DUALITY OF MEMRISTOR CIRCUITS

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Tutorials and Reviews International Journal of Bifurcation and Chaos, Vol. 23, No. 1 (2013) 1330001 (50 pages) c World Scientific Publishing Company DOI: 10.1142/S0218127413300012

DUALITY OF MEMRISTOR CIRCUITS MAKOTO ITOH 1-19-20-203, Arae, Jonan-ku, Fukuoka 814-0101, Japan [email protected]

Int. J. Bifurcation Chaos 2013.23. Downloaded from www.worldscientific.com by 119.175.29.193 on 03/05/13. For personal use only.

LEON O. CHUA Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, Berkeley, CA 94720, USA [email protected] Received July 15, 2011; Revised June 10, 2012 In this paper, we show that the dynamics of any memristor circuits can be simulated by a corresponding “dual” nonlinear RLC circuit where the memristor is substituted by a nonlinear resistor. They are in one-to-one correspondence, that is, they are duals of each other. We also propose a method for synchronizing these dual dynamic nonlinear circuits. We next define memory elements which can be characterized by charge and flux. The memory-element circuits can also be simulated by their corresponding dual nonlinear RLC circuits. We then define 2-terminal elements which are characterized by complementary pair of signals, and study them from the view point of one-to-one correspondence. We finally show an example of 2-terminal elements such that the terminal voltage and current are identical, and their time-derivatives of any order are also identical, however, their time-integrals are different. That is, these 2-terminal elements are in one-to-one correspondence except for the time-integral signals. Keywords: Memristor; memcapacitor; meminductor; dual; Chua’s circuit; pinched hysteresis loop; synchronization; 2-terminal element.

1. Introduction Memristor is a 2-terminal electronic device, which was postulated, formulated, and named by Leon O. Chua [1971]. It was recently fabricated as a nanometer-size solid state device by a team led by R. Stanley Williams from the Hewlett-Packard Company [Strukov et al., 2008]. The HP memristor exhibits an interesting nonvolatility property, that is, the memristor does not lose the value of the state when the power is switched off. Furthermore, memristor memory chips promise to run at least ten times faster and use ten times less power than equivalent flash memory chips. However, memristors are not yet available as off-the-shelf electronic components. In this paper, we show that the dynamics of any memristor circuit can be simulated by a

corresponding “dual” nonlinear RLC circuit where the memristor is substituted by a nonlinear resistor. We first study the circuit equation of Chua’s circuit [Madan, 1993]. It is the simplest chaotic circuit that can be easily built even by high school kids [Gandhi et al., 2009; Bilotta et al., 2010], and is one of few chaotic systems that has been proved rigorously to be chaotic [Chua et al., 1986]. We next study the memristor Chua’s circuit obtained by replacing the nonlinear resistor in Chua’s circuit with a corresponding memristor having the same nonlinearity. The circuit equation of Chua’s circuit can be defined on the (i, v)-space, and that of the memristor Chua’s circuit can be defined on the (q, ϕ)-space. Here, i, v, q, and ϕ denote current, voltage, charge, and flux, respectively. We will show that their dynamics are identical, the only

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M. Itoh & L. O. Chua

difference being in letter symbols of the state variables. That is, they are dual dynamic circuits. We then generalize this example and show that memristor circuits have identical dynamics as their corresponding dual nonlinear RLC circuits. That is, the dynamics of memristor circuits can be simulated by corresponding dual nonlinear RLC circuits. We then propose a method for synchronizing these dual dynamic nonlinear circuits. We next define memory elements which can be characterized by charge and flux. The memory-element circuits can also be simulated by their corresponding dual nonlinear circuits. We then define 2-terminal elements which are characterized by a complementary pair of signals [Chua, 2003], and study them from the view point of oneto-one correspondence. We finally show that there are many 2-terminal elements such that the terminal voltage and current signals are identical, and their time-derivatives of any order are also identical, however, their time-integrals are different. Hence, these elements are in one-to-one correspondence except for the time-integrals of terminal voltage and current signals.

2. Circuit Elements In this section, we describe the properties of basic circuit elements using fundamental circuit variables.

2.1. Fundamental circuit elements In electrical circuits, there are four fundamental circuit variables: current i, voltage v, charge q, and flux ϕ. It is well-known that four fundamental

circuit elements are defined by an appropriate pair of these fundamental circuit variables: • Resistor: v = fR (i)

(1)

Example: Linear Resistor v = Ri • Inductor: ϕ = fL (i)

(2)

Example: Linear Inductor: ϕ = Li, or v = L • Capacitor: q = fC (v)

di dt (3)

Example: Linear Capacitor: q = CV, or i = C • Memristor: ϕ = fM (q).

dv dt (4)

Here, the symbols R, L, C denote the resistance, inductance, and capacitance, respectively, and fR ( · ), fL ( · ), fC ( · ), and fM ( · ) denote scalar functions. We next explain the property of the fourth element “memristor ”.

2.2. Memristors The memristor shown in Figs. 1 and 2 is a 2terminal electronic device described by a nonlinear relation ϕ = f (q),

or

q = g(ϕ),

(5)

between the charge q and the flux ϕ. Its terminal voltage v and the terminal current i are obtained by differentiating Eq. (5) with respect to time variable t: v = M (q)i,

or

i = W (ϕ)v,

(6)

Fig. 1. The constitutive relation of a monotone-increasing piecewise-linear memristor: charge-controlled memristor (left) and flux-controlled memristor (right). 1330001-2

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Duality of Memristor Circuits

• Memductance W (ϕ) is uniquely defined via the constitutive relation q = g(ϕ). • Ohm’s law i = W (ϕ)v is constrained on the space of physical state q = g(ϕ).

Fig. 2. Charge-controlled memristor with the terminal voltage v = M (q)i (left) and flux-controlled memristor with the terminal current i = W (ϕ)v (right).

A memristor characterized by a differentiable q − ϕ (resp. ϕ − q) constitutive relation is locallypassive if, and only if, its memristance M (q) (resp. memductance W (ϕ)) is non-negative (see [Chua, 1971]). Since the instantaneous small-signal power δp(t) dissipated by the above memristor about an equilibrium point is given by


where dq dϕ , and i = . (7) dt dt Equation (6) can be interpreted as a statedependent Ohm’s law for the memristors, where the charge “q” is the state variable. The two nonlinear functions M (q) and W (ϕ), called the memristance and memductance, respectively, are defined by v=

df (q) , dq

(8)

W (ϕ)

dg(ϕ) , dϕ

(9)

and

representing the slope of a scalar function ϕ = f (q) and q = g(ϕ) (called the memristor constitutive relation), respectively. Differentiating both sides of ϕ = f (q) + K with respect to time t, we obtain the Ohm’s law (10)

where K is a nonzero constant. Thus, the characteristic curve ϕ = f (q) cannot be uniquely obtained from v = M (q)i (see Appendix A), that is, ==========⇒ ϕ = f (q) ⇐========= v = M (q)i.

