Complexity

Complexity / 2018 / Article

Research Article | Open Access

Volume 2018 |Article ID 5719397 | 15 pages | https://doi.org/10.1155/2018/5719397

Duality of Complex Systems Built from Higher-Order Elements

Academic Editor: Dan Selişteanu
Received19 Jul 2018
Accepted09 Sep 2018
Published07 Nov 2018

Abstract

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.

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 [27]. According to this classical approach, two fundamental quantities in electrical engineering, the voltage and current , 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 open-circuit. Two electrical networks that are governed by the same types of equation are called dual networks [3]. According to [7], two circuits and are said to be dual to one another if the equations describing the circuit are identical to those describing the circuit after substituting each term (for example, voltage, resistance, KVL, and series connection) for by the corresponding dual term (for example, current, conductance, KCL, and parallel connection) for .

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 [914].

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 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 well-known 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, voltage-charge, 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-CM-LM 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 versions, then the dynamics of the novel circuit in the charge-flux (q,φ) state space and the dynamics of the original circuit in the current-voltage (i,v) state space are identical. This knowledge can be utilized, for example, in computer simulations of circuits with mem-elements, where the models of such rather exotic elements can be replaced by their well-known classical nonlinear dual types. Another interesting area is that of building hardware emulators of applications containing memristors, memcapacitors, and meminductors with the help of conventional memoryless nonlinear components. This is because the duality of these systems makes it possible to observe all the attributes of the dynamic behavior of the emulated systems, even if in another than the original state space. As an example, let us mention the (i,v) pinched hysteresis loops of memristors, which can be observed in the (di/dt, dv/dt) coordinates for the dual resistive circuits [19]. This piece of knowledge is fully in agreement with the conclusions published in [21]: the pinched hysteretic loops, previously considered as a notable fingerprint of the memristors, can generate not only the memristor but also either classical [22] of fractional-order [23] memristive system, various other systems that emulate the memristors [24], and even every nonlinear, so-called higher-order element from Chua’s table [25] but in an exactly defined space of coordinates, which are derived from the location of a given element in Chua’s table [21].

Chua’s duality principle from [19] can be easily extended to circuits employing arbitrary higher-order elements and generalized so that it holds for both the autonomous and the externally driven systems. This study provides such a generalization with the help of the so-called storeyed structure of fundamental elements, which is an equivalent form of Chua’s table [26]. The basic facts about the storeyed structure are summarized in Section 2. Section 3 contains an example demonstrating the connection between the location of the fundamental elements in the storeyed structure, which are components of a given system, and the equations of motion of this system. Possible procedures of drawing up equations of motion of composite systems are generalized in Section 4. It is shown in Section 5 how the circuit elements move around the storeyed structure during their classical and Chua’s duality transformation. The respective generalizations of the currently known duality principles are made for systems compounded of arbitrary higher-order elements.

2. Storeyed Hierarchy of Higher-Order Elements

Two-terminal (α, β) higher-order elements (HOEs) as the building blocks of complex nonlinear systems were introduced in [20]. The HOEs are concurrently called the fundamental elements, which is to say that no fundamental element from the set of HOEs can be made as a combination of the other HOEs.

The (α, β) element is unambiguously defined via its algebraic and generally nonlinear constitutive relation

The symbols v(α) and i(β) denote multiple derivatives or integrals of the terminal voltage and current with respect to time according to the rule [19].

If the implicit constitutive relation (1) can be derived as an unambiguous function of the variable i(β) or v(α), thus where f( ) and g( ) are piece-wise differentiable functions, then the corresponding HOE is called the (α, β) element controlled by the variable i(β) or v(α). The variables i(β) and v(α) can be considered as generalized currents and voltages of orders β and α, and the elements with the constitutive relations (4) or (5) as elements controlled by (generalized) current or voltage. For the sake of brevity, the term “generalized” can be omitted.

Note that the variables v(0), v(−1), v(−2), and i(0), i(−1), i(−2) can be denoted by the symbols v, φ, ρ, and i, q, σ and called the voltage, flux, time integral of flux (TIF), and the current, charge, time integral of charge (TIQ) [27].

