Parasitic Effects on Memristor Dynamics

Transcription

1 International Journal of Bifurcation and Chaos, Vol. 26, No. 6 (2016) (55 pages) c World Scientific Publishing Company DOI: /S Makoto Itoh , Arae, Jonan-ku, Fukuoka , Japan itoh-makoto@jcom.home.ne.jp Leon O. Chua Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, Berkeley, CA 94720, USA chua@berkeley.edu Received December 1, 2015 In this paper, we show that parasitic elements have a significant effect on the dynamics of memristor circuits. We first show that certain 2-terminal elements such as memristors, memcapacitors, and meminductors can be used as nonvolatile memories, if the principle of conservation of state variables hold by open-circuiting, or short-circuiting, their terminals. We also show that a passive memristor with a strictly-increasing constitutive relation will eventually lose its stored flux when we switch off the power if there is a parasitic capacitance across the memristor. Similarly, a memcapacitor (resp., meminductor) with a positive memcapacitance (resp., meminductance) will eventually lose their stored physical states when we switch off the power, if it is connected to a parasitic resistance. We then show that the discontinuous jump that circuit engineers assumed to occur at impasse points of memristor circuits contradicts the principles of conservation of charge and flux at the time of the discontinuous jump. A parasitic element can be used to break an impasse point, resulting in the emergence of a continuous oscillation in the circuit. We also define a distance, a diameter, and a dimension, for each circuit element in order to measure the complexity order of the parasitic elements. They can be used to find higher-order parasitic elements which can break impasse points. Furthermore, we derived a memristor-based Chua s circuit from a three-element circuit containing a memristor by connecting two parasitic memcapacitances to break the impasse points. We finally show that a higher-order parasitic element can be used for breaking the impasse points on two-dimensional and three-dimensional constrained spaces. Keywords: Memristor; meminductor; memcapacitor; parasitic element; nonvolatility; jump behavior; impasse point; Chua s circuit; van der Pol oscillator; diameter; distance; dimension; complexity order; passivity; lossless; constrained space. 1. Introduction In electrical circuits, parasitic elements (resistance, inductance or capacitance) are unavoidable. They are usually undesirable because they can have a significant effect on the dynamics of the associated circuit. In this paper, we study the dynamics of memristors when they have parasitic elements. Major content of this paper is not new, but its approach provides some new perspective and insights on memristor dynamics. An ideal memristor is endowed with a nonvolatility property. We first show that 2-terminal elements, such as the memristor and the memcapacitor, can be used as nonvolatile memories, if the

2 M. Itoh & L. O. Chua principle of the conservation of state variables hold upon open-circuiting or short-circuiting their terminals. We also show that a passive memristor with a strictly-increasing constitutive relation will eventually lose its stored flux resulting from the effect of a parasitic capacitance when we switch off the power. Similarly, a memcapacitor (resp., meminductor) with a positive memcapacitance (resp., meminductance) will eventually lose their stored physical states resulting from the effect of a parasitic resistance when we switch off the power. The principles of conservation of charge and flux implies that the quantity of charge and flux is always conserved. We show that the dynamics of an inductor memristor circuit cannot exhibit a discontinuous jump behavior, because the charge and the flux of a memristor cannot change instantaneously. Hence, a higher-order parasitic element is necessary to define the dynamics for all times of an inductor memristor circuit, that is, a parasitic element can break an impasse point, resulting in the emergence of a continuous oscillation in the circuit. We also show that in some inductor memristor circuits, a parasitic linear element can be used to break an impasse point, however, it cannot give rise to a continuous oscillation, that is, all trajectories will tend to the origin as time t increases. Therefore, in order to break the impasse points and obtain a continuous oscillation, we must use a nonlinear parasitic element. We next define a distance, adiameter, anda dimension of circuit elements in order to measure the complexity order of the required parasitic elements. They can be used to identify higher-order parasitic elements for breaking impasse points. We then derive a memristor-based Chua s circuit from a three-element memristor circuit by connecting two parasitic memcapacitances to break the impasse point. Furthermore, we show that a higherorder parasitic element can be used for breaking the impasse points on two-dimensional and threedimensional constrained spaces. 2. Memristors Memristor is a 2-terminal electronic device, which was postulated in [Chua, 1971; Chua & Kang, 1976; Chua, 2012]. The memristor can be described by a constitutive relation between the charge q and the flux ϕ, ϕ = f(q) or q = g(ϕ), (1) where f( ) andg( ) are differentiable scalar-valued functions. Its terminal voltage v and terminal current i are described by where v = M(q)i or i = W (ϕ)v, (2) v = dϕ and i = dq, () which represent Faraday s induction law and its dual law, respectively. The two nonlinear functions M(q) and W (ϕ), called the small-signal memristance and small-signal memductance, respectively, are defined by and M(q) df (q) dq (4) W (ϕ) dg(ϕ) dϕ, (5) representing the slope of the scalar function ϕ = ϕ(q), and q = q(ϕ), respectively, called the memristor constitutive relation. The passivity criterion of the memristor is given by [Chua, 1971] 1 A memristor characterized by a differentiable q ϕ (resp., ϕ q) characteristic curve is passive if, and only if, its small-signal memristance M(q) (resp., small-signal memductance W (ϕ)) is non-negative. The ideal memristor is endowed with a nonvolatility property [Chua, 2012]. When we switch off the power of a flux-controlled memristor at t = t 0, such that, v(t) =0fort>t 0,theni(t) =0for t>t 0. But since the flux ϕ(t) satisfies dϕ = v =0, (6) 1 See also Appendix A

3 for t>t 0, it follows that ϕ(t) =ϕ(t 0 )fort t 0. The charge q(t) also holds the value q(t 0 )=g(ϕ(t 0 )) for t t 0. The ideal memristor does not lose the values of the flux ϕ(t 0 ) and the charge q(t 0 )when we switch off the power of the memristor at t = t 0. We next define a memcapacitor, andameminductor, via their integrated state variables [Chua, 2012]. The memcapacitor can be described by the constitutive relation σ = f(ϕ), (7) between the flux ϕ and the integrated charge σ, where f( ) is a differentiable scalar-valued function, and σ is defined by σ q(τ)dτ. (8) Differentiating both sides of Eq. (7), we obtain the following equivalent form reminiscent of a linear capacitor q = C(ϕ) dϕ = C(ϕ)v, (9) df (ϕ) where C(ϕ). Similarly, the meminductor dϕ can be described by the constitutive relation ρ = g(q), (10) between the charge q and the integrated flux ρ, where g( ) is a differentiable scalar-valued function, and ρ is defined by ρ ϕ(τ)dτ. (11) Differentiating both sides of Eq. (10), we obtain the following equivalent form reminiscent of a linear inductor ϕ = L(q) dq where L(q) dg(q) dq. = L(q)i, (12) The lossless property for a memcapacitor, and a meminductor (see Appendix A for more details), for a sinusoidal source, is as follow: A memcapacitor characterized by a differentiable ϕ σ constitutive relation σ = f(ϕ) is lossless and passive, if, and only if, C(ϕ) 0 and ϕ dc(ϕ) 0. Similarly, a meminductor dϕ characterized by a differentiable q ρ constitutive relation ρ = g(q) islossless and passive, if, and only if, L(q) 0andq dl(q) dq 0. Let us recall the following fundamental circuit law on the charge q, and the flux ϕ, called the principles of conservation of charge and flux [Chua, 1969]; respectively: Charge and flux can neither be created nor destroyed. The quantity of charge and flux is always conserved. We can restate this principle as follows: The charge q and the flux ϕ of a memristor cannot change instantaneously.. Memory Devices Many nonvolatile molecular and nanodevices have been reported [Chua, 2012]. Recall that the hysteresis loops of all memristors must pass through the origin, that is, they are pinched at the origin [Chua, 2014]. Thus, we have the following question: Do the hysteresis loops of all nonvolatile devices which are not memristors pass through the origin? Do they hold their physical states when we switch off the power? Let us consider a 2-terminal circuit element defined by y = f(x), (1) where x and y are physical states, and f is a differentiable scalar function of x. For example, some well-known 2-terminal circuit elements [Chua, 2012]

4 M. Itoh & L. O. Chua are described by where σ resistor: capacitor: inductor: memristor: memcapacitor: meminductor: q(τ)dτ and ρ i = f(v), q = f(v), ϕ = f(i), ϕ = f(q), σ = f(ϕ), ρ = f(q), ϕ(τ)dτ. Define next the rate of change over time by r dx, s dy (14). Differentiating both sides of Eq. (1) with respect to the time variable t, we obtain from Eq.(1) s = M(x)r, (15) df (x) where M(x) =.Equation(15) exhibits the dx same relationship as memristors. Hence, a hysteresis loop passes through the origin on the (r, s)-plane [Itoh & Chua, 201]. Furthermore, these elements do not lose the value of x when r became zero. That is, they remember the physical state x(t 0 ) when r(t) =0fort>t 0. Thus, we get the following result: An isolated 2-terminal circuit element defined by Eq. (1) can hold the value of f(x(t 0 )) when r(t) = 0 for t > t 0.Hysteresisloops of Eq. (1) are pinched at the origin on the (r, s)-plane. A 2-terminal element defined by Eq. (1) issaio be isolated if it is not connected to external circuit elements (called parasitics) which causes its initial state x(t 0 ) to decay and converge to the zero equilibrium state..1. Holding the memory Some 2-terminal circuit elements can hold their physical state x(t 0 )whenr(t) =0fort>t 0.It is to be noted that the condition r(t) =0maynot be realized by merely switching off the power. What kinds of operations are needed for a 2- terminal circuit element to hold its physical state? We next show how to hold the physical states of some 2-terminal circuit elements. (1) Resistor Consider a strictly passive nonlinear resistor defined by i = f(v) wherei v = f(v) v>0forv 0.In this case, r(t) = 0 implies dv/ =0,thatis,the terminal voltage v(t) must retain the value v(t 0 ) for t>t 0. This is possible only by applying a voltage source across the resistor for t t 0. Therefore, a nonlinear resistor retains its physical state only while it is powered, but the stored data is immediately lost upon switching off the power, since v(t 0 )=0fort>t 0. Hence, it can be used only as a volatile memory. (2) Capacitor and Inductor Consider a nonlinear capacitor defined by q = f(v). In this case, r = 0 implies dv/ =0,thatis,the terminal voltage v(t) can retain the value v(t 0 )for t>t 0 by forcing i =0viaopen-circuiting the terminals at t = t 0, since the charge can neither be created nor destroyed (the principles of conservation of charge and flux). Thus, the capacitor can be used as a nonvolatile memory. Similarly, the nonlinear inductor defined by ϕ = f(i) can hold its value ϕ(t 0 )=f(i(t 0 )) by short-circuiting the terminals at t = t 0. Thus, the capacitor and the inductor can be used as nonvolatile memories. () Memristor Consider the flux-controlled memristor defined by q = f(ϕ). In this case, r = 0 implies dϕ/ = v(t) = 0, that is, the terminal voltage v(t) has to keep the value v(t 0 )=0fort>t 0 by shortcircuiting the terminals at t = t 0. It follows that the flux-controlled memristor can hold the value of q(t 0 )=f(ϕ(t 0 )) for t>t 0 by the principles of conservation of charge and flux. Similarly, a chargecontrolled memristor defined by ϕ = f(q) canhold the value of ϕ(t 0 )=f(q(t 0 )) for t>t 0 by opencircuiting the terminals at t = t 0. Thus, both memristors cited above can be used as nonvolatile memories

5 (4) Memcapacitor and Meminductor Consider the memcapacitor defined by σ = f(ϕ). In this case, r = 0 implies dϕ/ = v(t) = 0, that is, the terminal flux ϕ(t) mustkeepthevalueϕ(t 0 )for t>t 0 by short-circuiting the terminals at t = t 0.It follows that the memcapacitor can hold the value of σ(t 0 ) = f(ϕ(t 0 )) for t > t 0 by the principles of conservation of charge and flux. Similarly, the meminductor defined by ρ = f(q) can hold the value of ρ(t 0 ) = f(q(t 0 )) for t > t 0 by open-circuiting the terminals at t = t 0. Thus, the memcapacitor and the meminductor can be used as nonvolatile memories. Let us summarize the above observations in the following table on the memory devices: Nonvolatile How to Hold Memory Device Memory? a Memory resistor no apply dc power inductor yes short-circuit capacitor yes open-circuit flux-controlled yes short-circuit memristor charge-controlled yes open-circuit memristor memcapacitor yes short-circuit meminductor yes open-circuit Note that some circuit elements with a higher-order state variable, for example, an element defined by v (n) = f(q) (or i (n) = f(ϕ)), (16) where n is an integer, can also hold the physical state v (n) (t 0 )=f(q(t 0 )) (or i (n) (t 0 )=f(ϕ(t 0 ))), (17) by open-circuiting (or short-circuiting) the terminals at t = t 0, because q(t) = = 0 i(τ)dτ if i(t) =0fort t 0 and i(τ)dτ = q(t 0 ) for t t 0, (18) ϕ(t) = = 0 v(τ)dτ v(τ)dτ = ϕ(t 0 ) for t t 0, (19) if v(t) =0fort t 0. Let us summarize the above observations as follow: Isolated 2-terminal circuit elements can hold their physical states, if the conservation of state variables hold by open-circuiting or short-circuiting their terminals. Furthermore, these kinds of 2-terminal elements can be used as nonvolatile memories. We can generalize the above observations as follow 2 : All higher-order (α, β) elements[chua, 200, 2012] withα = 1 (resp.,β = 1) are nonvolatile for arbitrary β (resp., α). Observe that if the above mentioned nonvolatile elements are passive, they will eventually lose the value of their physical states if connected to external parasitic elements when we switch off the power, as showninsec Parasitic Elements In electrical circuits, parasitic elements (small imbedded resistance, inductance or capacitance) are unavoidable. Let us first introduce the notations [Chua, 200, 2012] d k q(t) k, if k =1, 2,..., q(t), if k =0 q (k) t q(τ)dτ, if k = 1 τ k τ2 q(τ 1 )dτ 1 dτ 2 dτ k, if k = 2,,... (20) 2 A 2-terminal element characterized by a constitutive relation (v (α), i (β) )-plane is called an (α, β) element,wherev (α) and i (β) are complementary variables derived from the voltage v(t) and the current i(t). See Appendix F for more details on (α, β) elements

6 M. Itoh & L. O. Chua and ϕ (m) d m ϕ(t), if m =1, 2,..., m ϕ(t), if m =0 ϕ(τ)dτ, τ m if m = 1 τ2 ϕ(τ 1 )dτ 1 dτ 2 dτ m, if m = 2,,... (21) where k and m are positive, negative, or zero integers. Consider a 2-terminal circuit element ξ defined by q (m) ξ = g(ϕ (m) ξ ), (22) which includes resistors (m = 1) and memristors (m = 0) as special cases. Here, q ξ and ϕ ξ denote the charge and the flux of element ξ, andg( ) denotes a scalar function. Note that element ξ does not have dynamics, but they can define or constrain the physical space where the dynamics occur. In this paper, we discuss the limited case of parasitic elements. That is, we assume that the parasitic element has the same terminal voltage, that is, it has the same state variable ϕ ξ, and we assume that the parasitic element is characterized by q (l) p = f(ϕ(k) p )=f(ϕ(k) ξ ), (2) where q p and ϕ p (= ϕ ξ ) denote the charge and the flux of the parasitic element, l and k denote integers, and f( ) denotes a scalar function. Compare Eq. (2)withEq.(22). A broader class of 2-terminal circuit elements with parasitic elements is defined in Appendix B Examples of parasitic elements Substituting m =0intoEq.(22), we obtain the 2-terminal circuit element q ξ = g(ϕ ξ ), (24) which defines a flux-controlled memristor. We next present some examples of parasitic elements for m =0. (1) l = 1 andk =0. Substituting l = 1 andk =0intoEq.(2), we obtain q ( 1) p = f(ϕ p ), (25) where, ϕ p = ϕ ξ. This equation defines a memcapacitor. If f(ϕ p ) Cϕ p,theneq.(25) can be recast into q ( 1) p = Cϕ p. (26) Differentiating both sides of this equation, we obtain the definition of a linear capacitor q p = Cv p, (27) where v p dϕ p denotes the terminal voltage. (2) l =1andk =0. Substituting l = 1 and k = 0 into Eq. (2), we obtain q (1) p = f(ϕ p ), (28) which defines a nonlinear inductor. If f(ϕ p ) ϕ p L, then Eq. (28) can be recast into i p = ϕ p L, (29) which defines a linear inductor. Here, i p dq p denotes the terminal current of this element Higher-order linear elements We now show how linear elements can be trivially transformed into higher-order parasitic elements. Consider a linear capacitor defined by q p = Cϕ (1) p, (0) where C denotes its capacitance. Differentiating both sides of Eq. (0), m 1times(m>1), with respect to time t, we obtain the relationship q (m 1) p = Cϕ (m) p, (1) which defines a higher-order parasitic memcapacitor with a capacitance C. Notethatifm 1, we can obtain Eq. (1), by integrating Eq. (0), 1 m times. Similarly, a linear resistor defined by q p = ϕ p R, (2)

