, a memcapacitor and a meminductor were introduced (Di Ventra et al., 2009, pp. 1717-1724; Shen et al., 2016, p. 64028). Memristive elements allow constructing simple and reliable analog chaos generators suitable to work in a wide spectrum of electronic devices.

Promising electronic components, such as resistive switching, phase change and the other memory are considered as possible practical implementation of memristive devices (Pershin & Di Venta, 2011, pp. 145-227). The relevant models of the real memristive unit cells cannot be made up of a single memristor, therefore a virtual capacitive element, a linear capacitor or a nonlinear memcapacitor, must also be included in a model. Additionally, due to the internal chemical processes in ionic subsystem of the device, one can also expect an inclusion of a meminductive element as a battery-like component. Thus, memcapacitors and meminductors should receive no less attention than memristors during development of memristive circuit models.

New circuit elements require revision of existing simulation approaches in CAD/CAE software packages, most of which are based on the SPICE simulator. However, most of engineers and researchers do not consider the properties of finite difference schemes obtained by means of numerical methods built in simulators, thoroughly verified mostly for linearized or slightly nonlinear problems. During chaotic systems simulation, properties of the applied discrete operator can have a significant impact on the model behavior. Neglecting of this fact may lead to incorrect simulation results; therefore, a study of these approaches in the field of memristive circuit design is required.

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