Also, note that the implicit constitutive relation (1) is the most general description of the fundamental element, and that the explicit forms (4), (5) can be derived from (1) only under certain conditions. For example, the element controlled by the variable i(β) (4) has its version controlled by the variable v(α), if the g( ) from (5) exists as the inverse of f( ) in (4). If the function f( ) does not have its inverse function, then the given element can be derived only via the constitutive relation (1) or (4). A typical representative of this case is the tunnel diode with the constitutive relation i = f(v). Its inverse v = g(i) does not exist. It is shown in Section 4 that the necessary condition of the existence of just one explicit differential equation of motion of a system is that all its elements must be simultaneously controlled by either current or voltage.

Figure 1 shows a fragment of the storeyed structure of higher-order elements, formed by the so-called voltage and current nodes that represent the variables v(k) and i(k), by walls (vertical paths between nodes of the same type) and by reinforcements (horizontal and lateral links between walls). Let the order of the node of variables v(k) or i(k) be called the order of node. The baseline of the storeyed structure contains the zero-order nodes of variables v(0) and i(0), thus the voltage and current. The nodes of variables v(−1) and i(−1), thus the flux and charge, are located on the ceiling of the ground floor and simultaneously on the bottom of the first floor. The symbols of mathematical operations for transforming between the corresponding circuit variables are symbolically placed on the walls: the integrals for transforming voltage into flux and current into charge (bottom-up) and derivatives for backward transformations of flux into voltage and charge into current (top-to-bottom). Three HOEs are shown on the ground floor. Their constitutive relations are defined for pairs of the variables v(0) and i(0) (the (0,0) element-resistor), v(0) and i(−1) (the (0,-1) element-capacitor), and v(−1) and i(0) (the (−1,0) element-inductor). The first floor contains the memory versions of these elements, namely, the memristor (the (−1,−1) element), memcapacitor (the (−1,−2) element), and meminductor (the (−2,−1) element). Furthermore, some other elements transect several floors. Two of them, the (0,−2) and (−2,0) elements, are illustrated in Figure 1. It is obvious that the storeyed structure is unbounded both from below and from above, continuing upwards by next floors and downwards below the ground level.

The following specifics of the storeyed structure should be highlighted in connection with Figure 1: (1)The vertical parts (the walls) symbolize the links between voltage or current and its integrals and derivatives with respect to time. These links are given by physical definitions between the variables, for example, between the current and charge, and are of linear nature (the signal differentiation or integration with respect to time is a linear operation, expressed by a linear s-domain transfer function, namely, s for the differentiation and 1/s for the integration). Simultaneously, there are inertial transformations between the nodal variables(2)The horizontal and lateral links between the nodes generally represent nonlinear transformations between the nodal variables defined by nonlinear constitutive relations of the corresponding circuit elements. These transformations are algebraic(3)The horizontal line at the level no. k in Figure 1, i.e., at the level of the voltage or current node of the order , symbolizes generalized Kirchhoff’s law, which holds for the systems built from HOEs. Classical Kirchhoff’s current law (KCL) and Kirchhoff’s voltage law (KVL) hold at the levels of nodes of the variables v(0) and i(0). Integrating or differentiating the corresponding equations with respect to time yields, the generalized Kirchhoff’s flux law (KFLL) and Kirchhoff’s charge law (KCHL) for the levels of the nodes of variables v(−1) and i(−1) [19] and generally, Kirchhoff’s v(k) and i(k) laws (KV(k)L and KC(k)L) [28] for the levels of the nodes of variables v(k) and i(k). These dual forms of classical Kirchhoff’s laws can be used for the analysis of complex systems as will be shown in Section 3. The dual form means, in fact, that classical KVL and KCL hold independently of integrating and differentiating with respect to time, and thus their corresponding forms, which apply at various levels of the storeyed structure, represent mathematically equivalent links between the circuit variables. Note that the special cases for k = 0 and k = −1, the classical KVL and KCL for the voltage-current domain, and the KV(−1)L and KC(−1)L for the flux-charge domain can be advantageously used for describing classical RCL and memristive circuits [1214], but not for circuits containing general higher-order elements(4)The dual circuit elements (α,β) and (α + m,β + m), , having formally identical constitutive relationsare located in the storeyed structure “one below the other”, i.e., on the same places, but on different floors (5)If the constitutive relation of the (α,β) element is linear, then any other (α + m,β + m) element, , is dual to this element, because the systematic integration or differentiation of the linear constitutive relation generates identical linear constitutive relations of these other elements. In fact, it is just the same element which periodically appears on various vertical positions in the storeyed structure. For example, the linear resistor is simultaneously the linear memristor or the linear (k,k) element,

3. Example of Composite System and Its Dynamics

Let us seek for the equation of motion of the system from Figure 2 made up of a memristor, meminductor, linear capacitor, and driving voltage and current sources. It is obvious that the elements of the system appear on two various floors of the storeyed structure.

