Duality of Complex Systems Built from Higher-Order Elements

Hindawi Complexity Volume 2018, Article ID 5719397, 15 pages https://doi.org/10.1155/2018/5719397 Research Article Duality of Complex Systems Built from Higher-Order Elements Dalibor Biolek ,1,2 Zdeněk Biolek ,1 and Viera Biolková 1 2 1 Faculty of Electrical Engineering and Communication, Brno University of Technology, Brno, Czech Republic Faculty of Military Technologies, University of Defence, Brno, Czech Republic Correspondence should be addressed to Dalibor Biolek; Received 19 July 2018; Accepted 9 September 2018; Published 7 November 2018 Academic Editor: Dan Selişteanu Copyright © 2018 Dalibor Biolek et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 socalled 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. 1. Introduction Duality belongs to noticeable concepts of philosophy, social, natural, and also, engineering sciences. With regard to the duality of electrical circuits, introduced by Russell in 1904 [1], the fundamental ideas are summarized in [2–7]. According to this classical approach, two fundamental quantities in electrical engineering, the voltage v and current i, are termed dual since they are interconnected via two dual equations of Ohm’s law i = Gv and v = Ri, where G and R = 1/G are the resistance and conductance, respectively. Note that two equations are termed dual if one is the inverse of the other [8]. As some other examples of dual terms in electrical engineering, let us mention the KCL and KVL (Kirchhoff’s current and voltage laws), flux linkage and charge, impedance and admittance, inductance and capacitance, mesh and node, series and parallel, and short-circuit and opencircuit. Two electrical networks that are governed by the same types of equation are called dual networks [3]. According to [7], two circuits N and N ∗ are said to be dual to one another if the equations describing the circuit N are identical to those describing the circuit N ∗ after substituting each term (for example, voltage, resistance, KVL, and series connection) for N by the corresponding dual term (for example, current, conductance, KCL, and parallel connection) for N ∗ . Constructing dual circuits can be useful from the practical point of view. For example, passive LCR filters occur in two dual forms, differing in the number of inductors. One should select such a version which is more convenient for its implementation. The idea of dual circuits can also be applied to effective analysis or emulation of special networks. More details will be given below. Moreover, the duality principle can help with a thorough understanding of the current problems of modern circuit theory, for example, with the correct identification of dynamic properties of complex nonlinear networks that contain memristors [9–14]. The well-known procedure of finding a circuit which is dual to another circuit is a two-step transformation, topological and electrical [15]. Topological transformation starts from the nonseparable planar graph of the original circuit and transforms it via the rule, described for example in [6], into the graph of dual circuit. Electrical transformation 2 assigns the circuit elements to the branches of the transformed graph, which are dual to the elements of the branches of the original graph. Although the duality was originally introduced for linear passive circuits, it was subsequently generalized for circuits containing controlled sources [8] and for nonlinear circuits [15]. Based on the electrical-mechanical analogies, the paper [16] deals with the duality of mechanical systems and highlights the differences between the duality and analogy. In addition to the above concept of duality, there are also some alternative approaches to the term “dual” in electrical engineering. For example, a specific duality is studied in [17] for circuits which originate from original networks via skipping both the topological transformation and electrical transformation of voltage and current sources. The term “dual” may be also used for circuits generated by the wellknown Bruton’s transformation [18]. Although the two above cases have nothing to do with the classical concept of duality, they can be considered as its extension. In this sense, we will also understand the duality of memristor circuits introduced in [19] by Itoh and Chua, which is the starting point of this study. In order to discriminate between these different concepts, Section 3 deals with the classical and Chua’s duality. The current procedure of the electrical transformation starts from the knowledge that the fundamental R, C, and L elements are defined via voltage-current, voltagecharge, and flux-charge constitutive relations. Interchanging the dual quantities means that resistors are replaced by conductors, capacitors by inductors, and inductors by capacitors. As the memory versions of R, C, and L elements, i.e., the memristors RM, memcapacitors CM, and meminductors LM have arrived on the scene, a further elaboration of the rules of electrical transformations is necessary. Such a necessity is underlined by the fact that the above memory elements are merely a subset of the so-called higher-order elements from Chua’s table [20], which can also be potentially used for constructing the dual circuits. The paper [19] deals with a specific (Chua’s) duality of nonlinear R-C-L networks and networks with the RM-CMLM memory elements, thus memristors, memcapacitors, and meminductors. According to [19], dual R-C-L and RM-CM-LM circuits are described by formally the same differential equations but with different types of variables. Such circuits then exhibit the equivalent dynamic behavior. For example, circuits containing memristors, linear capacitors, and linear inductors are dual to circuits originating by replacing memristors with charge-flux constitutive relations by dual nonlinear resistors with formally the same current-voltage characteristics. This concept is then extended to other mem-elements: if all the nonlinear elements in the classical R-C-L circuit are replaced by their dual memory v (...truncated)