(12)

δp(t) = δv(t)δi(t) = W (ϕ(t))(δv(t))2 ≥ 0,

(13)

or

the energy flow into the memristor from time t0 to t satisfies t E(t0 ) + p(τ )dτ ≥ 0, (14) t0

M (q)

v = M (q)i,

δp(t) = δv(t)δi(t) = M (q(t))(δi(t))2 ≥ 0,

(11)

for all t ≥ t0 , where E(t0 ) is the energy of the memristor at t = t0 . Example 1. Consider the following example of a memristor [Chua, 2003]. The classical circuit model for a Josephson junction consists of a parallel connection of a linear capacitor C, a linear resistor R, and a nonlinear inductor with the constitutive relation

i = I0 sin(kϕ),

where I0 is a device parameter, k is the Josephson constant, and ϕ is the flux, that is, the integral of the voltage across the Josephson junction. A rigorous analysis of Josephson junction dynamics reveals the presence of an additional small current component represented approximately by

plus arbitrary constant

i = G cos(k0 ϕ)v,

In other words, the memristance M (q) is welldefined at every operating point on the constitutive relation ϕ = f (q). Hence, we conclude as follows: • Memristance M (q) is uniquely defined via the constitutive relation ϕ = f (q). • Ohm’s law v = M (q)i is constrained on the space of physical state ϕ = f (q).

(15)

(16)

where G and k0 are device constants. If we define a flux controlled memristor with the constitutive relation q = G0 sin(k0 ϕ),

(17)

where G0 = kG0 , then differentiating both sides of Eq. (17) would give us exactly Eq. (16) [Chua, 2003]. Note that the constitutive relation (17) cannot be uniquely obtained from Eq. (16), that is, 1330001-3

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q = G0 sin(k0 ϕ) ==========⇒ i = G cos(k0 ϕ)v ⇐========== Eq. (17) Eq. (16) plus arbitrary constant

In order to obtain the constitutive relation from Eq. (16), we have to solve a set of equations  dϕ   = v,   dt    dq (18) = i,    dt     i = G cos(k0 ϕ)v, Int. J. Bifurcation Chaos 2013.23. Downloaded from www.worldscientific.com by 119.175.29.193 on 03/05/13. For personal use only.

under the initial condition ϕ(0) = ϕ0 ,

q(0) = q0 ,

(19)

where q0 and ϕ0 are constants. Solving Eq. (18), we obtain q(t) = G0 sin(k0 ϕ(t)) + q0 − G0 sin(k0 ϕ0 ).

The instantaneous power dissipated by the memristor defined in (16) is given by p(t) = v(t)i(t) = G cos(k0 ϕ)v(t)2 .

If cos(k0 ϕ) < 0 and v(t) = 0, then p(t) < 0. Therefore, the Josephson memristor defined by (17) is locally-active, i.e. not locally-passive.

3. Circuit Laws In this section, we describe some basic circuit laws.

3.1. Kirchhoff circuit laws Applying Kirchhoff circuit laws [Chua, 1969] to an electronic circuit, we obtain a relation between the two fundamental circuit variables: the current and the voltage. (i, v)-relation

(20)

where v denotes the voltage across the capacitor C and i is the input current applied across the Josephson junction.

• Kirchhoff current law (KCL) The algebraic sum of all circuit element currents im flowing into the node is zero: im = 0. (23)

Hence, we can get Eq. (17) from Eq. (16), if q0 = G0 sin(k0 ϕ0 ). We next show the above refined circuit model of the Josephson junction in Fig. 3. The dynamics of this circuit can be written as   1 dv   + + G cos(k0 ϕ) v + I0 sin(kϕ) = i,  C  dt R   dϕ  = v,  dt (21)

(22)

m

• Kirchhoff voltage law (KVL) The algebraic sum of all circuit element voltages vn around any closed circuit is zero: vn = 0. (24)

n

If we integrate the Kirchhoff circuit laws with respect to time t (from t = −∞), we would obtain the following relation, dubbed conservation of “charge” and “flux”: (q, ϕ)-relation

• Kirchhoff charge law (KCHL) The algebraic sum of all circuit element charges qm flowing into the node is zero: qm = 0. (25) m

Fig. 3. Circuit model of Josephson junction consists of a parallel connection of four basic circuit elements, a linear resistor R, a linear capacitor C, a nonlinear inductor with a constitutive relation i3 = I0 sin(kϕ3 ), and a memristor with constitutive relation q4 = G0 sin(k0 ϕ4 ).

• Kirchhoff flux law (KFLL) The algebraic sum of all circuit element fluxes ϕn around any closed circuit is zero: ϕn = 0. (26)

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n

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(q, ϕ)-relation

Here, qm and ϕn are defined by    im (τ )dτ, qm (t) =   −∞

ϕn (t) =

• Capacitor

t

t

−∞

   vn (τ )dτ,  

(27)

C

dϕ =q dt

(30)

L

dq =ϕ dt

(31)

• Inductor

respectively. Applying the above principle to the circuit, we can obtain a relation between the two fundamental circuit variables: the charge and the flux. Recall the following principles of conservation of charge and flux [Chua, 1969]: Int. J. Bifurcation Chaos 2013.23. Downloaded from www.worldscientific.com by 119.175.29.193 on 03/05/13. For personal use only.

• 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 voltage vC across a capacitor cannot change instantaneously. • The flux ϕ and the current iL in an inductor cannot change instantaneously.

t

where q(t) = −∞ i(τ )dτ and ϕ(t) = −∞ v(τ )dτ . By using these relations and the conservation of charge and flux (KCHL and KFLL), we can also describe the dynamics of electrical circuits for the state variables (q, ϕ).

3.3. Characteristic of memristor and resistor The characteristic of a current-controlled (resp. voltage-controlled) resistor is given respectively by (i, v)-relation

or

i = g(v).

(32)

A linear capacitor, or inductor, does not impose a constraint between its current and voltage (or the charge and flux). However, it imposes a constraint on the rate of change of the state variable v for the capacitor, and i for the inductor. The relationship between the voltage v and the current i for a linear capacitor, and a linear inductor, is given by (i, v)-relation

• Nonlinear resistor v = f (i),

3.2. Dynamics of linear capacitors and inductors

t

The characteristic of a charge-controlled (resp. flux-controlled) memristor is given respectively by (q, ϕ)-relation

• Memristor ϕ = f (q),

or

q = g(ϕ).

(33)

4. Chua’s Circuit

• Capacitor

Consider Chua’s circuit in Fig. 4 [Madan, 1993; Gandhi et al., 2009; Bilotta et al., 2010]. It can (28) be easily built even by high school kids [Gandhi et al., 2009; Bilotta et al., 2010], and is one of few • Inductor chaotic systems that has been proved rigorously to di be chaotic [Chua et al., 1986]. Furthermore, Chua’s (29) L =v dt circuit is a fundamental and general tool for under standing and applying chaotic dynamics for appliUsing these relations and the Kirchhoff circuit laws cations in science and technology [Chua, 1993]. It (KCL and KVL), we can describe the dynamics of contains five circuit elements: two passive capacielectrical circuits for the state variables (i, v). tors, one passive inductor, one passive resistor, and Integrating Eqs. (28) and (29) with respect to one locally-active nonlinear resistor called Chua’s time t, we obtain the following equations: diode. dv =i C dt

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v4 , (36) R where v4 = v2 − v1 , and the v–i curve of Chua’s diode shown in Fig. 5 is given by i4 =

i = f (v) = bv + 0.5(a − b)(|v + 1| − |v − 1|), (37)


Fig. 4. 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.

where a and b are some constants. Substituting Eqs. (35)–(37) into Eq. (34), we obtain a set of three first order differential equations involving the state variables (v1 , v2 , i3 ) Chua’s circuit equation

  v2 − v1 dv1  = − f (v1 ), C1   dt R      v2 − v1 dv2 = i3 − , C2  dt R       di3   = −v2 . L  dt