The following concise notation of the circuit variables will be used below:

Consider two individual cases of the RM, LM, and C elements controlled by either voltage or current.

3.1. Voltage-Controlled HOEs

Let the constitutive relations of the memristor and meminductor be as follows:

The constitutive relation of the linear capacitor with the capacitance C is

The equations for generalized KCL and KVL, holding for arbitrary levels , , can be written for the nodes ①, ② and the loops :

Equations (8)–(15) represent a complete set of equations of motion of system from Figure 2: 8 equations for 8 circuit variables and their derivatives of various orders. Two of them, iv and vi, are the source variables, and 6 pair variables belong to three fundamental two-terminal elements of the system. Only one explicit differential equation of motion will be derived below for a selected circuit variable.

Since the RM and LM elements are controlled by voltage, their constitutive relations (8) and (9) can be directly substituted only into equations of generalized KC(kc)L (11) and (12), kc = −1, not into equations of KV(kv)L (13)–(15). The constitutive relation of the linear capacitor (10) is invertible, thus it does not represent any limitations.

Because the current nodes of all the elements appear at the same level of the storeyed structure (they represent the charges), their constitutive relations must be substituted into KCL equations for kc = −1:

The substitution yields

Note that equation (19) determines the charge passing through the driving voltage source. If this variable is of no interest, then equation (19) does not need to be considered anymore, because the variable qv appears only in this equation.

Voltage variables of different orders appear on the left sides of equations (18) and (19), namely, the voltage (v(0)), flux (v(−1)), and TIF (v(−2)). These variables are linked together via KVL equations (13)–(15), which have not yet been considered. The KVL order kv should be chosen to be equal to the lowest order of the voltage variables in (18) and (19), thus −2. Then KV(−2)L equations will represent the sum of TIFs in the loops. All other voltage variables in the equations can be derived as derivatives of the TIF, and these equations will be differential, not integro-differential.

Let us therefore rewrite equations (13)–(15) as follows:

Since the variable ρi appears only in equation (22), this equation does not need to be considered anymore if ρi is not the subject of subsequent analysis.

Expressing equation (18) in the form where ( )′ = d( )/dt, and substituting (20) and (21) into (23), the nonlinear differential equation of motion of the system is

3.2. Current-Controlled HOEs

Consider again the circuit in Figure 2, but now with the current-controlled RM, LM, and C elements. Their constitutive relations will be in the forms

KCL and KVL equations (11)–(15) hold for this circuit.

The voltage variables on the left sides of the element equations (25)–(27) are of various orders, namely, 0, −1, and −2. In order to substitute them into KVL equations, which use the unified order kv, it is necessary to state this order and also, to consolidate the orders in the element equations into kv.

It is important that the constitutive relations of the memristor and meminductor can be transformed via successive differentiation such that the orders of the voltage nodes of these elements can be increased (increasing the order of a node means shifting it downwards in the storeyed structure) up to an arbitrary level. The differentiation also enriches the right sides of the constitutive relations by the derivatives of the variables in current nodes. It is obvious that the differentiation should be performed with the aim to equalize the orders of voltage nodes to the level of the highest order in equations (25)–(27), thus 0 for the capacitor. It specifically means single differentiation of (25) and double differentiation of (26) with the results where is the memristance, is the meminductance, and is the first derivative of the meminductance with respect to charge.

In this way, the models of all the elements were transformed into a unified type of the voltage variable. Equations (14), (15), and (27) can then be used for the formulation of KVL equations (13)–(15) for kv = 0:

KVL equations are completed by KCL equations (11) and (12). With regard to the charge-type of equations (19)–(21), KCL equations must be written for kc = −1, thus in the form of (16) and (17).

The system can therefore be described by the set of charge-type equations of motion (16), (17), (32), (33), and (34). However, it seems that we cannot derive from them a single explicit differential equation such as in the case of the voltage-controlled elements in Section 3.1 (see the TIF-type equation (24)).