7 can be transformed into a higher-order parasitic element q (m) p = ϕ(m) p R, () where R has the unit of a resistance. Observe that Eq. () does not involve dynamics. 4.. Relationship between linear elements We have shown that ideal linear elements can be trivially transformed into higher-order parasitic elements. However, a nonlinear capacitor cannot be trivially transformed into a higher-order nonlinear memcapacitor. The converse is also true. The above trivial transformation seems to be a special case for ideal linear elements. For example, consider a nonlinear capacitor defined by q = Cv + ɛv, (4) where C>0and0 <ɛ 1. Integrating the above equation with respect to time t, weobtain σ = Cϕ + ɛ v(τ) dτ, (5) which is different from the dynamics of the memcapacitor. Consider next that a nonlinear memcapacitor defined by σ = Cϕ + ɛϕ, (6) where C > 0 and 0 <ɛ 1. Differentiating the above equation with respect to time t, we obtain q =(C +ɛϕ 2 )v, (7) which is different from the dynamics of the nonlinear capacitor. As we can easily see from the discussion in Sec. 4.1, the characteristic of a linear capacitor and memcapacitor is described by the same relation, that is, σ = Cϕ, q = Cv, on the (σ, ϕ)-plane, on the (q, v)-plane. (8) It means that we cannot distinguish these two elements from the above characteristics on the two planes, that is, the (σ, ϕ)-plane and the (q, v)-plane. However, we can identify them, if we can find their nonlinear terms, since they do not give any significant effect on the dynamics, these terms are neglected. That is, we can identify them if the characteristic of the element can be written as σ = Cϕ + ɛf(ϕ), (9) or q = Cv + ɛf(v), (40) where 0 < ɛ 1, C > 0, and f( ) is a nonlinear function. Here, Eqs. (9) and(40) representthe characteristic of the nonlinear memcapacitor and the nonlinear capacitor, respectively. 5. Decay of Charge and Flux The ideal memristor is endowed with the nonvolatility property. Consider the flux-controlled memristor defined by q = g(ϕ), (41) where g is a scalar function of the flux ϕ. Its terminal voltage v and terminal current i are defined by i = W (ϕ)v. (42) When we switch off the power of the flux-controlled memristor at t = t 0, such that, v(t) =dϕ(t)/ =0 for t>t 0, then the flux ϕ(t) helhevalueϕ(t 0 ) for t t 0, and the charge q(t) also held the value q(t 0 )=g(ϕ(t 0 )) for t t 0 by short-circuiting the terminals. That is, an isolated flux-controlled memristor retains its value ϕ(t 0 )andq(t 0 ) for all t>t 0 when v(t) is set to zero for t t Flux decay via a parasitic capacitance Consider the flux-controlled memristor with a parasitic capacitance shown in Fig. 1.Whenweseti = 0, the flux ϕ of the memristor satisfies the relation C dϕ = g(ϕ), (4) where C denotes a parasitic capacitance, and the flux-controlled memristor is defined by a scalar function q = g(ϕ), (44) where q and ϕ denote the charge and the flux of the memristor, respectively. If g(ϕ) is a strictlyincreasing function, then the characteristic curve A function f(x) is said to be strictly increasing if f(b) >f(a) for all b>a

8 M. Itoh & L. O. Chua C 2t + C, for ϕ(0) > 0, ϕ(0) 2 ϕ(t) = 0, for ϕ(0) = 0, C 2t + C, for ϕ(0) < 0, ϕ(0) 2 (49) Fig. 1. Flux-controlled memristor with a parasitic capacitance (magenta). The constitutive relation of the memristor is described by q = g(ϕ). q = g(ϕ) lies in the first and third quadrants, where we assume that g(0) = 0 without loss of generality (see Appendix C). Furthermore, Eq. (4) hasthe trivial solution ϕ(t) 0. It follows from Eq. (4), via the dynamic route theory [Chua, 1969], that for any initial flux ϕ(t 0 ), and ϕ(t) 0, as t (45) q(t) 0, as t. (46) For example, if we choose g(ϕ) = ϕ, then Eq. (4) isgivenby C dϕ = ϕ, (47) which can be recast into C dϕ = 1. (48) ϕ The solution of Eq. (47) can be written as where ϕ(0) is the initial condition at time t =0. Thus, if C 0, then we obtain and ϕ(t) 0, as t (50) q(t) =ϕ(t) 0, as t (51) (see Appendix D for the case where C 0). Hence, we conclude as follows: A passive flux-controlled memristor, which is defined by a strictly-increasing function q = g(ϕ), will eventually lose its initial value ϕ(t 0 )viaaparasitic capacitance (connected in parallel) when we switch off the power of the memristor at t = t 0 by open-circuiting the memristor terminals. A passive charge-controlled memristor, which is defined by a strictly-increasing function ϕ = f(q), will eventually lose its initial value q(t 0 ) via a parasitic inductance (connected in series) when we switch off the power of the memristor at t = t 0 by short-circuiting the memristor terminals. As an example, consider the flux ϕ(t) and charge q(t) of the memristor in Fig. 2, calculated from q 0.0 t 0.0 t Fig. 2. Flux and charge decay of a passive memristor. The flux ϕ(t) and the charge q(t) eventually lose the value ϕ(0) = 1 and q(0) = ϕ(0) = 1 via a parasitic capacitance when we switch off the power at t =

9 Fig.. Charge and flux of a locally active memristor. Initial condition: ϕ(0) = 2. The charge q(t) eventually loses the value q(0) = ϕ(0) ϕ(0) = 6 via a parasitic capacitance when we switch off the power at t = 0. However, the flux ϕ(t) tends to 1 as t. Eq. (49), where C =0.01 and ϕ(0) = 1. Observe that the passive flux-controlled memristor eventually loses its initial value ϕ(t 0 )andq(t 0 )viathe parasitic capacitance when we switch off the power at t = t 0. In the case of an active flux-controlled memristor, the flux may tend to a nonzero equilibrium state even if we switch off the power of the memristor at t = 0. However, the charge q(t) tends to zero when we switch off the power. For example, consider the flux ϕ(t) and the charge q(t) infig., wherethe active memristor is defined by q = ϕ ϕ. Inthis case, the charge q(t) tends to zero from q(0) = 6. However, the flux ϕ(t) tends to 1 from ϕ(0) = Flux decay by two parasitic elements It is well-known that a physical capacitor reduces its stored charge via a finite parallel resistance (parasitic resistance) when we switch off the power. Consider next the case where the memristor has both a parasitic capacitance and a parasitic resistance asshowninfig.4. Then the dynamics can be described by C dϕ = g(ϕ) ϕ R. (52) If g(ϕ) is a strictly-increasing function, then g(ϕ)+ ϕ R is also a strictly-increasing function. Thus, we obtain ϕ(t) 0, as t. (5) Hence, the passive flux-controlled memristor will eventually lose its initial flux ϕ(t 0 )viathetwoparasitic elements when we switch off the power at t = t Flux decay of memcapacitors We will show next that a memcapacitor with a positive memcapacitance eventually lost its initial flux ϕ(t 0 ) via a parasitic resistance. Consider a memcapacitor defined by σ = f(ϕ), (54) Fig. 4. Flux-controlled memristor with two parasitic elements (magenta): a parasitic capacitance C and a parasitic resistance R. Fig. 5. Memcapacitor with a parasitic resistance R (magenta). The parasitic resistance R causes the stored flux of a memcapacitor with a positive memcapacitance to reduce to zero. The constitutive relation of the memcapacitor is described by σ = f(ϕ)

10 M. Itoh & L. O. Chua t q t Fig. 6. Decay of the flux ϕ(t) and the charge q(t) of a memcapacitor with a positive memcapacitance. The memcapacitor eventually loses its initial flux ϕ(0) = 1, and initial charge q(0) = via a parasitic resistance when we switch off the power at t =0. where σ denotes the integrated charge defined by σ q(τ)dτ, (55) and f is a differentiable scalar function of the flux ϕ. Consider next a memcapacitor with a parasitic resistance R,asshowninFig.5. FromEq.(54), we obtain where q = C(ϕ) dϕ C(ϕ) = C(ϕ)v, (56) df (ϕ) dϕ. (57) The flux ϕ of the memcapacitor satisfies the relation If C(ϕ) > 0, then we obtain C(ϕ) dϕ = ϕ R. (58) ϕ(t) 0, as t. (59) Note that if C(ϕ) > 0, then f(ϕ) is a strictlyincreasing function. Furthermore, we can assume that f(0) = 0 without loss of generality. Thus, we obtain q(t) 0, as t. (60) We will now illustrate the decay of flux ϕ(t)and charge q(t) in Fig.6. The following parameters and initial condition are used in this example: C(ϕ) =0.01(ϕ 2 +1), R = 1000, ϕ(0) = 1. (61) Observe that the initial flux and the initial charge are eventually lost when we switch off the power. Hence, we conclude as follows: A memcapacitor with a positive memcapacitance C(ϕ), will eventually lose its initial flux ϕ(t 0 ) via a parasitic resistance (connected in parallel) when we switch off the power at t = t 0 by open-circuiting the terminals. A meminductor with a positive meminductance L(q), will eventually lose its initial charge q(t 0 ) via a parasitic resistance (connected in series) when we switch off the power at t = t 0 by short-circuiting the terminals. 6. Discontinuous and Instantaneous Jump Behavior The principles of conservation of charge and flux indicate that the charge q and the flux ϕ cannot change instantaneously [Chua, 1969]. Thus, we have the following question: Does the dynamics of memristor circuits describe a discontinuous relaxation oscillation? A detailed analysis of discontinuous (relaxation) oscillation is given in [Chua, 1980; Andronov et al., 1987]. In this section, we show that there exist some differences between the approximate dynamics of an inductor memristor circuit and an inductor resistor circuit Jump behavior of an inductor memristor circuit Consider the two-element circuit in Fig. 7, whichis composed of a linear inductor and a flux-controlled memristor [Itoh & Chua, 2011, 2014]. The fluxcontrolled memristor is characterized by q = g(ϕ) ϕ ϕ. (62)

11 Fig. 7. Two-element circuit composed of an inductor and a flux-controlled memristor. The dynamics of the circuit is given by L dq L = ϕ L, q = g(ϕ) = ϕ ϕ, (6) where q and q L denote the charge of the memristor and the inductor, ϕ and ϕ L denote the flux of the memristor and the inductor. Since q L = q and ϕ L = ϕ, Eq.(6) can be recast into L dq = ϕ, q = g(ϕ) = ϕ ϕ. (64) We say that the system (64) is constrained by the relation q = ϕ ϕ, (65) (see [Arnold, 1978] for more details). Differentiating both sides of Eq. (65) with respect to time t, weobtain Substituting into Eq. (66), we can obtain dq =(ϕ2 1) dϕ. (66) dq = ϕ L, (67) dϕ = ϕ L(ϕ 2 1), (68) where ϕ ±1. It follows from Eq. (68) that dϕ lim ϕ 1 + =, dϕ lim ϕ 1 =, (69) where ϕ 1 + denotes the one-sided limit where ϕ approaches 1 from the right, and ϕ 1 denotes ϕ approaches 1 fromtheleft.adynamic route analysis [Chua, 1969] ofeq.(64) inthe L dq = g 1 (q) versus q-plane, where g 1 (q) denotes the inverse of g(ϕ) ϕ ϕ, isshownbysingle arrow heads in Fig. 8. Observe that starting from any point on the blue curve q = g(ϕ), the trajectory must tend to either point 1, orpoint 2 at some finite time, where no further solution of Eq. (64) is possible. The same impasse situation occurs for any initial point on the upper, or the lower, red curve. One way to overcome this dilemma is to assume an instantaneous jump showninfig.8. Thus,thetrajectory of Eq. (64) oscillates by jumping from the points (ϕ, q) =(1, 2 ) to the new point ( 2, 2 ), horizontally, and from the point (ϕ, q) =( 1, 2 )to the point (2, 2 ),asshowninfig.8(a). However, this discontinuous and instantaneous jump contradicts the principle of conservation of flux [Chua, 1969]. The above points are called impasse points [Chua, 1989a, 1989b], and so the circuit is nonphysical. Higher-order element can be used to break impasse points [Chua, 1980, 2012]. As stated in Appendix B, we can consider a parasitic-augmented 2-terminal circuit element defined by q = h(ϕ, ϕ (1) ), (70) where q and ϕ denote the charge and flux of this element and h( ) is a scalar function. An example of the element (70) isgivenby q = ϕ ϕ + ɛϕ(1), (71) which consists of a flux-controlled memristor, augmented with the derivative term: ɛϕ (1) (0 <ɛ 1). If we replace the memristor with the above element, then the dynamics of the circuit in Fig. 7 can be recast into L dq = ϕ, ɛ dϕ = q ( ) ϕ ϕ. (72)

12 M. Itoh & L. O. Chua (a) (ϕ, q)-plane (b) (q, ϕ)-plane Fig. 8. Jump phenomenon. (a) Discontinuous jump (two-headed arrow) on the (ϕ, q)-plane, which contradicts the principle of conservation of flux. The symbols 1 and 2 denote impasse points. (b) Hysteresis loop on the (q, ϕ)-plane. Equation (72) is equivalent to that of an inductor memristor circuit with a parasitic capacitance ɛ. The elements defined by Eq. (71) can break the impasse points, since Eq. (72) is identical to the van der Pol oscillator (more details are given in the next section). In order for Eq. (72) torepresent a good approximation of the dynamics of Eq. (64), let us assume ɛ is sufficiently small, but not zero. The curve defined by q ( ϕ ϕ) = 0 is called a slow-curve, because ϕ varies rapidly except in its close neighborhood. It is well-known that the inductor resistor circuit in Fig. 9 exhibits the same behavior as the inductor memristor circuit. However, the dynamics of this circuit can be approximated by a discontinuous jump behavior [Andronov et al., 1987; Storti & Rand, 1987], since the instantaneous jump phenomenon does not contradict the principle of conservation of flux (see Appendix E for more details). Hence, we conclude as follows: The dynamics of an inductor memristor circuit cannot be approximated by a discontinuous behavior on the (ϕ, q)-plane, since it contradicts the principles of conservation of flux. However, the dynamics of the inductor resistor circuit can be approximated by a discontinuous jump behavior on the (v, i)-plane. (a) Two-element circuit (b) (v,i)-plane (c) (i, v)-plane Fig. 9. Relaxation oscillation. (a) Two-element circuit is composed of an inductor and a nonlinear resistor (Chua s diode). (b) Discontinuous oscillation of this circuit on the (v, i)-plane (heavy red line). The graph of the function i = g(v) = v v isdrawninblue.(c)hysteresislooponthe(i, v)-plane

13 An associated pair of waveforms (ϕ(t),q(t)), namely, a trajectory, must be continuous on the (ϕ, q)-plane, even though the associated pair of waveforms (v(t),i(t)) = ( ϕ(t), q(t)) are discontinuous on the (v, i)-plane. Since the charge q(t) is a continuous function of the time variable t, the flux of the memristor ϕ(t) = f(q(t)) must also be a continuous function of t. Hence, we get the following result about the characteristic curve of memristors: The q versus ϕ Lissajous figure of a chargecontrolled memristor defined by ϕ = f(q) must be continuous on the (q, ϕ)-plane. Similarly, the ϕ versus q Lissajous figure of a flux-controlled memristor defined by q = g(ϕ) must be continuous on the (ϕ, q)-plane. It follows that memristors cannot exhibit a Lissajous figure which exhibits an abrupt jump in ϕ, or q Jump behavior of a resistor memcapacitor circuit In the previous section, we have shown that the dynamics of an inductor memristor circuit cannot be approximated by a discontinuous oscillation without violating the principle of conservation of flux. This dilemma leads to the following question: Are there any other circuits whose dynamics cannot be approximated by a discontinuous oscillation? Consider the two-element circuit in Fig. 10, which is composed of a second-order linear resistor and a memcapacitor. The characteristic of the memcapacitor is given by σ = g(ϕ) = ϕ ϕ, (7) where ϕ denotes the flux of the memcapacitor, and σ denotes the integrated charge of the Fig. 10. Two-element circuit is composed of a second-order resistor and a memcapacitor. The second-order resistor is defined by ρ R = Rσ R. The dynamics of this circuit is identical to the dynamics of the circuit in Fig. 7. memcapacitor defined by σ q(t). (74) The second-order linear resistor is defined by the relation ρ R = Rσ R, (75) where ρ R and σ R denote the integrated flux and the integrated charge of the higher-order resistor, which are respectively defined by ρ R σ R R dσ R ϕ R (t), q R (t). The dynamics of this circuit is given by Since = ϕ R, σ = g(ϕ). σ R = σ, ϕ R = ϕ, q R (t) = q(t), Eq. (77) can be recast into R dσ = ϕ, σ = ϕ ϕ. (76) (77) (78) (79)