4.1. Dynamics of Chua’s circuit Applying Kirchhoff current law (KCL) and voltage law (KVL) to Chua’s circuit, we obtain a set of equations which define the relation among two fundamental circuit variables, namely, the current and the voltage: Kirchhoff circuit law

(38)

 • Apply KCL at node A: i1 = i4 − i,   • Apply KCL at node B: i2 = i3 − i4 ,   • Apply KVL at loop C: v3 = −v2 ,

We show the chaotic trajectory of Eq. (38) in Fig. 6(a), where the parameter values are as follows:1  1 1  , C1 = , C2 = 1, L = 10 14.87 (39) (34)   R = 1, a = −1.27, b = −0.68. where the symbols i1 , i2 , i3 , i4 , and i denote the current of the capacitors C1 , C2 , the inductor L, the resistor R, and Chua’s diode respectively, and the symbols v1 , v2 , v3 , and v denote the voltage of the capacitors C1 , C2 , the inductor L, and Chua’s diode, respectively. The inductor L and the two capacitors C1 , C2 impose the state changes in time, that is,  dv1   , i1 = C1  dt       dv2 (35) , i2 = C2 dt       di3    .  v3 = L dt The characteristics of the linear resistor R is given by 1

Fig. 5. v–i curve of Chua’s diode with a negative slope (a = −1.27, b = −0.68).

See [Gandhi et al., 2009] for the practical component values of Chua’s circuit. 1330001-6

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(a)

(b)

Fig. 6. Chaotic trajectories of Chua’s circuit. (a) The trajectory of Eq. (38) with initial condition: (v1 (0), v2 (0), i3 (0)) = dv1 (0) dv2 (0) di3 (0) , dt , dt ) = (1.27, 0.1, −1.487), which is obtained from the (0.1, 0.1, 0.1). (b) The trajectory of Eq. (40) satisfying ( dt 1 1 , C2 = 1, L = 14.87 , R = 1, a = −1.27, b = −0.68. initial condition of Eq. (38). Circuit parameters: C1 = 10

We also show the trajectory of the derivative of the state variables, that is, dv1 (t) dv2 (t) di3 (t) , , dt dt dt i1 (t) i2 (t) v3 (t) (40) , , = C1 C2 L

where (u1 (t), u2 (t), j3 (t))

Differentiating Eq. (38) with respect to time t, obtain Derivative of Eq. (38)

u2 − u1 du1 = −W C1 dt R C2

u2 − u1 du2 = j3 − , dt R

dj3 = −u2 , L dt

t

−∞

u1 (τ )dτ

   u1 ,       

v1 (t)

t −∞

u1 (τ )dτ,

df (v1 ) = W (v1 ) dv1

     

a, if |v1 | ≤ 1      b, if |v1 | ≥ 1.

(43)

The trajectory in Fig. 6(b) can be obtained from the computed numerical solutions of Eq. (41). Differentiating the v–i curve of Chua’s diode: we i = f (v) with respect to time t, we obtain the characteristic of Chua’s diode in terms of the state vari ables j(t) and u(t) t j(t) = W u(τ )dτ u(t) −∞

         

=

(41)

dv1 (t) dv2 (t) di3 (t) , , , (42) dt dt dt

and

in Fig. 6(b). Equation (40) can be obtained from Eq. (35).

4.2. Derivative of the circuit equation

where 1330001-7

     a u(t),

if

    b u(t),

if

t −∞ t −∞

u(τ )dτ ≤ 1 u(τ )dτ ≥ 1

(44)

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  di(t)  , j(t)  dt   dv(t)   u(t) .  dt

where q1

(45)

q2

Equation (44) appears on the right-hand side of Eq. (41). The detailed explanation of Eqs. (41) and (44) will be given in the next section.

q3


5. Memristor Chua’s Circuit

q4

Let us replace Chua’s diode in Fig. 4 by a fluxcontrolled memristor and obtain the memristor Chua’s circuit shown in Fig. 7 [Itoh & Chua, 2008; Muthuswamy, 2010]. Let us study next the dynamics of this circuit.

q

t −∞

i1 (t)dt, i2 (t)dt,

−∞

i3 (t)dt,

ϕ3

i4 (t)dt,

ϕ4

t

i(t)dt, −∞

ϕ

v1 (t)dt,

t

−∞

v2 (t)dt,

t

−∞

t −∞

ϕ2

t

t

−∞

t −∞

ϕ1

v3 (t)dt,

t

−∞

v4 (t)dt,

t −∞

v(t)dt = ϕ1 .

Here, the symbols q1 , q2 , q3 , q4 , and q denote the charge of the capacitors C1 , C2 , the inductor L, the 5.1. Dynamics of the memristor resistor R, and Chua’s memristor,3 respectively, and Chua’s circuit the symbols ϕ1 , ϕ2 , ϕ3 , ϕ4 , and ϕ denote the flux of the capacitors C1 , C2 , the inductor L, the resistor Applying Kirchhoff charge law (KCHL) and flux R, and Chua’s memristor, respectively. The induclaw (KFLL) to this circuit, we obtain a set of tor and two capacitors impose the state changes in 2 equations : time. Their relationship between the charge q and Kirchhoff circuit law the flux ϕ is given by   • Apply KCHL at node A: q1 = q4 − q,   dϕ1   , q1 = C1  • Apply KCHL at node B: q2 = q3 − q4 ,  dt        • Apply KFLL at loop C: ϕ3 = −ϕ2 , dϕ2 (47) , q2 = C2 (46)  dt      dq3   .  ϕ3 = L  dt The characteristics of the linear resistors R is given by q4 =

ϕ4 , R

(48)

where ϕ4 = ϕ2 − ϕ1 , and the ϕ–q curve of Chua’s memristor shown in Fig. 8 is given by Fig. 7. Memristor Chua’s circuit. The Chua’s diode in Fig. 4 is replaced by a flux-controlled memristor. 2

3

q = f (ϕ) = bϕ + 0.5(a − b)(|ϕ + 1| − |ϕ − 1|), (49)

Equation (46) can be obtained by integrating the following equations with respect to time t, 9 • Apply KCL at node A: i1 = i4 − i, > > = • Apply KCL at node B: i2 = i3 − i4 , > > ; • Apply KVL at loop C: v3 = −v2 . We use the terminology “Chua’s memristor” because it is the dual element of Chua’s diode. 1330001-8

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 1   C2 = 1, L = , 14.87   a = −1.27, b = −0.68.

1 C1 = , 10 R = 1,

(52)

Observe that the trajectory of the memristor Chua’s circuit in Fig. 9(a) is the same as that of Chua’s


(a)

Fig. 8. ϕ–q curve of Chua’s memristor with a negative slope (a = −1.27, b = −0.68).

where a and b are some constants. Substituting Eqs. (47)–(49) into Eq. (46), we obtain a set of three first order differential equations Memristor Chua’s circuit equation

  ϕ2 − ϕ1 dϕ1  = − f (ϕ1 ), C1   dt R      ϕ2 − ϕ1 dϕ2 = q3 − , C2  dt R       dq3   = −ϕ2 . L  dt

(b)

(50)

Observe that the dynamics (38) and (50) are identical since these two sets of equations differ only in the symbols of the three state variables: One-to-one correspondence of state variables

Eq. (38)

Eq. (50)

=⇒ (ϕ1 , ϕ2 , q3 ) (v1 , v2 , i3 ) ⇐=

(51) Fig. 9. Chaotic trajectories of the memristor Chua’s cir-

Here, the symbol ⇐= =⇒ indicates “one-to-one correspondence.” We next show the trajectories of Eqs. (50) in Fig. 9(a), where we choose the same parameters as Eq. (38), that is,

cuit. (a) The trajectory of Eq. (50) with initial condition (ϕ1 (0), ϕ2 (0), q3 (0)) = (0.1, 0.1, 0.1). (b) The trajectory of Eq. (55) with initial condition (v1 (0), v2 (0), i3 (0), ϕ1 (0)) = (1.27, 0.1, −1.487, 0.1), which is calculated from 1 1 , C2 = 1, L = 14.87 , Eq. (61). Circuit parameters: C1 = 10 R = 1, a = −1.27, b = −0.68.

