Two Element Chaotic and Hyperchaotic Circuits
The goals of this work are twofold: one is to illustrate the use of Field Programmable Gate Arrays (FPGAs) for emulating circuit elements with memory (memristors, memcapacitors and meminductors). The second goal is to use the FPGA emulation to realize two
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Abstract The goals of this work are twofold: one is to illustrate the use of Field Programmable Gate Arrays (FPGAs) for emulating circuit elements with memory (memristors, memcapacitors and meminductors). The second goal is to use the FPGA emulation to realize two element chaotic and hyperchaotic circuits. Such circuits utilize fully nonlinear models of memory devices in series-parallel configuration.
1 Introduction The memristor was postulated as the fourth fundamental circuit element by Leon O. Chua in 1971 [1]. It thus took its place along side the rest of the more familiar circuit elements such as the resistor, capacitor, and inductor. The common thread that binds these elements together as the four basic elements of circuit theory is the fact that the characteristics of these elements relate the four fundamental circuit variables (voltage, current, flux-linkage and charge). For over 30 years, the memristor was not significant in circuit theory. However in 2008, Strukov et al. [12] from Hewlett-Packard labs announced that they had fabricated a solid state implementation of the memristor. Ever since their announcement, a variety of circuit applications of memristors have been developed; refer to [6] for examples and further references. Chua and Kang [2] first extended the notion of the memristor to a general class of memristive systems. DiVentra et al. [4] incorporated capacitors and inductors into the notion of memory devices, as shown in Fig. 1. This “memory element” B. Muthuswamy () A. Przybylski C. Feilbach J. Mossbrucker Department of Electrical Engineering and Computer Science, Milwaukee School of Engineering, WI 53202, USA e-mail: [email protected]; [email protected]; [email protected]; [email protected] S.G. Stavrinides et al. (eds.), Chaos and Complex Systems, DOI 10.1007/978-3-642-33914-1 1, © Springer-Verlag Berlin Heidelberg 2013
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Fig. 1 We can generalize the four basic circuit elements to elements with memory. A resistor is a device that establishes a relation between current (i ) and voltage (v), a memristor is a device whose resistance depends on a state variable (z). Similarly memcapacitors establish a memory relationship between charge (q) and voltage; meminductors establish a relationship between flux-linkage () and current. Note that an ideal memristor is a special case of a general memristive system—the ideal memristor’s internal state is simply the charge flowing through or the flux-linkage across the device
abstraction enables us to model nanoscale systems [7] where the dynamical properties of electrons and ions strongly depend on the history of the system. In this work, we are concerned with utilizing memory elements to design two element chaotic and hyperchaotic circuits. We picked two elements because such circuits have been shown to correspond to models of physical systems [10]. Since memory elements are commercially unavailable as of this writing, we also show how one can use FPGAs to emulate such elements. Although microcontrollers have bee
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