14 M. Itoh & L. O. Chua Compare Eq. (79) witheq.(64). If L = R, then the dynamics of the circuits in Figs. 7 and 10 are identical. That is, they are isomorphic. 4 But we cannot model the dynamics of this circuit by invoking a jump phenomenon because it would violate the continuity property of ϕ(t) and σ(t). It follows that: The dynamics of the second-order resistor memcapacitor circuit in Fig. 10 cannot exhibit a discontinuous jump because it would contradict the continuity property of ϕ(t) and σ(t). 7. Parasitic Effects on Impasse Points The dynamics of the inductor memristor circuit in Fig. 7 has jump points. We have shown that they can be broken by replacing the memristor with a broader class of 2-terminal circuit elements. It is well-known that higher-order parasitic elements can be used to break impasse points [Chua, 1980, 2012]. This leads to the following question: What kinds of parasitic elements can break the impasse points of the inductor memristor circuit? 7.1. Diameter and distance of circuit elements Let us first introduce our notations on higher-order circuit elements [Chua, 200, 2012]. A 2-terminal element characterized by a constitutive relation in the (v (α), i (β) )-plane is called an (α, β) element, where v (α) and i (β) are complementary variables derived from the voltage v(t) and the current i(t). 5 Consider the inductor memristor circuit in Fig. 7, which consists of two elements: inductor: (α, β) =( 1, 0), memristor: (α, β) =( 1, 1). If we define the complexity metric of the circuit elements [Chua, 2012] by we obtain χ(α, β) α + β, (80) Element χ(α, β) inductor 1 memristor 2 Thus, the memristor has a higher complexity. In order to compare the complexity of two elements, we define the distance d between an (α 1,β 1 )element andan(α 2,β 2 ) element by 6 d α 1 α 2 + β 1 β 2. (81) Thus, the distance d between an inductor and a memristor is equal to d = 1 ( 1) + 0 ( 1) =1. (82) We next measure the complexity of a circuit by defining its diameter D as follow 7 : diameter D: The largest among all distances between all pairs of elements in the circuit. Since the circuit in Fig. 7 consists of two elements, the diameter D of this circuit is equal to 1. Consider next the circuit in Fig. 11, which has a memcapacitor as its parasitic element. The three elements in this circuit are identified as follows: inductor: (α, β) =( 1, 0), memristor: (α, β) =( 1, 1), memcapacitor: (α, β) =( 1, 2). Their complexity metric is given by Element χ(α, β) inductor 1 memristor 2 memcapacitor Thus, the memcapacitor has the highest complexity. The distance d between all pairs of circuit elements 4 We say two systems or phenomena are isomorphic if we can exhibit some kind of one-to-one correspondence between various quantities or attributes of the two systems. Two equations which differ only by symbols but are otherwise identical in form are said to be isomorphic equations [Lipschutz & Lipson, 2007]. 5 See Appendix F for more details on (α, β) elements. 6 The distance d can be used to compare the complexity of two elements. See Appendix F for more details. 7 The diameter D is defined as the largest among all distances between pairs of elements in the circuit. If there are the two elements with the largest distance, one of them is chosen to be the highest-order (derivative or integral) element. See Appendix F for more details

15 (a) (b) Fig. 11. Illustrations of adding a parasitic memcapacitor (magenta) to break impasse points. (a) Inductor memristor circuit with a parasitic memcapacitance C(ϕ). (b) (α, β)-element representation of the circuit. is calculated as follows: Pair of Elements Distance d (inductor, memristor) 1 (inductor, memcapacitor) 2 (memristor, memcapacitor) 1 From the above table, the diameter D of this circuit is 2. The diameter of the two circuits are summarized as follows: Circuit Diameter D two-element circuit in Fig. 7 1 three-element circuit in Fig Observe that the diameter of the circuit in Fig. 7 is increased from 1 to 2 by adding a parasitic memcapacitor. Furthermore, the memcapacitor is the highest-order element among the three elements. Therefore, we can expect that the memcapacitor can break the impasse points of the circuit in Fig Breaking impasse points by a parasitic element The dynamics of the circuit in Fig. 11 is given by L dq L = ϕ L, (8) C(ϕ) dϕ = q L g(ϕ), where q L, ϕ, andϕ L denote the charge of the inductor, the flux of the memristor, and the flux of the inductor, respectively. Assume the flux-controlled memristor is characterized by q = g(ϕ) = ϕ ϕ, (84) where q and ϕ denote the charge and flux of the memristor. Since ϕ L = ϕ, weobtainfromeq.(8) L dq L = ϕ, (85) C(ϕ) dϕ = q L ϕ + ϕ. Here, the terminal charge q and the terminal flux ϕ of the parasitic memcapacitance satisfies C(ϕ) dϕ = q, (86) which can be obtained from the characteristic curve defined by where σ C(ϕ) df (ϕ) dϕ σ = f(ϕ), (87) q(t),, (0 <C(ϕ) 1). (88) If we set Q L = q L, that is, the current direction of the inductor is reversed, we can obtain from Eq. (85) L dq L = ϕ, (89) C(ϕ) dϕ = Q L ϕ + ϕ

16 M. Itoh & L. O. Chua L dq L = ϕ L, C dϕ C = q L ϕ + ϕ, (91) Fig. 12. Inductor memristor circuit with a parasitic capacitance (magenta). This circuit exhibits a continuous oscillation on the (ϕ, q)-plane, that is, the parasitic capacitance can break the impasse points. The slow-curve of Eq. (89) is defined by Q L ϕ + ϕ =0. (90) Observe that the singular points of the slow-curve (90), that is, (ϕ, Q L ) = (1, 2 ), ( 1, 2 )arenot equilibrium points of Eq. (85). Figure 1(a) shows that Eq. (85) withl =1andC(ϕ) =0.05(1 + ϕ 2 ) exhibits a continuous oscillation on the (ϕ, Q L )- plane. Thus, the memcapacitor can be used to break the impasse points of the circuit. As stated in Sec. 4, a linear capacitor works as a higher-order element on the (q, ϕ)-plane, that is, it works as a memcapacitor. Thus, we can consider the circuit in Fig. 12. The dynamics of this circuit is given by where q L denotes the charge of the inductor, ϕ L, ϕ C, and ϕ denote the flux of the inductor, the capacitor, and the memristor, respectively, and C denotes the parasitic capacitance (0 < C 1). Note that if the parasitic capacitor is not ideal, for example, its characteristic curve has a nonlinear term, then the dynamics cannot be given by Eq. (91). If we set Q L = q L,Eq.(91) can be recast into the van der Pol oscillator L dq L = ϕ, C dϕ = Q L ϕ + ϕ, (92) where ϕ L = ϕ C = ϕ. Equation(92) withl =1and C = 0.05 exhibits a continuous oscillation on the (ϕ, Q L )-plane as shown in Fig. 1(b) [Itoh & Chua, 201]. Observe that the limit cycle in Fig. 1(a) is less stiff than that in Fig. 1(b). We will study the nonlinear effect of parasitic elements in Sec It is well-known that the limit cycle of Eq. (92) tends toward the red curve drawn in Fig. 8(a) as C 0, if we replace the symbol q with Q L in Fig. 8(a) [Rozov, 2011; Andronov et al., 1987]. The (a) Fig. 1. Continuous oscillation on the (ϕ, Q L )-plane. Observe that the limit cycle in Fig. 1(a) is less stiff (roundish) than that in Fig. 1(b). The trajectory in the two-headed arrow portion moves much faster than that in the one-headed arrow portion. In this figure, it does not mean an instantaneous jump. (a) Oscillation from Eq. (85) withl =1andC(ϕ) =0.05(1+ϕ 2 ). Initial condition: (ϕ(0),q L (0)) = (0.1, 0.1). (b) Oscillation from Eq. (92) withl =1andC(ϕ) =C =0.05. Initial condition: (ϕ(0),q L (0)) = (0.1, 0.1) (b)

17 above result leads to the following question: Which parasitic element causes an oscillation in physical inductor memristor circuits? A parasitic memcapacitor? A parasitic capacitor? 7.. Internal and external parasitic elements It is necessary to take into account linear and nonlinear, internal and external parasitic elements when describing the sufficiently complicated behavior of a memristor circuit. We next study the previous question from this viewpoint. Consider a parasitic-augmented 2-terminal element with higher-order derivative terms defined by dϕ N q = f(ϕ)+c 0 + c k (ϕ) dk ϕ k, (9) k=1 where q and ϕ respectively denote the charge and flux of the element, c 0 is a constant, c k (ϕ) is a scalar function of ϕ, andn is a positive integer (see Appendix B). Equation (9) defines a parasiticaugmented memristor, where and f(ϕ) (94) (c 0 + c 1 (ϕ)) dϕ, (95) correspond to a term of a memristor and that of a parasitic memcapacitor, respectively. Thus, the parasitic nonlinear memcapacitor, or its linear dϕ part c 0, can be used to break impasse points of the inductor memristor circuit in Fig. 7. Theycan be considered as internal parasitic elements of the memristor. We can also consider an ideal linear capacitor (i.e. equivalently a linear memcapacitor), or a nonlinear memcapacitor as an external parasitic element connected across the circuit components. They can be used to break impasse points, too. Furthermore, a parasitic memcapacitor is the higher-order element, and the diameter of the circuit is increased from 1 to 2 by adding a parasitic memcapacitor. However, applying a parasitic nonlinear capacitor does not seem to be appropriate, since the dynamics of the circuit becomes more complicated than Eq. (85), thatis, thedynamicsisgiven by L dq L = ϕ, ( ) dϕ F = q L ϕ + ϕ. (96) Here, the characteristic of ( the nonlinear ) capacitor dϕ is given by q = F (v) =F,whereF(v) isa one-to-one nonlinear function of v. Consequently, the parasitic memcapacitor with the most significant effect (whether it is an internal element or whether it is an external element), substantially breaks the impasse points of the circuit in Fig. 7. In this case, a linear capacitor can be regarded as a linear memcapacitor. Similarly, consider a parasitic-augmented 2- terminal element with higher-order derivative terms defined by dv N i = f(v)+c 0 + c k (v) dk v k, (97) k=1 where i and v respectively denote the current and voltage of the element, c 0 is a constant, c k (v) isa scalar function of v, and N is a positive integer. Equation (97) indicates the generalized expression for the characteristic of a nonlinear resistor, where and f(v), (98) (c 0 + c 1 (v)) dv, (99) correspond to a term of a nonlinear resistorand that of a parasitic nonlinear capacitor, respectively. Thus, the parasitic nonlinear capacitor, dv or its linear part, c 0, can be used to break impasse points of the inductor resistor circuit in Fig. 9 (see also Appendix E). They are considered to be internal parasitic elements of the nonlinear resistor. As an external parasitic element, we can also consider linear and nonlinear capacitors. Thus, the parasitic capacitor with the most significant effect (whether it is an internal element or whether it is an external element), breaks the impasse points of the inductor resistor circuit in Fig. 9. Whenwe add a parasitic capacitor to this circuit, it becomes the higher-order element, and the diameter of this circuit is increased from 1 to 2. However, applying a

18 M. Itoh & L. O. Chua parasitic nonlinear memcapacitor does not seem to be appropriate, since the dynamics of this circuit becomes more complicated, that is, the diameter of the circuit is increased from 1 to, and its dynamics is given by d 2 F (ϕ) ( ) dϕ ( ) dϕ + ϕ =0, (100) L where, L denotes the inductance, ϕ denotes the flux of the memcapacitor, and the characteristic of the nonlinear memcapacitor is given by σ = F (ϕ), where F (ϕ) is a nonlinear function of ϕ. Consequently, the parasitic capacitor with the most significant effect (whether it is an internal element or whether it is an external element), substantially breaks the impasse points of the circuit in Fig. 9. In this case, a linear memcapacitor can be regarded as a linear capacitor. Then the result is stated as follows: A parasitic memcapacitor can break the impasse points of the inductor memristor circuit in Fig. 7. Similarly, a parasitic capacitor can break the impasse points of the inductor resistor circuit in Fig. 9. The above breaking of impasse points is made possible by an internal, or an external, parasitic element, which cannot be ignored, and gives the most significant effect. As stated in Sec. 5, a passive flux-controlled memristor will eventually lose its initial value ϕ(t 0 ) via a parasitic element when we switch off the power of the memristor at t = t 0 by open-circuiting the memristor terminals. The above flux decay is caused by an internal or external parasitic element, which cannot be ignored Nonlinear effect of parasitic elements We have shown that a linear memcapacitor can break the impasse points of the inductor memristor circuit. In this section, we study the nonlinear effect of parasitic memcapacitors. We show that in some inductor memristor circuits, in order to break impasse points to obtain a continuous oscillation, we must use a nonlinear memcapacitor. Consider the circuit in Fig. 11. The dynamics of the circuit in Fig. 11 is given by L dq L = ϕ L, C(ϕ) dϕ = q L g(ϕ, µ), (101) where q L, ϕ, and ϕ L denote the charge of the inductor, the flux of the memristor, and the flux of the inductor, respectively. Assume that the fluxcontrolled memristor is characterized by ( ) ϕ q = g(ϕ, µ) = ϕ (ϕ 2 µ), (102) where µ is a parameter, and q and ϕ denote the charge and flux of the memristor. The characteristic curve q = g(ϕ, µ) forµ =1.4 andµ =1.6 is shown in Fig. 14. Observe that the function g(ϕ, µ) has four extrema. If we set Q L = q L, that is, the current direction of the inductor is reversed, we can obtain from Eq. (101) L dq L = ϕ, (10) C(ϕ) dϕ = Q L g(ϕ, µ), equivalently L dq L = ϕ, C(ϕ) dϕ = Q L ( ϕ ) ϕ (ϕ 2 µ), (104) where ϕ L = ϕ. The slow-curve of Eq. (104) is defined by ( ϕ ) Q L = g(ϕ, µ) = ϕ (ϕ 2 µ). (105) Consider next the point in the neighborhood of the slow-curve: (ϕ, Q L )=(ϕ 0,g(ϕ 0,µ)+ Q L ), (106) where 0 < Q L 1. The vector field at this point is given by dϕ = Q L C(ϕ 0 ), (107) dq L = ϕ 0 L