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circuit in Fig. 6(a), since they are obtained from the numerical integration of the identical sets of differential equations (38) and (50). Thus, the two nonlinear circuits in Figs. 4 and 7 are duals.4 Let us summarize this “duality” result as follows:

equations [Itoh & Chua, 2008]: Circuit equation B

 v2 − v1 dv1   C1 = − W (ϕ1 )v1 ,   dt R       v2 − v1 dv2   = i3 − , C2   dt R

Theorem 1. The dynamics of Chua’s circuit in Fig. 4 is identical to that of the memristor Chua’s circuit in Fig. 7. They are duals.

              

di3 L = −v2 , dt

==⇒ Memristor Chua’s circuit ⇐= duals Chua’s circuit

dϕ1 = v1 , dt

(55)


5.2. Derivative of the circuit equation

where (v1 (t), v2 (t), i3 (t)) dϕ1 (t) dϕ2 (t) dq3 (t) , , . = dt dt dt

Differentiating Eq. (50) with respect to time t, we obtain a set of differential equations Circuit equation A

  v2 − v1 dv1  = − W (ϕ1 )v1 , C1   dt R       v2 − v1 dv2    = i3 − , C2   dt R       di3   = −v2 , L   dt   dϕ1   = v1 ,   dt       dϕ2    = v2 ,   dt       dq3   = i3 ,  dt

Furthermore, Eq. (55) can be recast into Eq. (57): Circuit equation C

dv1 dt

C2

dv2 dt

L

di3 dt

t   v2 − v1   = −W v1 (τ )dτ v1 ,   R −∞     v2 − v1 = i3 − ,  R         = −v2 ,  (57)

where

where df (ϕ1 ) . dϕ1

C1

(53)

W (ϕ1 ) =

(56)

(54)

Since ϕ2 and q3 do not appear in the first four equations in Eq. (53), the dynamics of Eq. (53) is described uniquely by the following four state

ϕ1 (t) =

t

−∞

v1 (τ )dτ.

(58)

Note that although Eqs. (38) and (57) are quite similar to each other, they are not identical.

5.3. Initial conditions and constrained space Since Eq. (50) can be described as

4

We say that in two systems or phenomena are duals of each other if we can exhibit some kind of one-to-one correspondence between various quantities or attributes of the two systems [Chua, 1969]. Two equations which differ only in symbols but are otherwise identical in form are said to be dual equations. 1330001-10

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Constrained space of physical states

Fig. 9(b) is slightly different from that of Fig. 6(b).

 ϕ2 − ϕ1  − f (ϕ1 ), C1 v1 =    R   ϕ2 − ϕ1 ,  C2 v2 = q3 −   R     Li3 = −ϕ2 ,


Although these trajectories are mathematically identical, they are computed numerically from different equations, that is, the former is computed numerically from Eq. (55), and the latter is com(59) puted numerically from Eqs. (38) and (40). Hence, the two trajectories starting from the same initial point will be significantly different in the distant future as shown in Fig. 10. However, their qualitative behaviors are the same. the solutions of Eqs. (53), (55), and (57) must satisfy Eq. (59), which defines the constrained space of physical states. It is due to this reason why we can (a) also get Eq. (55) by differentiating the equations  ϕ2 − ϕ1  − f (ϕ1 ) + D1 , C1 v1 =    R   ϕ2 − ϕ1 + D2 , C2 v2 = q3 −    R     Li3 = −ϕ2 + D3 ,

(60)

where Di (i = 1, 2, 3) are arbitrary constants. Furthermore, the initial conditions of Eqs. (53), (55), and (57) at t = 0 must satisfy Initial conditions

  ϕ2 (0) − ϕ1 (0)  − f (ϕ1 (0)), C1 v1 (0) =    R  

C2 v2 (0) = q3 (0) −

ϕ2 (0) − ϕ1 (0) , R

Li3 (0) = −ϕ2 (0).

(b)

        (61)

Hence, if the solutions of three distinct but identical equations (53), (55), and (57) do not satisfy Eq. (61), then they are different from the solution of Eq. (50).

5.4. Chaotic trajectories We show the trajectories of Eq. (55) in Fig. 9(b). It is well-known that the behavior of chaotic dynamical systems is highly sensitive to initial conditions. Hence, the error associated with a numerical integration scheme, namely, the truncation error and round-off error may lead to significantly different future behavior. Observe that the trajectory of

Fig. 10. Chaotic trajectories of (a) Chua’s circuit and (b) memristor Chua’s circuit. Two trajectories starting from the same initial point will be significantly different in the distant future.

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5.5. Relationship among state variables From Eqs. (40), (51), and (56), the relationship between the state variables of Chua’s circuit and the memristor Chua’s circuit can be described as follows: Relationship among state variables

Chua’s circuit in Fig. 4 Memristor Chua’s circuit in Fig. 7 dv1 dv2 di3 dϕ1 dϕ2 dq3 , , ⇐===⇒ ϕ1 , ϕ2 , q3 , , , v1 , v2 , i3 , dt dt dt dt dt dt (v1 , v2 , i3 , i1 , i2 , v3 )

⇐===⇒

(ϕ1 , ϕ2 , q3 , v1 , v2 , i3 )


Here, the symbol ⇐= =⇒ indicates the one-to-one correspondence.

5.6. Pinched hysteresis loops It is well-known that memristors exhibit distinctive fingerprints characterized by pinched hysteresis loops. Let us study next the pinched hysteresis loops of Chua’s circuit. The terminal voltage v(t) and current i(t) of Chua’s memristor satisfy t v(τ )dτ v(t). (62) i(t) = W −∞

Therefore, if the Chua’s memristor exhibits a pinched hysteresis loop, then Chua’s diode also exhibits the same loop. The former pinched hysteresis loop is defined on the (u, j)-plane, or the di ( dv dt , dt )-plane, and the latter one is defined on the (v, i)-plane.

The pinched hysteresis loop of Chua’s diode in Fig. 4 is identical to that of Chua’s memristor in Fig. 7. They are duals. Lemma 1.