Alternatively, the equations of motion can be also found with the help of the consideration that the constitutive relation of the linear capacitor can be modified via a linear transformation (i.e., by integrating with respect to time) into a relationship between the flux and the TIQ, thus φc = C−1σc. It corresponds to the vertical translation of the element in the storeyed structure in Figure 2 upwards from the position (0,−1) to (−1,−2). Then KVL equations can be formulated for fluxes, thus kv = −1, because the highest order of the voltage nodes of HOEs is now −1. The lowest order of the current nodes is −2, thus the resulting equations of motion will be the equations for the TIQ. The above procedure leads to the set of equations of motion

However, it is also difficult to derive just one explicit equation of motion from the set (35)–(39).

It is therefore obvious that the system containing HOEs cannot be sometimes modeled by a single explicit differential equation of motion. On the other hand, such systems can be always described by a set of particular equations. The respective generalizations are given in the next section.

4. Equations of Motion of Composite Systems

Consider a nonlinear dynamic system compounded of ne HOEs and nv and ni independent sources of the v(k) and i(k) types, . Suppose that the HOEs are distributed in the storeyed structure such that the lowest and highest orders of their voltage and current nodes are αmin, αmax, βmin, and βmax.

For example, the circuit in Figure 2 is driven by one current source i(0) and one voltage source v(0), thus ni = 1 and nv = 1, and it contains ne = 3 HOEs, with αmin = −2, αmax = 0, βmin = −2, βmax = −1 if the linear capacitor is considered as the (0, −1) element, or αmin = −2, αmax = −1, βmin, = −2, βmax = −1 for the linear capacitor as the (−1, −2) element.

Let the HOEs of the system be described by their constitutive relations

The element-element and element-source links are described by generalized Kirchhoff’s laws. If the circuit contains nn independent nodes, i.e., nodes with except for the datum nodes [7], then the corresponding equations of generalized KCL for variables of the i(kc) type are in the form where AI is a (nn × (ne + ni + nv)) current incidence matrix and is a ((ne + ni + nv) × 1) vector of the kc-th derivative of the current of HOEs and independent sources.

Similarly, a set of equations of generalized KVL for variables of the v(kv) type can be written for nl independent loops in the circuit where AV is a (nl × (ne + ni + nv)) voltage incidence matrix and V(kv) is a ((ne + ni + nv) × 1) vector of the kv-th derivative of the terminal voltages of HOEs and independent sources.

Equations (40)–(42) represent a complete system of equations of motion of the system.

If all implicit constitutive relations (40) cannot be transformed into the constitutive relations of either voltage- or current-controlled elements, then the constitutive relations cannot be directly substituted into the equations of KVL (42) or KCL (41). In such cases, equations (40), (41), and (42) form relatively independent submodels of the implicit model of the system.

Let us introduce the variables

Then the constitutive relation (40) can be written as

Then the constitutive relations of the elements form a set of differential equations of the variables xv and xi and their derivatives of the order of 0 to (αmax − αmin) for xv and 0 to (βmax − βmin) for xi.

Let us choose the orders of KCL (41) and KVL (42) as follows: kc = βmin, ki = αmin. Then the current and voltage vectors in (41) and (42) will contain the variables (43), and equations (41) and (42) will be in the form

These equations are algebraic, representing the coupling conditions for the variables (43). Together with (43), they form a system of algebraic-differential equations for variables of the type (43).

If all HOEs can be modeled as current-controlled elements, then the constitutive relations (40) can be written as

The procedures described in Section 3 can be generalized in the following steps: (a)The constitutive equations (47) are modified via successive differentiations such that the orders of the corresponding voltage nodes are equalized to the level of the circuit element with the maximum order of the voltage node, thus αmax. The differentiation always increments this order according to the rule

where

It is obvious that a successive differentiation of the constitutive relation (47) of the element with the voltage will result in expressing the variable as a function of the variables

The above functions are governed by the chain rules for higher derivatives described by Faà di Bruno’s formula [29] (b)The constitutive relations of all elements, modified in item (a), are substituted into generalized KVL (42) for kv = αmax. The resulting set of equations will contain circuit variables of the i(k) types for all the elements in the circuit. According to (50), the lowest order of these variables is equal to the lowest order β of all the elements, thus βmin. It is useful to choose the variables i(βmin) = xi (see equation (43)) as state variables of the equations of motion, because their derivatives will cover all other variables (50) appearing in the set of equations of generalized KVL (42). In other words, only quantities of the i(βmin) = xi types and their derivatives will appear in equation (42)(c)With a view to item (b), the equation (41) of generalized KCL must be considered for kc = βmin. In other words, only quantities of the i(βmin) = xi types will appear in equation (41)(d)The sets of equations for variables of the xi type from (b) and (c) are general model of motion of the system in the space of xi coordinates. The order of these KVL equations is αmax. In special cases, a single nonlinear equation of motion for a selected state variable can be derived from them