19 (a) (b) Fig. 14. Constitutive relation of a flux-controlled memristor defined by q = g(ϕ, µ) =( ϕ ϕ) (ϕ2 µ). (a) Constitutive relation for µ =1.4 (thick red curve), which has four extrema P i (i =1, 2,, 4) at ϕ = e i (i =1, 2,, 4), where e , e , e 0.608, and e (b) Constitutive relation for µ =1.6 (thick blue curve), which has four extrema P j (j = A, B, C, D) atϕ = e j (j = A, B, C, D), where e A 1.54, e B 0.689, e C 0.689, and e D Q L Thus, affects the horizontal speed of the trajectory in the neighborhood of the slow-curve on the C(ϕ 0 ) (ϕ, Q L )-plane. That is, the nonlinear memcapacitance C(ϕ) gives a nonuniform horizontal velocity. If the memcapacitance C(ϕ 0 ) is increased, then the horizontal speed dϕ is decreased, which gives rise to a strong effect on the jump-wise behavior. We next show the necessity for choosing a nonlinear memcapacitor to break impasse points to obtain a continuous oscillation. Case 1. µ =1.4 Let us consider Eq. (105) withµ =1.4. As shown in Fig. 14(a), the function g(ϕ, 1.4) has four extrema P i at ϕ = e i (i =1, 2,, 4), where e ,e ,e 0.608,e Since the sign dq L of does not change in the right-half plane and in the left-half plane, Eq. (104) hastwofor- ward impasse points P 1 =(e 1,g(e 1, 1.4)) and P 4 = (e 4,g(e 4, 1.4)), and two backward impasse points P 2 =(e 2,g(e 2, 1.4)) and P =(e,g(e, 1.4)). 8 We show the trajectories of Eq. (104)inFigs.15 and 16. Observethatifa linear parasitic memcapacitance (that is, a linear parasitic capacitance) is sufficiently small, or if the coefficient of the nonlinear term of the nonlinear memcapacitor is sufficiently small, then all trajectories will tend to the origin as time t increases, though it may break the impasse points as shown in Figs. 15(a) and 16(a). However, if a linear parasitic memcapacitance (linear capacitor) becomes large, or if the coefficient of the nonlinear term of the nonlinear parasitic memcapacitor is increased from 0.04 to 0.06, the circuit can exhibit a periodic oscillation as shown in Figs. 15(b) and 16(b). Therefore, in order to break the impasse points of Eq. (104) with µ =1.4, by the emergence of a continuous oscillation, we must use a nonlinear parasitic memcapacitor, or a large linear memcapacitor (linear capacitor). Case 2. µ =1.6 Let us consider Eq. (105) withµ =1.6. As shown in Fig. 14(b), the function g(ϕ, 1.6) has four extrema P j at ϕ = e j (j = A, B, C, D), where e A 1.54,e B 0.689,e C 0.689,e D Compare the two characteristic curves in Fig. 14. Equation (104) with µ = 1.6 has two forward impasse points P A = (e A,g(e A, 1.6)) and P D = (e D,g(e D, 1.6)), and two backward impasse points P B =(e B,g(e B, 1.6)) and P C =(e C,g(e C, 1.6)). We next show the trajectories of Eq. (104) in Figs. 17 and 18. Observe that the circuit with a 8 When a forward trajectory arrives at some point (x 0,y 0 )onthe(x, y)-plane at some time t 0 but cannot be continued for t>t 0,then(x 0,y 0 )isaforward impasse point. Similarly, when a backward trajectory retracts to a point (x 0,y 0 )att 0 but cannot be continued backward for t<t 0,then(x 0,y 0 )isabackward impasse point. See [Chua, 1989a, 1989b] for more details about the forward and backward impasse points

20 M. Itoh & L. O. Chua (a) µ = 1.4, L =1,C = 0.05 (b) µ = 1.4, L =1,C = 0.67 Fig. 15. Behavior of the solutions of Eq. (104) forparameterc =0.05, 0.67, where µ =1.4, L = 1. The constitutive relation of a memristor is illustrated in Fig. 14(a). Trajectories of Eq. (104) are drawn in blue, brown, green, and purple. The slow-curve of Eq. (104) is drawn in red. The trajectory in the two-headed arrow portion moves much faster than that in the one-headed arrow portion. In this figure, it does not mean an instantaneous jump. (a) All trajectories of Eq. (104) will tend to the origin as t,whenc = Initial conditions: (ϕ(0),q L (0)) = (1.1, 1.1), ( 1.1, 1.5). The jump-wise behavior (two-headed arrow) is observed. A linear parasitic memcapacitor (linear parasitic capacitor) can be used to break an impasse point, however, it does not give rise to a continuous oscillation. (b) Equation (104) has two limit cycles, one (green) is stable and the other (purple) is unstable, if C is increased to a large value, compared to L =1,thatis,C =0.67. Therefore, in order to break the impasse points and obtain a periodic oscillation, we must use a large linear memcapacitor (linear capacitor). But in this case, it would not be appropriate to call C a parasitic element. (a) µ = 1.4, L =1,C(ϕ) =0.05( ϕ 2 ) (b) µ = 1.4, L =1,C(ϕ) =0.05( ϕ 2 ) Fig. 16. Nonlinear effect of the memcapacitance C(ϕ) defined by a quadratic polynomial. Trajectories of Eq. (104) aredrawn in blue, brown, green, and purple. The slow-curve of Eq. (104) is drawn in red. The trajectory in the two-headed arrow portion moves much faster than that in the one-headed arrow portion. In this figure, it does not mean an instantaneous jump. (a) All trajectories of Eq. (104) will eventually tend to the origin as t,ifµ =1.4, L =1,andC(ϕ) =0.05( ϕ 2 ). The linear part of the memcapacitance C(ϕ) is the same as that of Fig. 15(a). Initial conditions: (ϕ(0),q L (0)) = (1.1, 1.1), ( 1.1, 0.4). (b) Equation (104) has two limit cycles, one (green) is stable and the other (purple) is unstable, if µ =1.4, L =1,and C(ϕ) =0.05( ϕ 2 ). A nonlinear parasitic memcapacitance can be used to break an impasse point, resulting in the emergence of a continuous oscillation in the circuit. Note that the linear part of the memcapacitance C(ϕ) is0.05 1, which is the same as that of Fig. 15(a), and the coefficient of the nonlinear term ϕ 2 is = In this case, it is indeed appropriate to call the memcapacitor a parasitic element

21 (a) µ = 1.6, L =1,C = 0.05 (b) µ = 1.6, L =1,C = 0.67 Fig. 17. Behavior of the solutions of Eq. (104) forparameterc =0.05, 0.67, where µ =1.6, L = 1. The constitutive relation of the memristor is shown in Fig. 14(b). Trajectories of Eq. (104) are drawn in blue and brown. The slow-curve of Eq. (104) is drawn in black. The trajectory in the two-headed arrow portion moves much faster than that in the one-headed arrow portion. In this figure, it does not mean an instantaneous jump. (a) All trajectories of Eq. (104) will tend to the origin as t, when C = Initial conditions: (ϕ(0),q L (0)) = (1.1, 1.1), ( 1.1, 1.5). The jump-wise behavior (two-headed arrow) is observed. Although the linear parasitic memcapacitor (linear parasitic capacitor) can be used to break an impasse point, it does not give rise to a continuous oscillation. (b) All trajectories of Eq. (104) will also tend to the origin as t,evenifc is increased to a large value. In this case, C =0.67, and the initial conditions: (ϕ(0),q L (0)) = (1.1, 1.1), ( 0.5, 1.2). A linear parasitic memcapacitor can be used to break an impasse point, however, it does not give rise to a continuous oscillation. Note that two limit cycles in Fig. 15(b) merge and disappear if the parameter µ is increased from 1.4 to1.6. (a) µ = 1.6, L =1,C(ϕ) =0.05( ϕ 4 ) (b) µ = 1.6, L =1,C(ϕ) =0.05( ϕ 4 ) Fig. 18. Nonlinear effect of the memcapacitance C(ϕ) defined by a quartic polynomial C(ϕ) =0.05( ϕ 4 ). This example shows the necessity of a nonlinear memcapacitor. The constitutive relation of a memristor is illustrated in Fig. 14(b). Trajectories of Eq. (104) are drawn in blue, brown, green, and purple. The slow-curve of Eq. (104) is drawn in black. The trajectory in the two-headed arrow portion moves much faster than that in the one-headed arrow portion. In this figure, it does not mean an instantaneous jump. (a) All trajectories of Eq. (104) will eventually tend to the origin as t,forµ =1.6, L =1andC(ϕ) =0.05( ϕ 4 ). The linear part of the memcapacitance C(ϕ) is the same as that of Fig. 15(a). Initial conditions: (ϕ(0),q L (0)) = (1.1, 1.1), ( 1.1, 0.4). (b) Equation (104) has two limit cycles, one (green) is stable and the other (purple) is unstable, if µ =1.6, L =1,andC(ϕ) =0.05( ϕ 4 ). A nonlinear parasitic memcapacitance can be used to break an impasse point, and gives rise to a continuous oscillation in the circuit. This example shows that in order to break the impasse points of Eq. (104) with µ = 1.6 and obtain a continuous oscillation, we must use a nonlinear parasitic memcapacitor. A linear memcapacitor (or a linear capacitor) cannot break the impasse points as shown in Fig. 17. Note that the linear part of the memcapacitance C(ϕ) is0.05 1, which is the same as that of Fig. 15(a), and the coefficient of the nonlinear term ϕ 4 is =

22 M. Itoh & L. O. Chua linear parasitic memcapacitance (that is, a linear parasitic capacitance) cannot oscillate, that is, all trajectories will tend to the origin as the time t increases as shown in Fig However, if the coefficient of the nonlinear term ϕ 4 is increased from 0.04 to 0.06, the circuit with a nonlinear parasitic memcapacitance can exhibit a periodic oscillation as shown in Fig. 18. In our computer study, even if C is large, Eq. (104) does not oscillate. It is due to the reason why the two limit cycles in Fig. 15(b) merge and disappear if the parameter µ is increased from 1.4 to Therefore, in order to break the impasse points of Eq. (104) withµ =1.6 toobtainacontinuous oscillation, we must use a nonlinear parasitic memcapacitor. That is, this example shows the necessity of a nonlinear memcapacitor. Furthermore, we note that different initial conditions cause different behaviors as shown in Fig. 19. It implies that in order to break impasse points to obtain a continuous oscillation, we must choose the initial condition carefully. The nonlinear effect of C(ϕ) foreq.(89) is also shown in Fig. 20. Observe that the periodic trajectory in Fig. 20(b) is not symmetrical about the Q L -axis. That is, the nonlinear memcapacitance causes an asymmetry. Hence, a linear capacitor is not the best choice to break impasse points, since we would not be able to observe a variety of nonlinear effects Breaking impasse points of an isomorphic circuit Recall that the two circuits in Figs. 7 and 10 are isomorphic. We next show that in order to break the impasse point of the circuit of Fig. 10, a(0, 2) parasitic element can be used. Consider the resistor memcapacitor circuit in Fig. 21, which has a parasitic (0, 2) element defined by Kv A = σ A, (108) where K is a constant, and v A and σ A denote the flux and the integrated charge of this parasitic element, respectively. Here, we assume that the second-order resistor is defined by the nonlinear relation ρ R = R h(σ R ), (109) where ρ R and σ R denote the integrated flux, and the integrated charge, respectively, and h( ) isadiffer- entiable strictly-monotonically increasing function with a positive slope. For example, if we choose h(σ R )=σ R + σ R, then the slope is positive, that is, H(σ R ) dh(σ R) = σ 2 R +1> 0. dσ R (a) (b) Fig. 19. Dependence on initial conditions. Trajectories of Eq. (104) withl =1andC(ϕ) =0.05( ϕ 2 )aredrawnin blue. The slow-curve of Eq. (104) is drawn in red. The trajectory in the two-headed arrow portion moves much faster than that in the one-headed arrow portion. In this figure, it does not mean an instantaneous jump. (a) Equation (104) exhibits a periodic trajectory, if we choose the initial condition: (ϕ(0),q L (0)) = (1.1, 1.1). (b) The trajectory will tend to the origin as t, if we choose another initial condition, such as (ϕ(0),q L (0)) = ( 1.1, 0.6). 9 We chose the parameter µ and the function g(ϕ) (that is, the position of extrema) such that Eq. (104) does not oscillate when a linear parasitic capacitance C satisfies 0 C More rigorous computer study of the bifurcation parameter is needed, since the periodic orbit disappears at the bifurcation point

23 (a) (b) Fig. 20. Nonlinear effect of the memcapacitance C(ϕ) foreq.(89). Trajectories of Eq. (89) are drawn in blue. The slow-curve of Eq. (89) is drawn in red. The trajectory in the two-headed arrow portion moves much faster than that in the one-headed arrow portion. In this figure, it does not mean an instantaneous jump. (a) Oscillation of Eq. (89) withl =1andC(ϕ) =C =0.02. Initial condition: (ϕ(0),q L (0)) = (1.1, 1.1). If C(ϕ) is sufficiently small, the closed orbit is stiff. (b) Oscillation of Eq. (89) with L =1andC(ϕ) =0.02(1 + 5(e ϕ 1)κ(ϕ)), where κ(ϕ) represents the unit step function, κ(ϕ) =0forϕ<0and κ(ϕ) =1forϕ 0. Observe that the periodic trajectory in Fig. 20(b) is not symmetrical about the Q L -axis. The nonlinear memcapacitance causes such asymmetry. Initial condition: (ϕ(0),q L (0)) = (1.1, 1.1). The three elements in this circuit are characterized by the following properties: second-order resistor: (α, β) =( 2, 2), memcapacitor: (α, β) =( 1, 2), parasitic element: (α, β) =(0, 2). Their complexity metric χ(α, β) is as follows: Element χ(α, β) second-order resistor 4 memcapacitor parasitic element 2 The distance between two elements is given by Pair of Elements Distance d (second-order resistor, memcapacitor) 1 (second-order resistor, parasitic 2 element) (memcapacitor, parasitic element) 1 Observe that the diameter of the circuit is increased from1to2byaddinga(0, 2) parasitic element. The dynamics of the circuit in Fig. 21 is defined by R H(σ R ) dσ R = ϕ R, (110) K dϕ A = σ R g(ϕ), (a) (b) Fig. 21. (a) Resistor memcapacitor circuit with a (0, 2) parasitic element (aqua). The second-order resistor is denoted by a( 2, 2) element. (b) (α, β)-element representation of the circuit

24 M. Itoh & L. O. Chua (a) (b) Fig. 22. Nonlinear effect of the second-order resistor for Eq. (112). Trajectories of Eq. (112) aredrawninblue. Theslow-curve of Eq. (112) is drawn in red. The trajectory in the two-headed arrow portion moves much faster than that in the one-headed arrow portion. In this figure, it does not mean an instantaneous jump. (a) Oscillation of Eq. (112) withk =0.05, R =1and H( S R ) = 1. Initial condition: (ϕ(0),s R (0)) = (1.1, 1.1). If K is a sufficiently small constant, the closed orbit is stiff. (b) Oscillation of Eq. (112) withk =0.05, R =1andH( S R )=0.2+(e SR 1)κ( S R ), where κ(ϕ) represents the unit step function. In this case, the periodic trajectory is not symmetrical about the ϕ-axis. The nonlinear function H( S R )causes such asymmetry. Initial condition: (ϕ(0),s R (0)) = (1.1, 1.1). where the flux-controlled memcapacitor is characterized by σ = g(ϕ) = ϕ ϕ. (111) If we set σ R = S R,Eq.(110) can be recast into R ds R ϕ = H( S R ), (112) K dϕ = S R ϕ + ϕ, where ϕ A = ϕ R = ϕ. SinceH( S R ) > 0, Eq. (112) exhibits a continuous oscillation as shown in Fig. 22. Let us choose next R = L, K = C, and h(σ R )=σ R,thatis,H(σ R )=1.ThenEqs.(92) and (112) are identical. It implies that the secondorder (0, 2) parasitic element can be used to break the impasse points. The nonlinear function H( S R ) gives a nonuniform vertical velocity on the (ϕ, S R )- plane. Therefore, Eq. (112) can exhibit an asymmetrical closed orbit, as shown in Fig. 22(b). 8. Memristor-Based Chua s Circuit We usually use one parasitic element to break the impasse points. Thus, we have the following question: What kinds of oscillations occur if two parasitic elements are used to break the impasse points? 8.1. Three-element circuit Let us consider the three-element circuit in Fig. 2. The dynamics of this circuit can be described by L dq L = ϕ L = R q R + ϕ (11) = R q + ϕ = R g(ϕ)+ϕ, q = g(ϕ), where q L, q R, q denote the charge of the inductor, the charge of the resistor, and the charge of the memristor, respectively, and ϕ L and ϕ denote the flux of the inductor and the flux of the memristor, respectively. Note that q R = q = g(ϕ). Assume the flux-controlled memristor is characterized by q = g(ϕ) = ϕ 16 7ϕ 6, (114) (see [Hirsch et al., 200]). If R is sufficiently small, then the resistance R is considered to be a parasitic resistance of the inductor. Since q L = q, weobtainfromeq.(11) L dq ( ϕ 2 = (R g(ϕ)+ϕ) = 16 1 ) ϕ, 6 q = g(ϕ) = ϕ 16 7ϕ 6, (115) wherewehavechosenr =1,andq and ϕ denote the charge and flux of the memristor, respectively