If we compare Eq. (62) with Eq. (44), we would obtain the relation: One-to-one correspondence

We show the pinched hysteresis loops of Chua’s memristor and Chua’s diode in Fig. 11. Observe t that the computed numerical solutions are slightly u(τ )dτ u(t) j(t) = W −∞ different, since they are not obtained from the same equations. However, their qualitative behav (63) iors are the same. That is, from Eq. (44), the pinched hysteresis loop of Chua’s diode (resp. Chua’s memristor Chua’s memristor) made transitions between two t slopes j = au and j = bu (resp. i = av and v(τ )dτ v(t) i(t) = W i = bv). It follows that we can simulate the behav−∞ ior of the memristor Chua’s circuit by using Chua’s circuit. One-to-one correspondence If we add a small resistance (r = 0.35) in series with the inductor L, then a limit cycle, instead of a Chua’s chaotic trajectory, appears in the Chua’s circuit as Chua’s diode memristor shown in Fig. 12. In this case, the pinched hystere dv(t) di(t) sis loops of Chua’s memristor and Chua’s diode are , ⇐= =⇒ (v(t), i(t)) (u(t), j(t)) = dt dt identical as shown in Fig. 13, since the behavior of (64) the dynamical systems in this “non-chaotic” case is not sensitive to initial conditions. Chua’s diode

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(a)

(b)

Fig. 11. Pinched hysteresis loops of the active Chua’s diode and Chua’s memristor. The pinched hysteresis loops lie in the second and the fourth quadrants because Chua’s diode and Chua’s memristor are active. (a) Locus of the derivatives of the terminal voltage v(t) and current i(t) of Chua’s diode. (b) Locus of the terminal voltage v(t) and current i(t) of Chua’s memristor.

(a)

(b)

Fig. 12. Limit cycles of (a) Chua’s circuit and (b) memristor Chua’s circuit. A small resistance (r = 0.35) is added in series with the inductor L.

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M. Itoh & L. O. Chua (b)


(a)

Fig. 13. Pinched hysteresis loops of the active Chua’s diode and Chua’s memristor. (a) Locus of the derivatives of the terminal voltage v(t) and current i(t) of Chua’s diode. (b) Locus of the terminal voltage v(t) and current i(t) of Chua’s memristor.

6. Canonical Chua’s Circuit Consider the canonical Chua’s circuit [Chua & Lin, 1990; Itoh & Tomiyasu, 1989] in Fig. 14. It contains five circuit elements: two passive capacitors, one passive inductor, one negative resistor, and one active Chua’s diode. Applying KCL to nodes A, B and applying KVL to loop C, we obtain the following set of equations: Canonical Chua’s circuit equation

  dv1  C1 = i3 − g(v1 ),    dt      dv2 C2 = −i3 + Gv2 ,  dt       di3  = v2 − v1 ,  L  dt

(65)

where the v–i curve of Chua’s diode shown in Fig. 15 is given by i = g(v) = bv + 0.5(a − b)(|v + 1| − |v − 1|), (66)

where a and b are the slopes of the inner and outer segments of the v–i curve in Fig. 15, respectively. We show the chaotic trajectory of Eq. (65) in Fig. 16(a), where the parameter values are as follows:  1 1  , L = 1, C1 = , C2 = 10 0.47 (67)   G = 1, a = −2.0, b = 4.0.

Fig. 14. Canonical Chua’s circuit. It contains five circuit elements: two passive capacitors, one passive inductor, one negative resistor, and one Chua’s diode.

Fig. 15. v–i curve of Chua’s diode with negative and positive slopes (a = −2, b = 4).

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(a)

(b)

Fig. 16. Chaotic trajectories of the canonical Chua’s circuit. (a) The trajectory of Eq. (65) with initial condition dv1 (0) dv2 (0) di3 (0) , dt , dt ) = (3, 0, 0), which is (v1 (0), v2 (0), i3 (0)) = (0.1, 0.1, 0.1). (b) The trajectory of Eq. (68) satisfying ( dt 1 1 , C2 = 0.47 , L = 1, G = 1, a = −2, b = 4. obtained from the initial condition of Eq. (68). Circuit parameters: C1 = 10

We also show the trajectory of the derivative of the state variables, that is, dv1 (t) dv2 (t) di3 (t) , , dt dt dt i1 (t) i2 (t) v3 (t) (68) , , = C1 C2 L in Fig. 16(b). If we replace a Chua’s diode in Fig. 14 by a fluxcontrolled memristor, we would obtain the memristor canonical Chua’s circuit in Fig. 17 [Itoh & Chua, 2008]. Applying KCHL to nodes A, B and applying KFLL to loop C, we obtain a set of equations:

Memristor canonical Chua’s circuit equation

  dϕ1  C1 = q3 − g(ϕ1 ),    dt      dϕ2 C2 = −q3 + Gϕ2 ,  dt       dq3  = ϕ2 − ϕ1 ,  L  dt

(69)

where the ϕ–q curve of Chua’s memristor shown in Fig. 18 is given by q = g(ϕ) = bϕ + 0.5(a − b)(|ϕ + 1| − |ϕ − 1|), (70) where a and b are the slopes indicated in the ϕ–q curve in Fig. 18. Observe that the dynamics of the two sets of state equations (65) and (69) are identical since they differ only in the symbols chosen for the state variables: One-to-one correspondence of state variables Eq. (65)

Eq. (69)

=⇒ (ϕ1 , ϕ2 , q3 ) (v1 , v2 , i3 ) ⇐= Fig. 17. Memristor canonical Chua’s circuit. A Chua’s diode in Fig. 17 is replaced by a corresponding “dual” fluxcontrolled memristor.

(71)

We show the trajectories of Eq. (69) in Fig. 19(a), where we choose the same parameters as Eq. (67),

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numerical integration of identical differential equations, namely, Eqs. (65) and (69). Differentiating Eq. (69) with respect to time t, we obtain a set of differential equations Derivative of Eq. (69)

C1

dv1 dt

C2

dv2 dt


di3 = v2 − v1 , L dt

Fig. 18. ϕ–q curve of Chua’s memristor with negative and positive slopes (a = −2, b = 4).

  = i3 − W (ϕ1 )v1 ,           = −i3 + Gv2 ,   

dϕ1 = v1 , dt

              

(73)

where

that is, C1 =

1 , 10

G = 1,

C2 =

1 , 0.47

a = −2.0,

  L = 1, b = 4.0.

 

W (ϕ1 ) = (72)

Observe that the chaotic trajectory of the memristor canonical Chua’s circuit in Fig. 19(a) is the same as that of the canonical Chua’s circuit in Fig. 16(a), since they are obtained from the

dg(ϕ1 ) , dϕ1

(74)

and (v1 (t), v2 (t), i3 (t)) dϕ1 (t) dϕ2 (t) dq3 (t) = , , . dt dt dt

(75)

Since Eq. (69) can be recast as

(a)

(b)

Fig. 19. Chaotic trajectories of the memristor Chua’s circuit. (a) The trajectory of Eq. (69) with initial condition (ϕ1 (0), ϕ2 (0), q3 (0)) = (0.1, 0.1, 0.1). (b) The trajectory of Eq. (73) with initial condition (v1 (0), v2 (0), i3 (0), ϕ1 (0)) = 1 1 , C2 = 0.47 , L = 1, G = 1, a = −2, b = 4. (3, 0, 0, 0.1). Circuit parameters: C1 = 10 1330001-16

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(a)

(b)

Fig. 20. Pinched hysteresis loops of the active Chua’s diode and Chua’s memristor in the canonical Chua’s oscillator. The pinched hysteresis loops lie in all quadrants because the elements are active. (a) Locus of the derivatives of the terminal voltage v(t) and current i(t) of Chua’s diode. (b) Locus of the terminal voltage v(t) and current i(t) of Chua’s memristor.

Constrained space of physical states

We show the chaotic trajectories of Eq. (73) in Fig. 19(b). Observe that the trajectory of Fig. 19(b) is slightly different from that of Fig. 16(b). (76) Although these trajectories are mathematically identical, they are computed numerically from different equations, that is, the former is computed numerically from Eq. (73), and the latter is comthe solution and the initial condition of Eq. (73) puted numerically from Eqs. (65) and (68). Hence, must satisfy Eq. (76), which defines the constrained the two trajectories starting from the same initial space of physical states.  C1 v1 = q3 − g(ϕ1 ),   C2 v2 = −q3 + Gϕ2 ,   Li3 = ϕ2 − ϕ1 ,

(a) Fig. 21.