If all the HOEs can be modeled as voltage-controlled elements, then the constitutive relations (40) can be arranged in the form

A procedure analogous to that for the current-controlled elements would result in the following: the orders of the current nodes must be equalized to the value of βmax via successive differentiations of (51). It is useful to choose variables of the  = xv type (see (44)) as state variables. Then the equations of motion are KCL equations of order βmax.

5. Duality in Circuits Employing HOEs in the Light of Storeyed Structure

The storeyed structure of higher-order elements can serve as a useful tool for demonstrating the principles of both the classical and Chua’s duality and for illustrating their common and different features. First, we show that the classical approach of transforming R, C, and L elements into their dual counterparts can be easily extended to all the elements in the storeyed structure. Then Chua’s concept of the duality of memristor circuits will be generalized for the complete set of nonlinear higher-order elements.

Note that both the generalized classical and Chua’s dualities can be combined. For example, the original circuit can be topologically and electrically transferred via the classical approach in the first step, with subsequent undergoing the procedure of Chua’s duality. In this way, a group of circuits with interesting duality properties can be generated.

The analysis given below will be restricted to circuits containing only one-port higher-order elements and independent sources. The graphs of the circuits are connected, planar, and nonseparable [15].

5.1. Classical Duality: Topological + Electrical “Flip-Type” Transformation

Consider a system A compounded of HOEs and independent sources of generalized voltages v(α) and currents i(β). It can be transformed into the dual system A′ in the following steps: (1)Nodes of the system A′ are placed within each mesh of the original system A. A wire is drawn around the system A, creating one additional node of the system A′(2)Each element from the system A is replaced by its dual element. This dual element is drawn between the nodes of the system A′ located on either side of the original element(3)The reference directions of the branches of the system A′ are given as follows [15]: for each branch appearing clockwise around a mesh in the system A, its corresponding branch in the system A′ will be directed into the corresponding node of A′(4)Based on the reference directions from (3), the dual elements of the system A′ are assigned to each element of the system A according to Table 1


System A        ↔     System A′

Short circuit: v(α) = 0Open circuit: i(α) = 0
Independent v(α) sourceIndependent i(α) source
Resistor: v = v(0), i = i(0)Conductor: v = v(0), i = i(0)
F(v,i) = 0 orF(i,v) = 0 or
v = F1(i) ori = F1(v) or
i = F2(v)v = F2(i)
Capacitor: v = v(0), q = i(−1)Inductor: φ = v(−1), i = i(0)
F(v,q) = 0 orF(i,φ) = 0 or
v = F1(q) ori = F1(φ) or
q = F2(v)φ = F2(i)
Memristor: φ = v(−1), q = i(−1)Memductor: φ = v(−1), q = i(−1)
F(φ,q) = 0 orF(q,v) = 0 or
φ = F1(q) orq = F1(φ) or
q = F2(φ)φ = F2(q)
(α,β) HOE: v(α), i(β)(β,α) HOE: v(β), i(α)
F(v(α), i(β)) = 0 orF(i(α), v(β)) = 0 or
v (α) = F1(i(β)) ori(α) = F1(v(β)) or
i(β) = F2(v(α))v(β) = F2(i(α))

Note that the first three steps describe the well-known topological duality, which also holds for nonlinear circuits containing multiport elements [15]. The nonlinear resistor↔conductor and capacitor↔inductor dualities from Table 1 are also well known: the corresponding dual quantities are voltage-current and flux-charge. Worth noting are the specific positions of these pairs in the storeyed structure from Figure 1: they are placed symmetrically with respect to the imaginary vertical line passing through the center of the structure. The constitutive relations of dual voltage- or current-controlled resistors-conductors and capacitors-inductors are inverse functions. This simple symmetry can be generalized into the duality of independent sources, whose special case is the well-known duality of short-open circuit, memristors-memductors, and (α,β)-(β,α) HOEs dualities in Table 1. Such dualities can be understood as a specific “flip-type” transformation of the elements in the storeyed structure as demonstrated in Figure 3.