25 (a) (b) Fig. 2. A basic three-element circuit. (a) The circuit consists of an inductor, a linear resistor, and a flux-controlled memristor. (b) (α, β)-element representation of the circuit. The system (115) is constrained by the relation q = g(ϕ). The two functions g(ϕ) and g(ϕ) + ϕ are illustrated in Fig. 24. Observe that (1) g(ϕ) = ϕ 16 7ϕ 6 has the extrema at ϕ = ± 2 14 ± (116) (2) g(ϕ) ϕ satisfies the following inequalities: g(ϕ) ϕ<0, if ϕ> 2 6 g(ϕ) ϕ>0, if ϕ< , 1.6. (117) Fig. 24. Two nonlinear functions. Red curve: g(ϕ) = ϕ 16 7ϕ ϕ 6.Bluecurve: g(ϕ) ϕ = 16 + ϕ 6. Observe that (1) g(ϕ) = ϕ 16 7ϕ 6 has the extrema (marked in fuchsia) at ϕ = ± 2 14 ± (2) g(ϕ) ϕ satisfies the following inequalities: g(ϕ) ϕ < 0, if ϕ > , g(ϕ) ϕ >0, if ϕ< The points ϕ = ± 2 6 on the ϕ-axis are marked in cyan. Since the constrained curve q = g(ϕ) has two extrema, and since the sign of dq does not change in the neighborhood of these extrema, Eq. (115) has the following two impasse points on q = g(ϕ): ( 2 ) ( ) 14 (ϕ, q) =, 14 14, 2 14, Breaking impasse points In order to break the impasse points of Eq. (11), we added a parasitic memcapacitance to the circuit as shown in Fig. 25. Let us first calculate the diameter and the distance of the circuit elements. The four circuit elements in Fig. 25 are coded by resistor: (α, β) =(0, 0), inductor: (α, β) =( 1, 0), memristor: (α, β) =( 1, 1), memcapacitor: (α, β) =( 1, 2). Their complexity metric is given by Element χ(α, β) resistor 0 inductor 1 memristor 2 memcapacitor The distance between different pairs of elements is given by Pair of Elements Distance d (resistor, inductor) 1 (resistor, memristor) 2 (resistor, memcapacitor) (Continued)

26 M. Itoh & L. O. Chua (a) (b) Fig. 25. Necessity for adding a parasitic memcapacitance (magenta) to break impasse points. (a) Memristor circuit with a parasitic memcapacitance C 1 (ϕ 1 ). (b) (α, β)-element representation of the circuit. Pair of Elements Distance d (inductor, memristor) 1 (inductor, memcapacitor) 2 (memristor, memcapacitor) 1 Thus, the diameter D of the circuit is equal to, and D is increased from 2 to by adding the parasitic memcapacitor, which is the highest-order element. The dynamics of the circuit in Fig. 25 is given by L dq L = Rq R + ϕ 1, C 1 (ϕ 1 ) dϕ 1 = q L g(ϕ) ( ϕ = q L 16 7ϕ 6 ), (118) where q L, q R, ϕ 1,andC(ϕ 1 ) denote the charge of the inductor, the charge of the resistor, the flux of the memcapacitor, and the memcapacitance of the memcapacitor, respectively. Let us assume the fluxcontrolled memristor is characterized by ( ϕ q = g(ϕ) = 16 7ϕ ). (119) 6 Since ϕ = ϕ 1, q R = q L, and assuming R =1,we obtain L dq L = q L + ϕ 1, C 1 (ϕ 1 ) dϕ ( 1 ϕ = q L ϕ ) (120) 1. 6 The equilibrium points of Eq. (120) aregivenby ( 2 6 (ϕ 1,q L )=(0, 0),, 2 ) 6, ( 2 ) 6 6, 2. (121) The slow-curve of Eq. (120) isgivenby ( ϕ q L = ϕ ) 1, (122) 6 which has two singular points at ( 2 ) 14 (ϕ 1,q L )=, 14 14, 27 ( ) 2 14, (12) 27 These points are not equilibrium points of Eq. (120). We show the trajectories of Eq. (120) infig.26. The following parameters are used in our computer study: (a) L =1,C 1 (ϕ 1 )=0.02( ϕ 2 1 ), (b) L =1,C 1 (ϕ 1 )=0.01(1 + (e ϕ 1 1)κ(ϕ 1 )), (c) L =1,C 1 (ϕ 1 )=0.02, where κ(ϕ 1 ) represents the unit step function, equal to0forϕ 1 < 0and1forϕ 1 0. Observe that Eq. (120) exhibits a continuous oscillation on the (ϕ 1,q L )-plane. If we choose C 1 (ϕ 1 )=0.01[1+(e ϕ 1 1)κ(ϕ 1 )], Eq. (120) exhibits an asymmetric closed orbit. Note that we cannot break the impasse points if the parasitic memcapacitor is connected across an

27 (a) (b) (c) Fig. 26. Continuous oscillation on the (ϕ 1,q L )-plane. Trajectories of Eq. (120) aredrawninblue.theslow-curveofeq.(120) is drawn in red. The trajectory in the two-headed arrow portion moves much faster than that in the one-headed arrow portion. In this figure, it does not mean an instantaneous jump. (a) Trajectory of Eq. (120) converges to a roundish limit cycle, where C 1 (ϕ 1 )=0.02(1+ϕ 2 1 )andl = 1. Initial condition: (ϕ 1(0),q L (0)) = (0.1, 0.1). (b) Trajectory of Eq. (120) converges to an asymmetric limit cycle, where C 1 (ϕ 1 )=0.01(1+(e ϕ1 1)κ(ϕ 1 )) and L = 1. Initial condition: (ϕ 1 (0),q L (0)) = (0.1, 0.1). (c) Trajectory of Eq. (120) converges to a stiff limit cycle, where C 1 (ϕ 1 )=0.02 and L = 1. Initial condition: (ϕ 1 (0),q L (0)) = (0.1, 0.1). inappropriate element, such as the inductor shown in Fig Parasitic effects of two memcapacitances Consider the circuit with two different memcapacitors in Fig. 28. Note that the diameter D of this circuit is, which is not increased, even if a new memcapacitor is connected to the circuit in Fig. 25. That is, D is not increased even if two different parasitic elements of the same kind are connected. However, the dimension of the space where the dynamics of the circuit takes place is increased. In order to show this, we define the dimension of physical space. Define first the subdimension λ j for each element by λ j α j β j. (124) The dimension λ of the circuit is defined by N N λ λ j = α j β j, (125) j=1 j=1 where N denotes the number of circuit elements. 11 Note that if the element satisfies α β, then the (a) Fig. 27. Example of a circuit whose impasse points cannot be broken by connecting a parasitic element at an inappropriate location. (a) Memristor circuit with a parasitic memcapacitance C 2 (ϕ 2 ) (magenta). (b) (α, β)-element representation of the circuit. (b) 11 See Appendix F for more details, where the dimension λ of Chua s circuit is calculated

28 M. Itoh & L. O. Chua (a) (b) Fig. 28. Memristor-based Chua s circuit. (a) A four-element circuit with a parasitic memcapacitance (magenta). (b) (α, β)- element representation of the circuit. element imposes a constraint on the rate of change of the state variables with respect to time. Thus, λ defines the dimension of the space where the dynamics of the circuit takes place. Roughly speaking, we can describe the diameter D and the dimension λ as follows: Diameter D is used to find the highestorder circuit element. Dimension λ is used to find the dimension of the space where the dynamics of the circuit takes place. The subdimension λ j of the circuit in Fig. 28 is given by Element (α j,β j ) λ j linear resistor (0, 0) 0 memristor ( 1, 1) 0 inductor ( 1, 0) 1 memcapacitor C 1 (0, 1) 1 memcapacitor C 2 (0, 1) 1 Thus, the dimension λ of this circuit is given by λ = =, (126) which implies that the circuit has a possibility to exhibit a chaotic oscillation. Compare the dimension λ of the following three circuits: Circuit Dimension λ three-element circuit in Fig. 2 1 four-element circuit in Fig five-element circuit in Fig. 28 by The dynamics of the circuit in Fig. 28 is given C 1 (ϕ 1 ) dϕ 1 C 2 (ϕ 2 ) dϕ 2 = ϕ 2 ϕ 1 g(ϕ 1 ), = ϕ 1 ϕ 2 + q L, L dq L = ϕ 2, (127) where R =1,andϕ i and C(ϕ i ) denote the flux and the memcapacitance of the memcapacitors, respectively, where i = 1, 2. The flux-controlled memristor is characterized by ( ϕ q = g(ϕ) = 16 7ϕ ). (128) 6 Note that the reference voltage polarity and the current direction of the inductor are different from those in Fig. 25. If the two memcapacitors are linear, then Eq. (127) is equivalent to the dynamics of the Chua s circuit with a cubic polynomial in Fig. 29 [Hirsch et al., 200; Madan, 199]. We show the trajectories of Eq. (127) infig.0. Observe that Eq. (127) exhibits a periodic and a chaotic oscillation. The following parameters are used in our computer study: (a) L =1, C 1 (ϕ 1 )=0.05( ϕ 2 1 ), C 2 =0.05, (b) L = 1 15, C 1(ϕ 1 )=0.1( ϕ 2 1), C 2 =1, (c) L = 1 14, C 1(ϕ 1 )=0.1, C 2 =

29 Fig. 29. Chua s circuit. It contains five circuit elements: two passive capacitors, one passive inductor, one passive resistor, and one nonlinear resistor called Chua s diode. (a) (b) (c) Fig. 0. Periodic and chaotic oscillations of Eq. (127) with the same initial condition: (ϕ 1 (0),ϕ 2 (0),q L (0)) = (0.1, 0.1, 0.1). The trajectory in the two-headed arrow portion moves much faster than that in the one-headed arrow portion. In this figure, it does not mean an instantaneous jump. (a) Limit cycle of Eq. (127) withl =1,C 1 (ϕ 1 )=0.05( ϕ 2 1), C 2 =0.05. (b) Chaotic trajectory of Eq. (127) withl = 1 15, C 1(ϕ 1 )=0.1( ϕ 2 1), C 2 = 1. (c) Chaotic trajectory of Eq. (127) with L = 1 14, C 1(ϕ 1 )=0.1, C 2 =1. Hence, we obtain the following result: The three-element circuit in Fig. 2 exhibits a periodic or chaotic oscillation when two memcapacitors are added to break the impasse points. points lying on a one-dimensional constrained curve. This leads to the following question: Is it possible to break impasse points on a two- or three-dimensional constrained space by adding higher-order parasitic elements? Note that we could not find a chaotic attractor when C 2 1andL =1. 9. Impasse Points on a Two-Dimensional Constrained Space In the previous section, we have shown that a parasitic element can be used to break the impasse In this section, we break the impasse points on a two-dimensional constrained space by adding a parasitic element Impasse points in higher-order circuits We first show the formal definition of impasse points in higher-dimensional systems [Chua, 1989a,

30 M. Itoh & L. O. Chua 1989b]. Consider a higher-dimensional system defined by dx = f(x, y), (129) h(x, y) =0, where x R n, y R m, f : R n+m R n,andh : R n+m R m. We assume that f(x, y) andh(x, y) are smooth functions, i.e. their partial derivative of allordersexist.wesaythesystem(129) isconstrained by the space S {(x, y) h(x, y) =0}. (10) Apoint(x 0, y 0 ) is a dynamic operating point of Eq. (129) ifitsatisfies h(x 0, y 0 )=0, (11) that is, (x 0, y 0 ) S. Given any initial condition (x 0, y 0 )ons at t = 0, the solution of Eq. (129) is simply a system trajectory lying on the constrained space S, parametrized by time t for <t<. When a trajectory of Eq. (129) arrives at some point (x 0, y 0 )ons at some time t 0 but cannot be continued for t>t 0, then a dynamic operating point (x 0, y 0 )ofeq.(129) is called an impasse point. If the Jacobian matrix h(x, y) J y1 y (12) (x 1,y 1 ) is nonsingular at a dynamic operating point (x 1, y 1 ) on S, then a solution of Eq. (129) exists in a neighborhood of (x 1, y 1 )byvirtueoftheimplicit function theorem and Peano existence theorem. Hence, (x 1, y 1 )cannotbeanimpassepoint.however,a dynamic operating point (x 2, y 2 )ons satisfying J y2 h(x, y) y = 0 (1) (x 2,y 2 ) can be an impasse point. The above singularity condition is only a necessary but not a sufficient condition for impasse points [Chua, 1989a, 1989b]. Therefore, impasse points are a proper subset of F defined by { ( ) } F (x, y) h(x, y) det =0, h(x, y) =0, y (14) equivalently { ( ) } F (x, y) S h(x, y) det =0. (15) y Fig. 1. Illustration of impasse points (yellow) and singular trajectories (red, magenta, brown, and dark brown) on a constrained surface. Almost all trajectories (green) arrive at a forward impasse point (yellow) on the fold line (black dotted line) at some time t 0,andcannot be continued for t>t 0.We can consider four kinds of singular trajectories. Two singular trajectories (red and magenta) move on the constrained surface passing through the point (marked in cyan) on the fold line. The two other singular trajectories (brown and dark brown) move on the constrained surface passing through the point (marked in cyan) on the fold line, but do not cross it. The two points marked in cyan are called pseudo equilibrium points. Backward impasse points are marked in white (if the time orientation is reversed, they become forward impasse points). The constrained surface is given by x 1 = y 1 y 1, whichisfoldedaty 1 = ±1,x 1 = 2/. Most trajectories of Eq. (129) cannot pass throughanimpassepointonf. However, there is a singular trajectory which arrives at some point in F at some finite time t 1, and can be continued for t>t 1. In fact, in a three-dimensional system, there are some singular trajectories which can move on a constrained surface passing through the point on F (see [Chua, 1989a, 1989b; Benoit, 198] and Appendix G). Some well-known singular trajectories are illustrated in Fig. 1. We will not discuss these singular trajectories further, since the purpose of this paper is to break impasse points. In higher-dimensional systems, impasse points need not be isolated. They often form an impasse set made of impasse curves, impasse surfaces, impasse manifolds, etc. [Chua, 1989a, 1989b] Basic circuit equation Let us consider the three-element circuit in Fig. 2. The dynamics of this circuit can be described

31 (a) (b) Fig. 2. Basic three-element circuit. (a) The circuit consists of an inductor, a capacitor, and a flux-controlled memristor. (b) (α, β)-element representation of the circuit. by L dq L = ϕ C ϕ, C dϕ C = q L, q = g(ϕ), (16) where q L, q, ϕ C,andϕ denote the charge of the inductor, the charge of the memristor, the flux of the capacitor, and the flux of the memristor, respectively. The flux-controlled memristor is characterized by q = g(ϕ) = ϕ ϕ. (17) Since q L = q, weobtainfromeq.(16) L dq L = ϕ C ϕ, C dϕ C (18) = q L, q L = g(ϕ), which has a two-dimensional constrained surface S defined by } S = {(q L,ϕ C,ϕ) R q L = g(ϕ) = ϕ ϕ. (19) The three-dimensional plot of the constrained surface S isshowninfig.. An initial point (q L (0),ϕ C (0),ϕ(0)) at t =0is a dynamic operating point of Eq. (18) if it satisfies (q L (0),ϕ C (0),ϕ(0)) S. Given any initial condition (q L (0),ϕ C (0),ϕ(0)) on S at t =0,the solution (q L (t),ϕ C (t),ϕ(t)) of Eq. (18) must lie on the constrained surface S. Equation (18) has impasse points on the set F defined by the singular points of S, namely, { F = (q L,ϕ C,ϕ) S Since dg(ϕ) dϕ =0 } { = (q L,ϕ C,ϕ) q L = g(ϕ), dg(ϕ) dg(ϕ) dϕ } dϕ =0. (140) = d ( ) ϕ dϕ ϕ = ϕ 2 1=0, (141) we obtain ϕ = ±1, and ( ϕ ϕ) q L = g(ϕ) = ϕ=±1 ϕ=±1 = 2. (142) Fig.. Three-dimensional plot of the constrained surface S, which is folded at ϕ = ±1 (thick black lines)