(b)

(a) v–i curve of passive Chua’s diode (a = 0.2, b = 5). (b) ϕ–q curve of passive Chua’s memristor (a = 0.2, b = 5). 1330001-17

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(a)

(b)

Fig. 22. Chaotic trajectories of the canonical Chua’s circuit. (a) The trajectory of Eq. (65) with initial condition dv1 (0) dv2 (0) di3 (0) , dt , dt ) = (0.2, 0, 0). Circuit (v1 (0), v2 (0), i3 (0)) = (0.1, 0.1, 0.1). (b) The trajectory of Eq. (68) satisfying ( dt parameters: C1 = 0.4, C2 = 2, L = 1, G = 1, a = 0.2, b = 5.

point will be significantly different in the distant future. However, their qualitative behaviors are the same. Thus, the two nonlinear circuits in Figs. 14 and 17 are duals. We next show the pinched hysteresis loops of Chua’s memristor and Chua’s diode in Fig. 20. The pinched hysteresis loop of Chua’s diode (resp. Chua’s memristor) made transitions between two slopes j = au and j = bu (resp. i = av and i = bv). Observe that the computed numerical solutions are slightly different, since they are not obtained from the same equations. However, their qualitative behaviors are also the same. We next study the case where Chua’s diode and memristor are passive as shown in Fig. 21. The chaotic trajectories and pinched hysteresis loops are shown in Figs. 22–24. Their circuit parameters are given by C1 = 0.4, C2 = 2, L = 1, (77) G = 1, a = 0.2, b = 5.0. Observe the following from Figs. 22–24: • The trajectory of Fig. 22(a) is the same as that of Fig. 23(a). • The trajectory of Fig. 22(b) is slightly different from that of Fig. 23(b), however, their qualitative behaviors are the same.

• The pinched hysteresis loops lie in the first and the third quadrants as shown in Fig. 24, because Chua’s diode and memristor are passive. It follows that the “negative” resistor −G is responsible for the chaotic phenomenon in these dual circuits. We summarize the above results as follow: Theorem 2. The dynamics of the canonical Chua’s circuit in Fig. 14 is identical to that of the memristor canonical Chua’s circuit in Fig. 17. They are duals. Canonical Chua’s dynamics Memristor canonical ⇐=====⇒ circuit Chua’s circuit duals

Lemma 2. The pinched hysteresis loop of Chua’s diode in Fig. 14 is identical to that of Chua’s memristor in Fig. 17. They are duals.

To sum up our results, we can simulate the behavior of the memristor canonical Chua’s circuit by using the canonical Chua’s circuit.

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(a)

(b)

Fig. 23. Chaotic trajectories of the memristor Chua’s circuit. (a) The trajectory of Eq. (69) with initial condition (ϕ1 (0), ϕ2 (0), q3 (0)) = (0.1, 0.1, 0.1). (b) The trajectory of Eq. (73) with initial condition (v1 (0), v2 (0), i3 (0), ϕ1 (0)) = (0.2, 0, 0, 0.1). Circuit parameters: C1 = 0.4, C2 = 2, L = 1, G = 1, a = 0.2, b = 5.

(a)

(b)

Fig. 24. Pinched hysteresis loops of the passive Chua’s diode and Chua’s memristor. The pinched hysteresis loops lie in the first and the third quadrants because the elements are passive. (a) Locus of the derivatives of the terminal voltage v(t) and current i(t) of Chua’s diode. (b) Locus of the terminal voltage v(t) and current i(t) of Chua’s memristor. Circuit parameters: C1 = 0.4, C2 = 2, L = 1, G = 1, a = 0.2, b = 5.

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7. Van der Pol Oscillator Consider the Van der Pol oscillator in Fig. 25. It contains three circuit elements: one passive capacitor, one passive inductor, and one nonlinear resistor. Applying KCL and KVL to node A and closed loop B, respectively, we obtain a set of two first order differential equations: Van der Pol oscillator equation


 di   L = v − f (i),  dt   dv   = −i, C dt

(78)

where L = C = 1 and the v–i curve of nonlinear resistor shown in Fig. 26 is given by v = f (i) =

Fig. 25.

i3 − i. 3

(a)

(79)

Van der Pol oscillator.

(b) Fig. 27. Trajectories of the Van der Pol oscillator. The initial transient segment is drawn in red and orange, and the limit cycle is drawn in blue. The orange trajectory following the red trajectory indicates the direction of increasing time. (a) The trajectory of Eq. (78) with initial condition (v(0), i(0)) = (0.1, 0.1). (b) The trajectory of Eq. (80) satis3 dv(0) di(0) fying ( dt , dt ) = (0.2 − 0.1 3 , −0.1) ≈ (0.199667, −0.1). Circuit parameters: C = 1, L = 1.

Fig. 26. v–i curve of nonlinear resistor with a cubic nonlin3 earity: v = i3 − i.

We show the trajectory of Eq. (78) in Fig. 27(a). We also show the trajectory of the derivative of the state variables, that is, i(t) v(t) − f (i(t)) dv(t) di(t) , = − , (80) dt dt C L in Fig. 27(b).

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where M (q) =

df (q) . dq

(85)

Since Eq. (81) can be described as Constrained space of physical states

Li = ϕ − f (q), Cv = −q,


Fig. 28.

Memristor Van der Pol oscillator.

(86)

the solution and the initial condition of Eq. (84) must satisfy Eq. (86), which defines the constrained space of physical states. We show the trajectories of Eqs. (81) and (84) in Fig. 30. Observe that the trajectory of the memristor Van der Pol oscillator in Fig. 30 is the same as that of the Van der Pol oscillator in Fig. 27(a). Memristor Van der Pol oscillator equation  We also show the pinched hysteresis loops of Chua’s  dq  memristor and Chua’s diode in Fig. 31. Observe L = ϕ − f (q),  dt that they are identical. We summarize the above (81)  results as follows:  dϕ   = −q, C dt Theorem 3. The dynamics of the Van der Pol where q and ϕ denote the charge of the inductor L oscillator in Fig. 25 is identical to that of the and the flux of the capacitor C, respectively, that is, memristor Van der Pol oscillator in Fig. 28.  They are duals. t   i(t)dt,  q(t)  Van der Pol Memristor Van der  ⇐=====⇒ −∞ oscillator Pol oscillator duals (82) t     v(t)dt, ϕ(t) Replace the nonlinear resistor by the nonlinear memristor with the same nonlinearity ϕ = f (q) as shown in Fig. 28. Applying KCHL and KFLL to node A and closed loop B, respectively, we obtain a set of two first order differential equations:

−∞

and the q–ϕ curve of the charge-controlled memristor shown in Fig. 29 is given by q3 − q. (83) 3 Observe that Eqs. (78) and (81) are identical, that is, they are duals. Differentiating Eq. (81) with respect to time t, we obtain a set of differential equations ϕ = f (q) =

Derivative of Eq. (81)

L C

di dt

dv dt dq dt

   = v − M (q)i,       = −i,          = i,

(84) Fig. 29. q–ϕ curve of a memristor with a cubic nonlinearity: q3 ϕ = 3 − q. 1330001-21

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(a)

(b)

Fig. 30. Trajectories of the memristor Van der Pol oscillator. The initial transient segment is drawn in red and orange, and the limit cycle is drawn in blue. The orange trajectory following the red trajectory indicates the direction of increasing time. (a) The trajectory of Eq. (81) with initial condition (ϕ(0), q(0)) = (0.1, 0.1, 0.1). (b) The trajectory of Eq. (84) with initial 3 condition (v(0), i(0), ϕ(0)) = (0.2 − 0.1 3 , −0.1, 0.1) ≈ (0.199667, −0.1, 0.1). Circuit parameters: C = 1, L = 1.