Demonstrating the classical procedure of finding circuits that are dual to circuits containing general higher-order elements, consider the network in Figure 4(a) and its step-by-step transformation as self-explained in Figures 4(b)–4(d). The reference directions for each element are indicated via arrows, and the constitutive relations of all elements are also included. The symbol I in Figure 4 denotes the nonlinear (1,0) electrical HOE called inerter [30], which is an electrical analogy of the nonlinear mechanical element with force-acceleration constitutive relation [31]. It is obvious from Figure 4 that the network can be transformed into a parallel combination of a nonlinear (0,1) HOE denoted as the inverse inerter (II), nonlinear conductor, and linear capacitor, driven by independent charge source.

5.2. Chua’s Duality: Electrical “Shift-Type” Transformation

Based on the analysis in Section 4, the following Duality Theorem can be formulated:

Consider a system A compounded of HOEs and independent sources of generalized voltages and currents . The system dynamics is described in the space , where αmin and βmin are the minimum orders of voltage and current nodes of the elements in the storeyed structure.

Consider a system Am derived from the system A via the following procedure: the constitutive relations of all HOEs remain unchanged, but their variables of the (v(α), i(β)) type are modified to (v(α + m), i(β + m)), .

Then the dynamics of the system A in the space will be equal to the dynamics of the system Am in the space . The systems A and Am are dual.

It is sufficient to provide the proof for the most general case of the system with the implicit constitutive relations (40) of the elements, which are linked by generalized KCL (41) and KVL (42).

From the point of view of the storeyed structure, the systems A and Am differ only in the shift of the HOEs of the system Am by m floors down (for m > 0) or up (for m < 0) with respect to the HOEs of the system A, while preserving the constitutive relations of all the elements. Similar shifts hold for the character of the voltages and currents of the driving sources. It is obvious that these shifts do not affect the system equations, but only the equation variables.

The shift of the elements causes a modification of the variables in their constitutive relations (40):

The above shift of all the elements results in the modification of the parameters αmin, αmax, βmin, βmax to αmin + m, αmax + m, βmin + m, βmax + m. The original variables xv and xi, whose space was used for the analysis of the system dynamics, are modified to

Equally, the orders kc = βmin, ki = αmin of KCL (45) and KVL (46) are changed by m. Together with the shifts of HOEs in the storeyed structure, this means that the corresponding equations (45) and (46) remain unchanged, but the vectors Xi and Xv will now contain variables modified by the transformations (52):

Equations (52), (54), and (55) describe a system dual to the system (40), (45), and (46). The dynamics of both systems are identical.

It is obvious from the above that the generalized Chua’s duality can be considered as a “shift-type” transformation in the storeyed structure indicated in Figure 5. Recall that if the element has a linear constitutive relation, then we can shift it in the structure independently of the shift m of the nonlinear elements: in the limiting case, it can stay in its original position.

Generalized Chua’s duality will be demonstrated on the circuit in Figure 6, which was created from the circuit in Figure 2 by shifting vertically all the HOEs one floor down, thus m = 1. Although this circuit does not contain any memory elements, it exhibits the same dynamics in the (φ,q) coordinates as the original circuit in the (ρ,σ) coordinates.

Let the constitutive relations of the elements in Figure 6 be dual to the constitutive relations (8)–(10):

The procedure described in Section 4 leads to the equation of motion

The duality of both systems is obvious from a comparison of their equations (57) and (24).

Another example of dual circuits is shown in Figure 7. The original circuit in Figure 7(a) is the circuit from Figure 4(d) generated in Figure 4 via classical duality transformation. Figure 7 demonstrates its subsequent transformation via the shift-type m = −1 procedure. The final circuitry is a nonlinear-inductor parallel resonance tank circuit with nonlinear memristive damping. It is interesting that for σ = 0 and with m, n, and h parameters as specified in [30], the circuit in Figure 7(b) models the spin-torque nano-oscillator from [32]. It should be stressed that all the circuits in Figures 4 and 7 represent dual models of this nanodevice, and we can choose a circuit that best meets our needs for the device simulation or emulation.