32 M. Itoh & L. O. Chua Hence, the set F is given by { F = (q L,ϕ C,ϕ) ϕ = ±1,q L = 2 }. (14) Furthermore, the set F is a fold, since d 2 g(ϕ) dϕ 2 =2ϕ = ±2. (144) ϕ=±1 ϕ=±1 That is, the constrained surface S is folded at F, which consists of two one-dimensional lines. As stated before, most trajectories starting from an operating point (q L (0),ϕ C (0),ϕ(0)) on the constrained surface S at t = 0 cannot pass through the fold F. However, a singular trajectory can exist, which moves on the constrained surface S passing through the fold F (see Appendix G and [Chua, 1989a, 1989b; Benoit, 198; Itoh, 1990]). Let us explain these behaviors briefly. The dynamics on the (q L,ϕ)-plane is described by dq L = ϕ C ϕ, L (145) q L = g(ϕ) = ϕ ϕ. Let us compare Eq. (145) witheq.(64). The derivative dq L points on F : in Eq. (145) becomes zero at the following (q L,ϕ C,ϕ)= ( 2 ) ( ) 2, 1, 1,, 1, 1. (146) These points are called pseudo equilibrium points (see Appendix G and [Benoit, 198; Chua, 1989a, 1989b]) because they are not equilibrium points of Eq. (18), since dϕ C = q L 0 on F. Thus, C we can consider a singular trajectory, which moves on the constrained surface S passing through these points and satisfying the relation dϕ C = q L C = 2 C or 2 C, (147) where q L = ± 2 on F. Note that the uniqueness of solutions does not hold at these points. Furthermore, most points on F do not satisfy Eq. (146), and thus they are impasse points. Recall the singular trajectories passing through the cyan point in Fig. 1. In the case of Eq. (64), the sign of dq = ϕ L does not change in the neighborhood of the singular points (q,ϕ) = ( 2 ) ( ) 2, 1,, 1. (148) It follows that they are also impasse points. 9.. Breaking impasse points via a memcapacitor In order to break the impasse points of Eq. (16), let us add a parasitic memcapacitor as shown in Fig. 4. Let us first calculate the diameter and the distance of the circuit elements. The four circuit elements in Fig. 4 are coded by capacitor: (α, β) =(0, 1), inductor: (α, β) =( 1, 0), memristor: (α, β) =( 1, 1), memcapacitor: (α, β) =( 1, 2). Their complexity metric is given by Element χ(α, β) capacitor 1 inductor 1 memristor 2 memcapacitor The distance between different pairs of elements are as follows: Pair of Elements Distance d (capacitor, inductor) 2 (capacitor, memristor) 1 (capacitor, memcapacitor) 2 (inductor, memristor) 1 (inductor, memcapacitor) 2 (memristor, memcapacitor) 1 Thus, the diameter D of the circuit is equal to 2, and D is not increased by adding the parasitic memcapacitor. That is, the memcapacitor is not the highest-order element, though it has the highest complexity metric. However, the dimension λ of the circuit is increased. That is, the subdimension λ j

33 (a) (b) Fig. 4. Necessity for adding a parasitic memcapacitance (magenta) to break one-dimensional impasse points. (a) Memristor circuit with a parasitic memcapacitance C 1 (ϕ 1 ). (b) (α, β)-element representation of the circuit. of the circuit in Fig. 4 is given by Element (α j,β j ) λ j capacitor (0, 1) 1 memristor ( 1, 1) 0 inductor ( 1, 0) 1 memcapacitor ( 1, 2) 1 Thus, the dimension λ of this circuit is given by λ = =, (149) which increased from 2 to upon adding the parasitic memcapacitor. In order to break the onedimensional impasse points, the dimension λ must be greater than 2. In physical circuits, there exists a parasitic resistance or conductance. Furthermore, the complexity metric χ(α, β) of an element(α, β) can be defined as its distance from the resistor [Chua, 2012]. Thus, it is not so unusual to add a parasitic conductance in order to identify the highest-order element without contradiction. If we add a parasitic conductance as shown in Fig. 5, then the diameter D of the circuit is increased from 2 to, since the distance between a resistor and a memcapacitor is (see the table below): Pair of Elements Distance d (resistor, capacitor) 1 (resistor, inductor) 1 (resistor, memristor) 2 (resistor, memcapacitor) Thus, the memcapacitor becomes the highest-order element by adding a parasitic conductance. Note that the dimension λ of this circuit is not increased, even if we add a parasitic conductance. The dynamics of the circuit in Fig. 5 is given by the third-order differential equation L dq L = ϕ C ϕ 1, C dϕ C = q L Gϕ C, (150) C 1 (ϕ 1 ) dϕ 1 = q L g(ϕ 1 ), where q L, ϕ C, ϕ 1,andC 1 (ϕ 1 ) denote the charge of the inductor, the flux of the capacitor C, the flux of the memcapacitor, and the memcapacitance, respectively, and the memristor is characterized by g(ϕ 1 )=ϕ 1 ϕ 1. (151) The slow-surface (slow-manifold) is given by S 1 = {(q L,ϕ C,ϕ 1 ) R q L = g(ϕ 1 )}, (152) where ϕ 1 varies rapidly except in a small neighborhood of S 1. We show the trajectories of Eq. (150) with L =1,C =1/0.45, in Figs. 6 and 7. The following parameters are used in our numerical simulations: (a) G =0, C 1 (ϕ 1 )=0.05( ϕ 4 1 ), (b) G =0, C 1 (ϕ 1 )=0.05, (c) G =0.05, C 1 (ϕ 1 )=0.05. Observe that Eq. (150) exhibits a relaxation oscillation on the (q L,ϕ C,ϕ 1 )-space, even if G =0and C 1 (ϕ 1 )=C 1 =0.05 as shown in Figs. 6 and 7. Furthermore, if we set C 1 (ϕ 1 ) = 0.5(1 + (e ϕ 1 1)κ(ϕ 1 )), Eq. (150) exhibits an asymmetric closed orbit

34 M. Itoh & L. O. Chua (a) (b) Fig. 5. Necessity for adding a parasitic conductance (magenta). (a) Memristor circuit with a parasitic conductance G and a parasitic memcapacitance C 1 (ϕ 1 ). (b) (α, β)-element representation of the circuit. 2 (a) (b) (c) Fig. 6. Relaxation oscillation of Eq. (150), where L = 1,C = 1/0.45 and the initial condition is given by (q L (0),ϕ C (0),ϕ 1 (0)) = (0.1, 0.1, 0.1). (a) G =0,C 1 (ϕ 1 )=0.05( ϕ 4 1). (b) G =0,C 1 (ϕ 1 )=0.05. (c) G =0.05,C 1 (ϕ 1 )= The trajectory in the two-headed arrow portion moves much faster than that in the one-headed arrow portion. In this figure, it does not mean an instantaneous jump. (a) (b) (c) Fig. 7. Oscillation of Eq. (150) onthe(ϕ 1,q L )-plane, where L =1,C =1/0.45. The trajectory in the two-headed arrow portion moves much faster than that in the one-headed arrow portion. In this figure, it does not mean an instantaneous jump. The initial condition is given by (q L (0),ϕ C (0),ϕ 1 (0)) = (0.1, 0.1, 0.1). The slow-surface (blue) is given by q L = g(ϕ 1 )= ϕ 1 ϕ 1. (a) G =0,C 1 (ϕ 1 )=0.05( ϕ 4 1 ). (b) G =0,C 1(ϕ 1 )=0.05. (c) G =0.05,C 1 (ϕ 1 )=

35 Hence, Eq. (150) is well defined and has no impasse points. They are broken by adding the parasitic memcapacitance C 1 (ϕ 1 ). If G is negative and C 1 (ϕ 1 ) is a constant, then the circuit becomes the memristor-based canonical Chua s circuit shown in Fig. 9, and it exhibits a chaotic oscillation [Itoh & Chua, 201]. Fig. 8. The green trajectories of Eq. (150) will tend to two stable equilibrium points as time t increases. A stable limit cycle (red) also coexists with these two stable equilibrium points. Parameters: L =1,C =1/0.45,G =0.05,C 1 (ϕ 1 )= 0.05( ϕ 4 1 ). Initial conditions: (q L(0),ϕ C (0),ϕ 1 (0)) = (2, 2, 1), ( 2, 2, 1), (0.1, 0.1, 0.1) correspond to the right green trajectory, the left green trajectory, and the red limit cycle, respectively. The trajectory in the two-headed arrow portion moves much faster than that in the one-headed arrow portion. In this figure, it does not mean an instantaneous jump. Equation (150) with a sufficiently small G has two stable equilibrium points in the neighborhood of the points (q L,ϕ C,ϕ 1 )=(0,, ), (0,, ), as shown in Fig. 8. The origin is an unstable equilibrium point. These equilibrium points coexist with a stable limit cycle. Thus, no equilibrium points lie on the set of singular points of the slow-surface S 1, which is defined by { } dg(ϕ 1 ) F 1 = (q L,ϕ C,ϕ 1 ) S 1 =0 dϕ 1 where { = (q L,ϕ C,ϕ 1 ) q L = g(ϕ 1 ), dg(ϕ 1) { = (q L,ϕ C,ϕ 1 ) ϕ 1 = ±1,q L = 2 } =0 dϕ 1 }, (15) dg(ϕ 1 ) dϕ 1 = d dϕ 1 (ϕ 1 ϕ 1 )=ϕ 2 1 1=0, (154) that is, ϕ 1 = ±1. Thus, we obtain q L = g(±1) = 2. (155) The set F 1 is a fold, since d 2 g(ϕ 1 ) ϕ1 dϕ 2 =2ϕ 1 = ±2. (156) 1 ϕ1 =±1 =± Coupled memristor circuit Let us consider the circuit in Fig. 40, wherethe flux-controlled memristor in Fig. 2(a) is replaced with a coupled memristor. The characteristic of the coupled memristor is defined by q = g(ϕ, ϕ C )= ϕ + ϕ ϕ C, (157) where q, ϕ, andϕ C denote the charge of the memristor, the flux of the memristor, and the flux of the capacitor C, respectively. Since this element is formally characterized as a ( 1, 1) element, its metrics are the same as those of the circuit in Fig. 2. The dynamics of this circuit is given by L dq L = ϕ C ϕ, C dϕ C = q L, q L = g(ϕ, ϕ C ), which has a constrained surface S C defined by } S C = {(q L,ϕ C,ϕ) R q L = ϕ + ϕ ϕ C. (158) (159) Fig. 9. Memristor-based canonical Chua s circuit. It contains five circuit elements: two passive capacitors, one passive inductor, one active resistor, and one charge-controlled memristor

36 M. Itoh & L. O. Chua (a) (b) Fig. 40. Necessity for adding parasitic elements (magenta) to break one-dimensional impasse points. (a) Three-element circuit which consists of a capacitor, an inductor, and a coupled memristor. (b) A parasitic memcapacitance C 1 (ϕ 1 ) and a parasitic conductance G are added to break the impasse points of the three-element circuit in Fig. 40(a). We show the three-dimensional plot of the constrained surface S C in Fig. 41(a). Equation (158) has impasse points located on the set of the singular points of the constrained surface S C defined by { } g(ϕ, ϕ C ) F C = (q L,ϕ C,ϕ) S C =0 ϕ where = {(q L,ϕ C,ϕ) S C ϕ 2 + ϕ C =0}, (160) g(ϕ, ϕ C ) ϕ = ( ) ϕ ϕ + ϕ ϕ C = ϕ 2 + ϕ C =0. (161) If the discriminant D c defined by of the cubic polynomial ϕ + ϕ ϕ C q L =0, (162) is positive, that is, D c = 4ϕ4 C +9q2 L > 0, (16) then Eq. (162) has three distinct real roots. 12 Furthermore, the curve defined by 4ϕ C +9q 2 L =0, (164) forms a cusp curve on the (q L,ϕ C )-plane as shown in Fig. 41(b). It corresponds to the projection of the (a) (b) Fig. 41. Constrained surface S and its fold curve F C. (a) Three-dimensional plot of the constrained surface S. ThefoldF C is indicated by the red curve. (b) Projection of the fold F C onto the (q L,ϕ C )-plane, which forms the cusp curve. 12 The discriminant D c of the cubic polynomial ax + bx 2 + cx + d, isgivenbyd c = b 2 c 2 4ac 4b d 27a 2 d 2 +18abcd. In the case of Eq. (162), a = 1,b=0,c= ϕc,d = q L

37 set F C onto the (q L,ϕ C )-plane. Note that the origin, which is situated on the set F C,isnot an impasse point, but it is an equilibrium point of Eq. (158). We next consider the singular trajectories of Eq. (158). The dynamics on the (q L,ϕ)-plane is described by dq L = ϕ C ϕ, L (165) q L = ϕ + ϕ ϕ C. Let us compare Eq. (165) with Eq. (145). Since ϕ C = ϕ 2 on F C, the derivative dq L dq L = ϕ C ϕ L is given by = ϕ2 ϕ. (166) L Thus, it becomes zero at the following two points on F C : ( ) 2 (q L,ϕ C,ϕ)=(0, 0, 0),, 1, 1. (167) The origin is an equilibrium point of Eq. (158), since dϕ C = dq L = 0. The other point is a pseudo equilibrium point since dϕ C = 2 0.Thus,wecan consider two singular trajectories passing through this point (see the illustration in Fig. 42). Note that the uniqueness of solutions does not hold at this point. We next break the impasse points of Eq. (158) by adding a parasitic memcapacitance and a parasitic conductance as shown in Fig. 40(b). The dynamics of this circuit can be described by L dq L = ϕ C ϕ 1, C dϕ C C 1 (ϕ 1 ) dϕ 1 = q L Gϕ C, = q L ϕ 1 ϕ 1 ϕ C, (168) where q L, ϕ C, and ϕ 1 denote the charge of the inductor, the flux of the capacitor C, and the flux of the memcapacitor, respectively, and G, andc 1 (ϕ 1 ) denote the conductance of the resistor and the memcapacitance of the memristor, respectively. The slow-surface (slow-manifold) is given by Fig. 42. Illustration of impasse points (yellow) and singular trajectories (brown and dark brown) on the constrained surface. Almost all trajectories (green) arrive at a forward impasse point (yellow) on the fold curve (red) at some time t 0,andcannot be continued for t>t 0. However, two singular trajectories move on the constrained surface passing through the point (marked in cyan) on the fold curve. We can also consider singular trajectories which move on the constrained surface crossing the fold curve as shown in Fig. 1. Thepoint marked in cyan is a pseudo equilibrium point. The origin is an equilibrium point. Backward impasse points are marked in white. The constrained surface is given by q L = ϕ + ϕ ϕ C, whichisfoldedatϕ 2 + ϕ C =0. S 1 = { } (q L,ϕ C,ϕ 1 ) R q L = ϕ 1 + ϕ 1 ϕ C, (169) where ϕ 1 varies rapidly except in a small neighborhood of S 1. If we choose the parameters L =1, C =1/0.45, G =0.05, C 1 (ϕ 1 )=0.05( ϕ 2 1 ), then Eq. (168) has three equilibrium points: O =(0, 0, 0), P (0.1475, 2.949, 2.949), Q ( , , ), (170) on the (q L,ϕ C,ϕ 1 )-space. The equilibrium points O and P are stable, but the equilibrium point Q is unstable. All eigenvalues of the origin O have a negative real part if 0 <G 1. If G = 0, the origin is a degenerate equilibrium point, and all eigenvalues have a zero real part. This is another reason why we add the parasitic conductance G( 0). Note that

38 M. Itoh & L. O. Chua (a) Fig. 4. Two different views of the trajectories of Eq. (168). Two trajectories exhibit jump-wise motions (two-headed arrow), and they will tend to the origin as time t increases. The trajectory in the two-headed arrow portion moves much faster than that in the one-headed arrow portion. In this figure, it does not mean an instantaneous jump. Parameters: L =1,C =1/0.45,G=0.05,C 1 (ϕ 1 )=0.05(1+0.05ϕ 2 1 ). Initial condition for the blue orbit: (q L(0),ϕ C (0),ϕ 1 (0)) = (10, 40, 20). Initial condition for the red orbit: (q L (0),ϕ C (0),ϕ 1 (0)) = ( 10, 50, 20). the equilibrium points P and Q do not lie on the set of the singular points of the slow-surface S 1,which is defined by where F 1 = { (q L,ϕ C,ϕ 1 ) S } g(ϕ 1,ϕ C ) 1 =0 ϕ 1 = {(q L,ϕ C,ϕ 1 ) S 1 ϕ ϕ C =0}, (171) g(ϕ 1,ϕ C ) = ( ϕ ) 1 ϕ 1 ϕ 1 + ϕ 1 ϕ C = ϕ ϕ C =0. (172) (b) 9.5. Breaking impasse points via a higher-order element In this section, we show another method for breaking the impasse points of Eq. (16). That is, in order to break the impasse points, we add a parasitic (0, 2) element to the circuit as shown in Fig. 45. The (0, 2) element is defined by i 2 = ɛv, (17) Hence, Eq. (168) is well defined and has no impasse points. They are broken by adding the parasitic memcapacitance C 1 (ϕ 1 ). We show the trajectories of Eq. (168) infigs.4 and 44. Observe that the trajectories exhibit jumpwise motions at the set F 1, and they will tend to the origin O as time t increases as shown in Fig. 4. For the other initial condition, the trajectory will tend to the stable equilibrium point P as time t increases asshowninfig.44. Similar results are obtained if we choose C 1 (ϕ 1 )=C 1 =0.05. In this example, a parasitic memcapacitance can be used to break an impasse point, however, it does not give rise to a continuous oscillation, even if the memcapacitor is a nonlinear element. Fig. 44. Trajectory of Eq. (168) will tend to a different stable equilibrium point P. Parameters: L = 1,C = 1/0.45, G =0.05,C 1 (ϕ 1 )=0.05(1+0.05ϕ 2 1). Initial condition for the purple orbit: (q L (0),ϕ C (0),ϕ 1 (0)) = (4,, ). Initial condition for the green orbit: (q L (0),ϕ C (0),ϕ 1 (0)) = (5, 4, 5). The trajectory in the two-headed arrow portion moves much faster than that in the one-headed arrow portion. In this figure, it does not mean an instantaneous jump