(a)

(b)

Fig. 31. Pinched hysteresis loops of the active nonlinear diode and memristor. The pinched hysteresis loops lie in all quadrants because the elements are active. (a) Locus of the derivatives of the terminal voltage v(t) and current i(t) of Chua’s diode. (b) Locus of the terminal voltage v(t) and current i(t) of Chua’s memristor. Circuit parameters: C = 1, L = 1.

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Memristor circuit equation A

The pinched hysteresis loop of Chua’s diode in Fig. 25 is identical to that of Chua’s memristor in Fig. 28. They are duals. Lemma 3.

To sum up our results, we can simulate the behavior of the memristor Van der Pol oscillator by using Van der Pol oscillator.

L


8.1. Memristor Hamiltonian circuit Consider the memristor described by a constitutive relation ϕ = f (q),

(87)

between the charge q and the flux ϕ. By differentiating Eq. (87) with respect to time variable t, we obtain v = M (q)i,

(88)

dϕ dt

and

i=

dq , dt

df (q) . dq

If i = 0, Eq. (92) can be written as   1 dq   = 1,   i dt  M (q)  1 di  =− .  i dt L

(93)

This equation can be recast into the form of Pfaff ’s equation describing a conservative system: Pfaff’s equation

 dq ∂H   = ,   dt ∂i   ∂H  1 di  =− ,  i dt ∂q

where H(q, i) =

(a) Fig. 32.

(92)

(89)

Consider next the circuit in Fig. 32(a). Its differential equation is given by either the

  di  L = −M (q)i.  dt

1 i

(90)

     

dq = i, dt

and M (q)

(91)

Memristor circuit equation B

where v=

dq = −f (q), dt

or the

8. Hamiltonian Circuits In this section, we study the duality of memristor Hamiltonian circuits [Itoh & Chua, 2011].

(94)

M (q) dq + L

1di =

f (q) + i, L

(95)

(b)

Two-element Hamiltonian circuits. (a) Memristor Hamiltonian circuit and (b) nonlinear resistor Hamiltonian circuit. 1330001-23

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and the symbol denotes an “indefinite integral.” The Pfaff’s equation (94) has the solution H(q, i) = H0 (H0 is any constant), since ∂H dq ∂H di dH(q, i) = + dt ∂q dt ∂i dt dq 1 dq di 1 di + =− i dt dt i dt dt

and M (i)

Nonlinear resistor circuit equation A

(96)

After time scaling by dτ = i dt, Eq. (94) assumes the equivalent form


Hamiltonian system

∂H dq = , dτ ∂i

di = −f (i), dt

(104)

Nonlinear resistor circuit equation B

di = j, dt

    

  dj  L = −M (i)j. dt

(97)

or the

    

di ∂H   =− ,  dτ ∂q

(103)

Consider next the circuit in Fig. 32(b). Its differential equation is given by either the

L = 0.

df (i) . di

(105)

If j = 0, Eq. (105) can be written as

where i = 0. The Hamiltonian of Eq. (97) is defined by: Hamiltonian

1 j

H(q, i) =

f (q) + i. L

1 j

(98)

f (q) + i = 0. L

(106)

 M (i)   =− .  L

Pfaff’s equation

1 j 1 j where

v = f (i),

(100)

where f ( · ) is a scaler function. By differentiating Eq. (100) with respect to time variable t, we obtain u = M (i)j,

di dt dj dt

∂H = , ∂j

     

 ∂H   =− ,  ∂i

(107)

(101)

di , dt

f (i) M (i) di + 1 dj = + j, (108) H(i, j) = L L and the symbol denotes an “indefinite integral.” Pfaff’s equation (107) has the solution H(i, j) = H0 (H0 is any constant), since

where and j =

= 1,

     

Consider Chua’s diode described by a nonlinear relation

dv dt

dj dt

equation describing a conservative system:

(99)

8.2. Nonlinear resistor Hamiltonian circuit

u=

di dt

This equation can be recast into the form of Pfaff ’s

The Hamiltonian H(q, i) is conserved on each trajectory, q(τ ) and i(τ ) with initial condition (q(0), i(0)). It follows from Eq. (91) that the trajectories of Eqs. (92) and (97) are constrained on the space: H(q, i) =

(102) 1330001-24

∂H di ∂H dj dH(i, j) = + dt ∂i dt ∂j dt di 1 di dj 1 dj + =− j dt dt j dt dt = 0.

(109)

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After time scaling by dτ = j dt, Eq. (107) assumes the equivalent form

The Hamiltonian H(i, j) is conserved on each trajectory, i(τ ) and j(τ ) with initial condition (i(0), j(0)). It follows from Eq. (104) that the traHamiltonian system jectories of Eqs. (105) and (110) are constrained on  ∂H  di the space: = ,   dτ ∂j  f (i) + j = 0. (112) H(i, j) = (110) L ∂H  dj   =− , The Hamiltonian systems (97) and (110) differ dτ ∂i only in symbols, but are identical in form. They are duals. Hence, we obtain the following relation where j = 0. The Hamiltonian of Eq. (110) is between the corresponding state variables: defined by: One-to-one correspondence Hamiltonian Eq. (97) Eq. (110) f (i) (113) (q, i) ⇐= =⇒ (i, j) + j. (111) H(i, j) = L

(a)

(b)

(c)

(d)

Fig. 33. Trajectories from (a) Eq. (115), (b) Eq. (116), (c) Eq. (122), and (d) Eq. (123). These trajectories are duals, and they tend to the origin as t → ∞. Initial conditions for Eqs. (115) and (116): q(0) = −1.9, i(0) = −f (−1.9) ≈ 4.18633. Initial conditions for Eqs. (122) and (123): i(0) = −1.9, j(0) = −f (−1.9) ≈ 4.18633. In this case, H(q, i) = 0 and H(i, j) = 0. 1330001-25

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We summarize the above results as follows:

The Hamiltonian of Eq. (118) is given by H(q, i) = f (q) + i

Theorem 4. The dynamics of the memristor

Hamiltonian circuit in Fig. 32(a) is identical to that of the nonlinear resistor Hamiltonian circuit in Fig. 32(b). They are duals.