It is shown in [30] that the flux φGM of the memristor in Figure 7(b) models the so-called in-plane (azimuthal) angle of rotation ϕ, and the memristor voltage its derivative with respect to time, thus the speed of the rotation. The simulations of the spin-torque oscillator model reveal the full-rotation regime of self-oscillations (see Figure 5(c) in [30]) for the initial angle φGM(0) = ϕ(0) = 65°, i.e., 1.134 rads, and for the following numerical values of the parameters from Figure 7(b):

All the nonlinear phenomena of the oscillator behavior can be also studied via its dual models. For example, the inerter, resistor, and inductor in series from Figure 4(a) generate nonlinear oscillations, observable via the waveforms of loop current (dual to the angle of rotation) and its derivative which is equal to the inductor voltage divided by the inductance (dual to the speed of the rotation). The SPICE simulation of dual inerter-based circuit from Figure 4(a) provides the waveforms in Figure 8 which are identical to the results from [30].

6. “Shift-Type” Duality in a Broader Context

The idea of dual HOEs as elements that move vertically in the storeyed structure while preserving their constitutive relations is related to the well-known Bruton transformation of linear two-terminal elements [18]. The linear operation connected with the Bruton division by the s operator and modifying the nature of the element is generalized here for transforming the nonlinear two-terminal element “one floor up” in the storeyed structure.

The above duality principle is a potential tool for solving problems that seem to be complicated in circuits containing “exotic” circuit elements such as memristors, memcapacitors, or meminductors. With the use of duality, the task can be transferred into the area of classical circuits, where the solution may be easier. In such a way, for example, the controversial questions of the orders of dynamical systems built from classical elements and their memory versions [911] can be solved. It follows from the equation of motion (24) that the circuit in Figure 2 is of the 2nd order, although it contains a capacitor, meminductor, and memristor. After the circuit transformation into its dual form in Figure 6, it is obvious that the circuit with identical dynamics contains only one resistor and two classical accumulating elements, the capacitor and inductor, and that the 2nd-order dynamics is concerned here.

The duality principle can also be useful in the opposite direction. It can substantiate atypical phenomena in classical circuits on the basis of analogies with circuits containing the higher-order elements, where such phenomena are common. A typical example is the generation of pinched hysteresis loops in the state space (dv/dt, di/dt) via simple nonlinear resistive circuits [21], which is a mere consequence of the duality of such circuits with memristive networks, where such a well-documented phenomenon can be observed in the (v,i) space.

Note that the methodology of drawing up equations of motion of systems built from HOEs, described in Section 4, can conversely be used for the synthesis of the so-called predictive models of systems or phenomena of most varied nature from HOEs as the building blocks [30]. For example, the nonlinear equation of motion of torsion pendulum from [33] is where θ is the torsion angle, J is the moment of inertia, C1 and C2 are the linear and nonlinear damping coefficients, and K1 and K3 are the linear and nonlinear stiffness coefficients of the torsion bar. Taking into account the electro-mechanical analogies of the through-across type, where the voltage corresponds to the velocity and the flux to the angle declination [34], then the equation of motion of the pendulum (59) corresponds to the equation (57) of the electric circuit in Figure 2 under the following conditions:

The circuit in Figure 6 and the given pendulum therefore provide the same dynamics if the driving sources do not act and if the nonlinear resistor and inductor have the constitutive relations (62) and (63). The circuit from Figure 2, containing the memristor and meminductor, also exhibits the same dynamics but in the state space of a variable of the TIF type.

7. Conclusions

Chua’s concept of the duality of R-C-L and RM-CM-LM circuits from [19] is extended here to general nonlinear systems built from arbitrary higher-order elements from Chua’s table or from a storeyed structure and independent sources. The dual circuits consist of different components, but they exhibit identical dynamics in different state spaces. The work starts from the general equations of motion of the system, consisting of the constitutive relations of system elements and a set of coupling conditions given by generalized KVL and KCL. The state space for describing the system dynamics is derived from the element distribution in the storeyed structure. This space enables a comparison of the dynamic properties of potentially dual systems. It turns out that the necessary condition of the transformation of the general equations of motion into a single compact equation of motion is the existence of explicit constitutive equations of all HOEs constituting the system. As another condition, all the HOEs must be controlled by the same type of circuit variable, namely, the generalized current or voltage. Then the model of the system dynamics can be reduced via a technique of equaling the orders of voltage or current nodes. The duality of circuits described in [19] is a special case of such reductions.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Acknowledgments

This work has been supported by the Czech Science Foundation under grant no. 18-21608S. For research, the infrastructure of K217 Department, UD Brno, was also used. The authors also wish to express their sincere thanks to the Open Access Fund of Brno University of Technology for covering the APC.

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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.

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