39 (a) (b) Fig. 45. Necessity for adding a parasitic element (magenta) to break one-dimensional impasse points. (a) Memristor circuit with a parasitic (0, 2) element. (b) (α, β)-element representation of the circuit. or equivalently σ = ɛv, (174) where v, i, andσ denote respectively the voltage, the current, and the integrated charge of this element, and ɛ is a parameter. Differentiating both sides of Eq. (174) with respect to time t, weobtain q = ɛ d2 ϕ 2, (175) where q and ϕ denote the charge and the flux of this element, respectively. Let us first calculate the diameter and the distance of the circuit elements. The four circuit elements in Fig. 45 are coded by capacitor: (α, β) =(0, 1), inductor: (α, β) =( 1, 0), memristor: (α, β) =( 1, 1), parasitic: (α, β) =(0, 2). Their complexity metric is given by Element χ(α, β) capacitor 1 inductor 1 memristor 2 parasitic 2 The distance between each pair of elements is given by Pair of Elements Distance d (capacitor, inductor) 2 (capacitor, memristor) 1 (capacitor, parasitic) 1 (Continued) Pair of Elements Distance d (inductor, memristor) 1 (inductor, parasitic) (memristor, parasitic) 2 Thus, thediameter D of the circuit is increased from 2 to by adding the parasitic (0, 2) element, which means that the parasitic element is considered to be the highest-order element. The subdimension λ j of the circuit in Fig. 45 is given by Element (α j,β j ) λ j capacitor (0, 1) 1 memristor ( 1, 1) 0 inductor ( 1, 0) 1 parasitic (0, 2) 2 Thus, the dimension λ of this circuit is given by λ = =4. (176) Since the dimension λ and the diameter D are increased, we can expect that the impasse points of the fold are broken by the parasitic element. The dynamics of the circuit in Fig. 45 is given by the fourth-order differential equation L dq L = ϕ C ϕ, C dϕ C ɛ d2 ϕ 2 where 0 <ɛ 1and = q L, = q L g(ϕ), q = g(ϕ) = ϕ (177) ϕ. (178)

40 M. Itoh & L. O. Chua Equation (177) can be written as L dq L = ϕ C ϕ, C dϕ C = q L, dϕ = ζ, ɛ dζ = q L g(ϕ). (179) The slow-constrained space (three-dimensional slow-manifold) of Eq. (179) isgivenby S ζ = {(q L,ϕ C,ϕ,ζ) R 4 q L = g(ϕ)}, (180) where ζ varies rapidly except in a small neighborhood of S ζ. Since the divergence of the vector field of Eq. (179) is zero, the vector field is incompressible. Furthermore, Eq. (177) can be written as two coupled oscillators: LC d2 ϕ C 2 + ϕ C = ϕ, ɛ d2 ϕ 2 + g(ϕ) = C dϕ C. (181) Thus, if g(ϕ) =ϕ, Eq.(181) can be interpreted as two coupled harmonic oscillators. We show one of the trajectories of Eq. (179) in Fig. 46, where L = 1,C = 1/0.45,ɛ = Observe that Eq. (179) exhibits continuous oscillations on the (q L,ϕ C,ϕ)-space. Observe that the trajectory does not exhibit jump-wise motions on the (ϕ, q L )-plane as shown in Fig. 46(b). Note that Eq. (179) has three equilibrium points; namely, (q L,ϕ C,ϕ,ζ)=(0, 0, 0, 0), (0, ±, ±, 0). (182) They do not lie on the set of singular points of the slow-constrained space S 1 ζ defined by { } dg(ϕ) F ζ = (q L,ϕ C,ϕ,ζ) S ζ dϕ =0 { = (q L,ϕ C,ϕ,ζ) q L = g(ϕ), dg(ϕ) } dϕ =0 { = (q L,ϕ C,ϕ,ζ) ϕ = ±1,q L = 2 }, (18) where dg(ϕ) dϕ = d ( ϕ ) dϕ ϕ = ϕ 2 1=0, (184) that is, ϕ = ±1, and thus q L = g(±1) = 2.The set F ζ consists of a two-dimensional surface in R 4. Hence, Eq. (177) is well defined and has no impasse points. They are broken by adding the parasitic (0, 2) element. (a) (b) Fig. 46. Continuous oscillation of Eq. (179) withɛ =0.05. Initial condition: (q L (0),ϕ C (0),ϕ(0),ζ(0)) = (0, 0.1, 0.1, 0.1). (a) Trajectory on the (q L,ϕ C,ϕ)-space. (b) Very rapid motions are not observed on the (ϕ, q L )-plane. 1 In this case, the slow-constrained space S ζ defined by Eq. (180) is a three-dimensional slow-manifold. That is, it is not a surface (two-dimensional manifold), but a three-dimensional space (three-dimensional manifold)

41 10. Impasse Points on a Three-Dimensional Constrained Space In this section, we break the impasse points on a three-dimensional constrained space by adding a parasitic element Basic circuit equation Let us consider the five-element circuit in Fig. 47. The dynamics of this circuit can be described by dq 1 L 1 = ϕ 1 Rq 1, dq 2 L 2 = ϕ + ϕ 1, C 1 dϕ 1 = q q 1, q = g(ϕ), (185) where q 1, q 2, q denote the charge of the inductor L 1, the charge of the inductor L 2, the charge of (a) the memristor, respectively, and ϕ 1 and ϕ denote the flux of the capacitor C 1 and the flux of the memristor, respectively. The flux-controlled memristor is characterized by q = g(ϕ) = ϕ ϕ. (186) Since q 2 = q, weobtainfromeq.(185) dq 1 L 1 = ϕ 1 Rq 1, L 2 dq 2 = ϕ + ϕ 1, C 1 dϕ 1 = q 2 q 1, q 2 = g(ϕ) = ϕ ϕ, (187) which has the three-dimensional constrained space defined by } A = {(q 1,q 2,ϕ 1,ϕ) R 4 q 2 = g(ϕ) = ϕ ϕ. (188) Furthermore, Eq. (187) has impasse points located on the set of singular points of the constrained space A, which is defined by { } B = (q 1,q 2,ϕ 1,ϕ) A dg(ϕ) dϕ =0 = = { (q 1,q 2,ϕ 1,ϕ) q 2 = g(ϕ), dg(ϕ) { (q 1,q 2,ϕ 1,ϕ) ϕ = ±1,q 2 = 2 } dϕ =0 }, (189) where dg(ϕ) dϕ = d ( ) ϕ dϕ ϕ = ϕ 2 1=0, (190) (b) Fig. 47. Basic five-element circuit. (a) The circuit consists of two inductors, a capacitor, a resistor, and a flux-controlled memristor. (b) (α, β)-element representation of the circuit. that is, ϕ = ±1 and thus q 2 = g(±1) = 2.Observe that the set B consists of a two-dimensional surface in the (q 1,q 2,ϕ 1,ϕ)-space. Almost all trajectories of Eq. (185) cannot pass through the impasse points in B. However, a singular trajectory can exist, which moves on the constrained space A, passing through the set B. We can explain this as follows. The dynamics on the (q 2,ϕ)-plane is described by

42 M. Itoh & L. O. Chua (a) (b) Fig. 48. Necessity for adding a parasitic memcapacitance (magenta) to break impasse points. (a) Memristor circuit with a parasitic memcapacitance C (ϕ ). (b) (α, β)-element representation of the circuit. dq 2 = ϕ 1 ϕ L 2, q 2 = g(ϕ) = ϕ ϕ. (191) Observe that Eqs. (145) and(191) arethesameif we replace (ϕ 1,q 2 )with(ϕ C,q L ). Thus, the sign of dq 2 = ϕ 1 ϕ L changes at the points satisfying (q 2,ϕ 1,ϕ)= ( 2 ) ( ) 2, 1, 1,, 1, 1, (192) on B. These points consist of a one-dimensional line in the (q 1,q 2,ϕ 1,ϕ)-space. Then, a singular trajectory can move on the constrained surface A passing through the above points on B. Since most points on B do not satisfy Eq. (192), they are impasse points. We do not need to discuss these singular trajectories further, since the purpose of this paper is to break impasse points. In order to break the impasse points of Eq. (187), we add a parasitic memcapacitor as shown in Figs. 48 and 49. The circuit elements in Figs. 48 and 49 are coded by resistor: (α, β) =(0, 0), capacitor: (α, β) =(0, 1), inductor: (α, β) =( 1, 0), memristor: (α, β) =( 1, 1), memcapacitor: (α, β) =( 1, 2). Their complexity metric is given by Element χ(α, β) resistor 0 capacitor 1 inductor 1 memristor 2 memcapacitor (a) (b) Fig. 49. Necessity for adding a parasitic memcapacitance (magenta) to break impasse points. (a) Memristor circuit with a parasitic memcapacitance C 2 (ϕ 2 ). (b) (α, β)-element representation of the circuit

43 Thus, the memcapacitor has the highest complexity metric. The distance between different pairs of elements are as follows: Pair of Elements Distance d (resistor, capacitor) 1 (resistor, inductor) 1 (resistor, memristor) 2 (resistor, memcapacitor) (capacitor, inductor) 2 (capacitor, memristor) 1 (capacitor, memcapacitor) 2 (inductor, memristor) 1 (inductor, memcapacitor) 2 (memristor, memcapacitor) 1 The diameter D of the circuit is increased from 2 to by adding the memcapacitor. Thus, the memcapacitor is the highest-order element. The subdimension λ j of the circuit is given by Element (α j,β j ) λ j resistor (0, 0) 0 capacitor (0, 1) 1 inductor ( 1, 0) 1 memristor ( 1, 1) 0 inductor ( 1, 0) 1 memcapacitor ( 1, 2) 1 The dimension λ of this circuit is given by λ = =4, (19) which is increased from to 4 by adding the parasitic memcapacitor. Let us study first the circuit in Fig. 48. The dynamics of the circuit is given by dq 1 L 1 = ϕ 1 Rq 1, L 2 dq 2 = ϕ 1 + ϕ, C 1 dϕ 1 C (ϕ ) dϕ = q 1 q 2, = g(ϕ ) q 2, (194) where ϕ denotes the flux of the memcapacitor, C (ϕ ) denotes the memcapacitance, and g(ϕ )= ϕ ϕ. (195) The slow-space (slow-manifold) of Eq. (194) isgiven by S = {(q 1,q 2,ϕ 1,ϕ ) R 4 q 2 = g(ϕ )}, (196) where ϕ varies rapidly except in a close neighborhood of S.Equation(194) has one equilibrium point at the origin, which is unstable. Furthermore, the origin does not lie on the set of singular points of the slow-space S, which is defined by { } dg(ϕ ) F = (q 1,q 2,ϕ 1,ϕ ) S =0 dϕ { = (q 1,q 2,ϕ 1,ϕ ) q 2 = g(ϕ ), dg(ϕ } ) =0 dϕ { = (q 1,q 2,ϕ 1,ϕ ) ϕ = ±1,q 2 = ± 2 }, (197) where dg(ϕ ) = d ( ) ϕ dϕ dϕ ϕ = ϕ 2 1=0, (198) that is, ϕ = ±1 and thus q 2 = g(±1) = ± 2. The set F consists of a two-dimensional surface in the (q 1,q 2,ϕ 1,ϕ )-space. Hence, Eq. (194) iswell defined and has no impasse points. They are broken by adding the parasitic memcapacitance C (ϕ ). We show the trajectories of Eq. (194) infig.50. The following parameters are used in our computer study: C 1 =0.5, C (ϕ )=0.05( ϕ 2 ), L 1 =1, L 2 = 1 1.5, R =1. Observe that Eq. (194) exhibits a relaxation oscillation on the (q 1,q 2,ϕ 1,ϕ )-space. Furthermore, the trajectory exhibits very rapid motions on the (ϕ,q 2 )-plane as shown in Fig. 50(c). We can obtain a similar result when we choose C (ϕ ) = C = If we choose C (ϕ ) = 0.05(1+(e ϕ 1)κ(ϕ )), then an asymmetric closed orbit can be observed. Let us study next the circuit in Fig. 49. The dynamics of the circuit is given by

44 M. Itoh & L. O. Chua (a) (b) (c) Fig. 50. Relaxation oscillation of Eq. (194). Parameters: C 1 =0.5,C (ϕ )=0.05( ϕ 2 ),L 1 =1,L 2 =1/1.5,R =1. Initial condition: (q 1 (0),q 2 (0),ϕ 1 (0),ϕ (0)) = (0.1, 0.1, 0.1, 0.1). (a) Trajectory on the (q 1,q 2,ϕ 1 )-space. (b) Trajectory on the (ϕ 1,q 2,ϕ )-space. (c) Very rapid motions (denoted by a two-headed arrow) on the (ϕ,q 2 )-plane. The trajectory in the two-headed arrow portion moves much faster than that in the one-headed arrow portion. However, it does not mean an instantaneous jump. The slow-space is given by q 2 = ϕ + ϕ. L 1 dq 1 = ϕ 1 Rq 1, L 2 dq 2 = ϕ 2, C 1 dϕ 1 C 2 (ϕ 2 ) dϕ 2 = g(ϕ 1 ϕ 2 ) q 1, = g(ϕ 1 ϕ 2 ) q 2, (199) where ϕ 2 denotes the flux of the memcapacitor, C 2 (ϕ 2 ) denotes the memcapacitance, and g(ϕ 1 ϕ 2 ) is described by g(ϕ 1 ϕ 2 )= (ϕ 1 ϕ 2 ) (ϕ 1 ϕ 2 ). (200) The slow-space (slow-manifold) of Eq. (199) isgiven by S 2 = {(q 1,q 2,ϕ 1,ϕ 2 ) R 4 q 2 = g(ϕ 1 ϕ 2 )}, (201) where ϕ 2 varies rapidly except in a close neighborhood of S 2. Equation (199) has one equilibrium point at the origin, which is unstable. Furthermore, the origin does not lie on the set of singular points of the slowspace S 2, which is defined by { } g(ϕ 1 ϕ 2 ) F 2 = (q 1,q 2,ϕ 1,ϕ 2 ) S 2 =0 ϕ 2 = = where { (q 1,q 2,ϕ 1,ϕ 2 ) q 2 = g(ϕ 1 ϕ 2 ), } g(ϕ 1 ϕ 2 ) =0 ϕ 2 { (q 1,q 2,ϕ 1,ϕ 2 ) ϕ 1 ϕ 2 = ±1,q 2 = 2 }, g(ϕ 1 ϕ 2 ) ϕ 2 = ϕ 2 ( (ϕ1 ϕ 2 ) (202) ) (ϕ 1 ϕ 2 ) =(ϕ 1 ϕ 2 ) 2 1=0, (20) that is, ϕ 1 ϕ 2 = ±1 and thus q 2 = g(±1) = 2. The set F 2 consists of a two-dimensional surface in the (q 1,q 2,ϕ 1,ϕ 2 )-space. Hence, Eq. (199) iswell defined and has no impasse points. They are broken by adding the parasitic memcapacitance C 2 (ϕ 2 ). We show the trajectories of Eq. (199) infig.51. The following parameters are used in our computer study: C 1 =0.5, C 2 (ϕ 2 )=0.05( ϕ 2 2), L 1 =1, L 2 = 1 1.5, R =1. Observe that Eq. (199) exhibits a relaxation oscillation on the (q 1,q 2,ϕ 1,ϕ 2 )-space. Furthermore, the trajectory exhibits very rapid motions on the (ϕ, q 2 )-plane as shown in Fig. 51(c), where

45 (a) (b) (c) Fig. 51. Relaxation oscillation of Eq. (199). Parameters: C 1 =0.5,C 2 (ϕ 2 )=0.05( ϕ 2 2),L 1 =1,L 2 =1/1.5,R =1. Initial condition: (q 1 (0),q 2 (0),ϕ 1 (0),ϕ 2 (0)) = (0.1, 0.1, 0.1, 0.1). (a) Trajectory on the (q 1,q 2,ϕ 1 )-space. (b) Trajectory on the (q 1,q 2,ϕ 2 )-space. (c) Very rapid motions (denoted by a two-headed arrow) on the (ϕ, q 2 )-plane, where ϕ = ϕ 1 ϕ 2.The trajectory in the two-headed arrow portion moves much faster than that in the one-headed arrow portion. However, it does not mean an instantaneous jump. The slow-space is given by q 2 = g(ϕ) = ϕ ϕ. ϕ = ϕ 1 ϕ 2. Note that the slow-space S 2 is described by q 2 = g(ϕ) = ϕ ϕ on the (ϕ, q 2)- plane. We can obtain a similar result when we choose C 2 = If we set C 2 (ϕ 2 ) = 0.5(1 + (e ϕ 2 1)κ(ϕ 2 )), then an asymmetric closed orbit can be observed. In this section, we have shown that the impasse points on the three-dimensional constrained space are broken by a memcapacitor. We can also consider the following problem: How many parasitic elements are necessary to break impasse points on the high-dimensional constrained space which are defined by a set of implicit functions? 11. Conclusion In this paper, we have studied the dynamics of memristors with parasitic elements. We have shown that a passive memristor with a strictly-increasing characteristic curve will eventually lose a stored physical state via a parasitic resistance when we switch off the power. An inductor memristor circuit cannot exhibit a discontinuous behavior on the (ϕ, q)-plane. A higher-order parasitic element can be used to break impasse points of memristor circuits. Acknowledgment This paper is supported in part by AFOSR grant FA References Andronov, A. A., Vitt, A. A. & Khaikin, S. E. [1987] Theory of Oscillators (Dover, NY). Arnold, V. I. [1978] Mathematical Methods of Classical Mechanics (Springer-Verlag, NY). Baatar, C., Tamas Roska, T. & Porod, W. (eds.) [2009] Cellular Nanoscale Sensory Wave Computing (Springer, NY). Benoit, E. [198] Systèmes lents-rapides dans R et leurs canards, Astérisque , Chua, L. O. [1969] Introduction to Nonlinear Network Theory (McGraw-Hill, NY). Chua, L. O. [1971] Memristor The missing circuit element, IEEE Trans. Circuit Th. CT-18, Chua, L. O. & Kang, S. M. [1976] Memristive devices and systems, Proc. IEEE 64, Chua, L. O. [1980] Device modeling via nonlinear circuit elements, IEEE Trans. Circuit Th. 17, Chua, L. O. [1989a] Impasse point. Part I: Numerical aspects, Int. J. Circ. Th. Appl. 17, Chua, L. O. [1989b] Impasse point. Part II: Analytical aspects, Int. J. Circ. Th. Appl. 17, Chua, L. O. [200] Nonlinear circuit foundations for nanodevices. I. The four-element torus, Proc. IEEE 91, Chua, L. O. [2012] The fourth element, Proc. IEEE 100,

46 M. Itoh & L. O. Chua Chua, L. O. [2014] If it s pinched it s a memristor, J. Semiconductor Sci. Technol. Special Issue on Memristor Devices 29, Hirsch, M. W., Smale, S. & Devaney, R. L. [200] Differential Equations, Dynamical Systems & An Introduction to Chaos, Second edition (Elsevier Academic Press, Amsterdam). Itoh, M. [1990] On the systems dropping canardsolutions, Proc. JTC-CSCC 90, Cheju, pp Itoh, M. & Murakami, H. [1994] Chaos and canards in the Van der Pol equation with periodic forcing, Int. J. Bifurcation and Chaos 4, Itoh, M. & Chua, L. O. [2011] Memristor Hamiltonian circuits, Int. J. Bifurcation and Chaos 21, Itoh, M. & Chua, L. O. [201] Duality of memristor circuits, Int. J. Bifurcation and Chaos 2, Itoh, M. & Chua, L. O. [2014] Dynamics of memristor circuits, Int. J. Bifurcation and Chaos 24, Kanamaru, T. [2007] Van der Pol oscillator, Scholarpedia 2, Lipschutz, S. & Lipson, M. [2007] Theory and Problems of Discrete Mathematics (McGraw-Hill, NY). Madan, R. N. [199] Chua s Circuit: A Paradigm for Chaos (World Scientific, Singapore). Rozov, N. Kh. [2011] Relaxation oscillation, Encyclopedia of Mathematics (Kluwer Academic Publishers). Storti, D. & Rand, R. H. [1987] A simplified model of coupled relaxation oscillators, Int. J. Non-Lin. Mech. 22, Appendices q = C(ϕ)v, (A.5) Appendix A df (ϕ) where C(ϕ) Passivity and Losslessness dϕ. We first consider the linear memcapacitor, that The passivity condition for memristors is described is, C(ϕ) = C, wherec is a constant. Define the as follows [Chua, 1971]: energy of the memcapacitor delivered by the sinu- source with period T Passivity condition soidal by A memristor characterized by a differentiable q ϕ (resp., ϕ q) characteristic curve ϕ = f(q) (resp., q = g(ϕ)) is passive if, and only if, df (q) its small-signal memristance M(q) ( dq resp., small-signal memductance W (ϕ) ) dg(ϕ) is non-negative. dϕ Note that if M(q) 0(resp.,W (ϕ) 0), then f(q) (resp., g(ϕ)) is an increasing function. We show the outline of the proof, which is given in [Chua, 1971]. The instantaneous power dissipated by a charge-controlled memristor is given by p(t) =v(t)i(t) =M(q(t))i(t) 2. (A.1) If M(q) 0, then the energy flow into the memristor from time 0 to t satisfies 0 p(τ)dτ 0, (A.2) for all t 0, and the memristor is obviously passive. To prove the converse, suppose that there exist a point q 0 such that M(q 0 ) < 0. Then there exist an ɛ > 0 such that M(q 0 + q) < 0 and q <ɛ. Let us drive the memristor with the current i(t) which is zero for 0 t t 0 andsuchthat q(t) =q 0 + q(t) fort 0 t 1 t where q(t) <ɛ. Then, 0 p(τ)dτ < 0 for sufficiently large t, and hence the memristor is active. We next obtain the lossless condition of the memcapacitor. Consider the memcapacitor defined by σ = f(ϕ), (A.) where σ denotes the integrated charge defined by σ q(τ)dτ. The memcapacitor satisfies the relation w mc (t) = = = 0 q(t) q(0) v(t) v(0) u(t) u(0) v(τ)i(τ)dτ v(τ)dq(τ) Cvdv Cdu (A.4) = C(u(t) u(0)) (A.6)

47 where dq = idτ = Cdv, u v2,andu(t )=u(0). 2 Let us assume u(0) = 0. Then we obtain from Eq. (A.6): (1) If C 0, then w mc (t) 0 for t 0, and w mc (t) = 0 for t = T. Thus, the linear memcapacitor is passive and lossless. (2) If C < 0, then w mc (t) 0 for t 0, and w mc (t) = 0 for t = T. Thus, the linear memcapacitor is lossless, but active. For example, define the sinusoidal voltage source with period T by { a sin(ωt) t 0 s(t) = 0 t<0 (A.7) where ω = 2π and a is a positive constant. Then T the energy of the memcapacitor delivered by the sinusoidal source from 0 to t is given by w mc (t) = = = = Ca2 4 v(τ) i(τ)dτ s(τ) C ds(τ) dτ dτ a sin(ωτ) Caωcos(ωτ)dτ Ca 2 ω sin(2ωτ)dτ 2 ( ( 4πt 1 cos T From Eq. (A.8), we obtain the same result: )). (A.8) (1) If C 0, then w mc (t) 0 for t 0, and w mc (t) = 0 for t = T. Thus, the linear memcapacitor is passive and lossless. (2) If C < 0, then w mc (t) 0 for t 0, and w mc (t) = 0 for t = T. Thus, the linear memcapacitor is lossless, but active. We next study the lossless property of nonlinear memcapacitors. The energy flow into the memcapacitor from time 0 to t satisfies w mc (t) 0 v(τ)i(τ)dτ = v(τ)dq(τ), (A.9) where q(τ) = C(ϕ(τ))v(t) and dq(τ) = i(τ)dτ. Apply the sinusoidal voltage source defined by Eq. (A.7). Then, we obtain v(τ) =a sin(ωτ), and a(1 cos(ωτ)) ϕ(τ) =, ω ( ) a(1 cos(ωτ)) C(ϕ(τ)) = C, ω ( ) T v(0) = v 2 ( ) T q(0) = q 2 = v(t )=0, = q(t )=0. Furthermore, ϕ(τ) andc(ϕ(τ)) satisfy ( ) ( ) T T ϕ 2 τ = ϕ 2 + τ, ( ( )) ( ( )) T T C ϕ 2 τ = C ϕ 2 + τ, (A.10) (A.11) (A.12) where 0 τ T 2. That is, they are symmetric with respect to t = T 2. Consider next the hysteresis loop defined by (q(τ),v(τ)) = (C(ϕ(τ))v(τ),v(τ)) [Baatar et al., 2009]. If C(ϕ) 0, a Lissajous figure is located in the first and third quadrants, which is symmetric with respect to the origin as shown in Fig. 52. For the period satisfying C(ϕ) = 0, the hysteresis loop moves on the v-axis, since q =0. Consider first a Lissajous figure located in the first quadrant. Assume df (ϕ) C(ϕ) = dϕ 0, ϕ dc(ϕ) dϕ = ϕd2 f(ϕ) dϕ 2 0. (A.1) Then C(ϕ) is monotonically increasing for ϕ 0. Choose the time t 1 and the time t 2 satisfying q(t 1 )=q(t 2 ), for 0 <t 1 <t 2 < T 2,thatis, C(ϕ(t 1 ))v(t 1 )=C(ϕ(t 2 ))v(t 2 ), for 0 <t 1 <t 2 < T 2. Then we obtain v(t 1 ) v(t 2 ), (A.14) (A.15) (A.16)

48 M. Itoh & L. O. Chua (a) (b) Fig. 52. Lissajous figures for v(t) =a sin(ωt), q(t) =C(ϕ(t))v(t), and C(ϕ) =ϕ Parameters: a =1,ω =1,T = 2π ω =2π. (a) Hysteresis loop on the (v, q)-plane. (b) Hysteresis loop on the (q, v)-plane. Two points satisfying q(t 1 )=q(t 2 ) are indicated in green. Blue and red curves indicate Lissajous figures for 0 t T 2 and T 2 t T, respectively. since C(ϕ(t 1 )) C(ϕ(t 2 )) [see Fig. 52(b)]. Note that ϕ(t) is monotonically increasing for 0 t T 2. Thus, the area enclosed by two oppositely oriented pinched hysteresis loop, that is, w mc (t) 0 v(τ)i(τ)dτ = v(τ)dq(τ), (A.17) is non-negative for 0 t T. Furthermore, w mc (t) =0fort = T, since a Lissajous figure is symmetric with respect to the origin. It follows that the memcapacitor is lossless and passive. The above result can be confirmed by recasting Eq. (A.9) into: w mc (t) 0 v(τ)i(τ)dτ = v(τ)dq(τ) Note that the charge q(τ) = C(ϕ(τ))v(t) is equal to zero and does not change during the period satisfying C(ϕ(τ)) = 0. Similarly, we obtain dq(τ) = 0, and 1 C(ϕ) dq = q dq = vdq =0, C(ϕ) (A.21) during the above period. Thus, Eq. (A.18) isnot singular even if C(ϕ(τ)) = 0. We next study the passivity of the memcapacitor. If C(ϕ) 0andϕ dc(ϕ) 1 0, then dϕ C(ϕ) is monotonically decreasing for ϕ 0. Choose the time t 1 and the time t 2 satisfying Q(t 1 )=Q(t 2 ) for 0 <t 1 <t 2 < T 2, (A.22) where = Q(t) Q(0) 1 C(ϕ(τ)) dq(τ), (A.18) Q(τ) C(ϕ(τ))2 v(τ) 2, (A.19) 2 which satisfies ( ) T Q(0) = Q = Q(T )=0. (A.20) 2 Note that C(ϕ(τ)) and Q(τ) are symmetric to t = T 2. That is, the two Lissajous curves in Fig. 5 are identical except for the time orientation. Hence, if C(ϕ) 0, then w mc (T ) = 0, that is, the memcapacitor is lossless. which corresponds to the points P 1 and P 2 in Fig. 5(a). Then we obtain 1 C(ϕ(t 1 )) 1 C(ϕ(t 2 )). (A.2) Therefore, w mc (t) 0 for 0 t T, and the memcapacitor is passive. However, if C(ϕ) 0, and ϕ dc(ϕ) < 0 for ϕ 0, then the memcapacitor is not passive. Furthermore, if C(ϕ) < 0, dϕ then w mc (T ) = 0, but the memcapacitor becomes active. Note that we may have a different result if a nonsinusoidal source is applied. Thus, the lossless property for a sinusoidal source is described

49 (a) (b) Fig. 5. Lissajous figures for v(t) =a sin(ωt), q(t) =C(ϕ(t))v(t), and C(ϕ) =ϕ Parameters: a =1,ω =1,T = 2π ω =2π. (a) Lissajous curve on the (Q, C(ϕ))-plane for 0 t T 2.TwopointsP 1 and P 2 represent the points (Q(t 1 ),C(ϕ(t 1 ))) and (Q(t 2 ),C(ϕ(t 2 ))), respectively, where Q(t 1 )=Q(t 2 ). (b) Lissajous curve on the (Q, C(ϕ))-plane for T 2 t T. as follows: Lossless property for a sinusoidal source A memcapacitor characterized by a differentiable ϕ σ constitutive relation σ = f(ϕ) is lossless and passive, if, and only if, C(ϕ) df (ϕ) 0 and ϕ dc(ϕ) 0. Similarly, a dϕ dϕ meminductor characterized by a differentiable q ρ constitutive relation ρ = g(q) islossless and passive, if, and only if, L(q) dg(q) dq and q dl(q) 0. dq 0 If the characteristic curves σ = f(ϕ) andρ = g(q) are piecewise-linear functions with positive slopes for all segments, then the above conditions are automatically satisfied. Appendix B Broader Class of 2-Terminal Circuit Elements Consider a 2-terminal circuit element defined by where q p and ϕ p denote the charge and the flux of the parasitic element (ϕ ξ = ϕ p ), l and k denote integers, and f( ) denotes a scalar function. Let us consider the case where Eq. (B.2) satisfies l = m, thatis, q p (m) = f ( ϕ p (k) ). (B.) From Eqs. (B.1) and (B.), the characteristic of the 2-terminal element with a parasitic element is defined by q (m) η = q (m) ξ + q (m) p = g ( ϕ (m) η ) ( ) + f ϕ (k) η, (B.4) where q η and ϕ η denote the charge and the flux of this element and ϕ η = ϕ p. Therefore, we can define a parasitic-augmented 2-terminal element with parasitic elements by q (m) η = g ( ϕ (m) η ϕ (m 1) η ) ( + k ϕ (m K) η,ϕ η (m K+1),...,,ϕ (m+1) η,...,ϕ (m+n) η ), (B.5) or a broader class of 2-terminal elements with parasitic elements by q (m) η = h ( ϕ (m K) η,ϕ η (m K+1),...,ϕ (m 1) η, q (m) ξ = g ( ϕ (m) ξ ), (B.1) where q ξ and ϕ ξ denote the charge and the flux of the element ξ, and g( ) denotes a scalar function. Assume that its parasitic element has the same state variable ϕ ξ, and assume that it is characterized by q (l) p = f( ϕ (k) p ) ( (k)) = f ϕ ξ, (B.2) ϕ (m) η,ϕ (m+1) η,...,ϕ (m+n) η ), (B.6) where h( ) and k( ) denote scalar functions, and K and N are positive integers. An example of the element (B.5) isgivenby q η = ϕ η ϕ η + ɛϕ (1) η, (B.7)