=

q3 + q + i = H0 , 3

duals

Example 2 [Memristor Hamiltonian circuit]. Choose


(119)

where i = 0. Thus, Eq. (118) has the integral

Eq. (97) ⇐====⇒ Eq. (110)

  q3 + q, f (q) = 3   L = 1.

q3 + q + i, 3

(114)

(120)

where H0 is a constant and i = 0. It follows from Eq. (115) that the trajectories of Eqs. (116) and (118) are constrained on the space H(q, i) = 0. We show the trajectory of Eq. (116) in Fig. 33. The Hamiltonian (120) is shown in Fig. 34. Observe

In this example, Eqs. (91) and (92) assume the form Memristor circuit equation A

3 q dq =− +q , dt 3

(115)

and Memristor circuit equation B

dq = i, dt

      (116)

  di  = −(q 2 + 1)i.  dt

(a)

If i = 0, Eq. (116) can be transformed into the form of Pfaff’s equation Pfaff’s equation

dq = 1, dt   1 di  = −(q 2 + 1).  i dt

1 i

      (117)

After time scaling by dτ = idt, we obtain the Hamiltonian system: Hamiltonian system

∂H dq = = 1, dτ ∂i

     

 ∂H  di  =− = −(q 2 + 1).  dτ ∂q

(118)

(b)

Fig. 34. Hamiltonian (119). (a) Three-dimensional plot of the Hamiltonian (119). Red line denotes the trajectory with the elevation H(q, i) = 0. (b) Contours of the Hamiltonian (119). Several curves along which the Hamiltonian has a constant value are illustrated on the (q, i)-plane. Red curve denotes the trajectory with H(q, i) = 0. The horizontal axis is the q-axis and the vertical axis is the i-axis. Labels on contour lines denote the elevation (height). 1330001-26

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that the trajectories are constrained on the space H(q, i) = 0, and they tend to the origin as t → ∞.

The Hamiltonian of Eq. (125) is given by i3 + i + j, (126) 3 where i = 0. Thus, Eq. (125) has the integral H(i, j) = f (i) + j =

Example 3 [Nonlinear resistor Hamiltonian circuit]. Choose

  i3 f (i) = + i, 3   L = 1.

(121)

In this example, Eqs. (104) and (105) assume the form


Nonlinear resistor circuit equation A

3 di i =− +i , dt 3

i3 + i + j = H0 , (127) 3 where H0 is a constant and i = 0. It follows from Eq. (122) that the trajectories of Eqs. (123) and (125) are constrained on the space H(i, j) = 0. We show the trajectories of Eqs. (122) and (123) in Fig. 33. Observe that the trajectories are identical, and they tend to the origin as t → ∞.

9. Nonvolatile Circuits

(122)

Memristors exhibit nonvolatility property. We

examine whether a memristor circuit and its dual

circuit have this property.

and Nonlinear resistor circuit equation B

     

di = j, dt

9.1. Memristor circuit Consider the circuit in Fig. 35(a). The q–ϕ curve of the memristor is given by

(123)

  dj  = −(i2 + 1)j.  dt

ϕ = f (q),

(128)

where f ( · ) is a scalar function. We obtain the fol-

lowing equations from Fig. 35(a):

     v(t) = M (q(t))i(t),      t  i(τ )dτ,  q(t) =  0       dq(t)   i(t) = , dt

If j = 0, Eq. (123) can be transformed into the form of Pfaff’s equation Pfaff’s equation

1 j 1 j

di dt dj dt

     

= 1,

ϕ = f (q),

(124)

   = −(i + 1).  2

(129)

where M (q) dfdq(q) , and we assume that i(t) = 0 for t ≤ 0. From Eq. (129), the terminal variables i(t) and v(t) can be described as After time scaling by dτ = jdt, we obtain the Hamiltonian system: Hamiltonian system

∂H di = = 1, dτ ∂j

     

  ∂H dj  =− = −(j 2 + 1).  dτ ∂i

(i(t), v(t)) = (i(t), M (q(t))i(t)).

(130)

This memristor circuit is endowed with the following nonvolatility property: (125)

1330001-27

dq = i = 0, dt

when i = 0.

(131)

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Fig. 35. Nonvolatile circuits. (a) Memristor circuit driven by a current source i(t), and (b) dual nonlinear resistor circuit where the current entering the Chua’s diode is h(t) and the input to the differentiator on top is V (t).

That is, (q, i) = (q, 0) is an equilibrium point of the equation, for any value of q. This means that this circuit does not lose the value of q when the current i becomes zero at the instant when the power is switched off. If the power is switched off at t = t0 , then q(t) and M (q(t)) hold the values unchanged at q(t0 ) and M (q(t0 )), respectively.

t ≤ 0. From Eq. (133), the terminal variables i(t) and u(t) can be described as (i(t), u(t)) = (i(t), M (h(t))i(t))

This circuit is also endowed with the following nonvolatility property:

9.2. Nonlinear resistor circuit

dh = i = 0, dt

Consider the circuit in Fig. 35(b). The v–i curve of Chua’s diode is given by i = f (v),

(132)

where f ( · ) is a scalar function. We obtain the following equations from Fig. 35(b):  v = f (i),      u(t) = M (h(t))i(t),        dv(t)   , u(t) =   dt (133) t     i(τ )dτ,  h(t) =   0       dh(t)   , i(t) =  dt where the current entering Chua’s diode is h(t), and (h) , and we assume that i(t) = 0 for M (h) dfdh

(134)

when i = 0.

(135)

That is, if the power is switched off at t = t0 , then h(t) and M (h(t)) hold the values unchanged at h(t0 ) and M (h(t0 )), respectively. Hence, we obtain the relationship between the state variables: One-to-one correspondence

Eq. (130)

Eq. (134)

(i(t), q(t), v(t)) ⇐= =⇒ (i(t), h(t), u(t))

(136)

Note that i(t) must be identical since it is a current source. We summarize the above results as follows: Theorem 5. The two circuits shown in Fig. 35

exhibit nonvolatility property. They are duals.

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10. Synchronization of Dual Circuits It was known that it is possible to construct a set of chaotic systems so that their common signals will have identical, or synchronized behavior [Pecora & Carroll, 1990]. In this section, we show that it is possible to synchronize dual Chua’s circuits. Consider the nonlinear circuit in Fig. 36. In this circuit, two Chua’s circuits are coupled by a voltage follower and a resistor Rx . The dynamics of the circuit is given by a set of differential equations


Coupled Chua’s circuits

  dv1 v2 − v1  = − f (v1 ), C1   dt R      v2 − v1 dv2 = −i3 − , C2  dt R       di3   L = v2 ,  dt C1

dva dt

C2

dvb dt

L

dic dt

Chua’s circuit (A)

 vb − va v1 − va   = − f (va ) + ,  R Rx      vb − va = −ic − ,   R        = vb ,

(137)

Chua’s circuit (B)

where f (v) = bv + 0.5(a − b)(|v + 1| − |v − 1|).

(138)

Here, a and b are some constants. If Rx = 0, then the dynamics can be described as Master-slave systems

  v2 − v1 dv1  C1 = − f (v1 ),   dt R      v2 − v1 dv2 C2 = −i3 − ,  dt R       di3   = v2 , L  dt  vb − v1  dvb  = −ic − , C2   dt R   dic   = vb . L  dt

Masters circuit (A)

(139)

Slave circuit (B)

1330001-29

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Fig. 36. Schematic diagram for synchronizing Chua’s circuits. Two Chua’s circuits are coupled by a voltage follower and a resistor Rx (top).

It is well-known that the master circuit (A) and the slave circuit (B) would synchronize [Pecora & Carroll, 1990], that is, |v2 (t) − vb (t)| → 0, for t → ∞ (140) |i3 (t) − ic (t)| → 0. If Rx is sufficiently small, and if the initial conditions of circuits (A) and (B) in Eq. (137) are sufficiently close, the two Chua’s circuits would also synchronize [Chua et al., 1992]. We show the chaotic trajectories and Lissajous curves of the

synchronized attractors in Figs. 37 and 38. Observe that the two Chua’s circuits are synchronized, and the Lissajous curves consist of 45◦ lines, since the two variables have the same value as t → ∞. The circuit parameters and the initial conditions are given by   1 1  ,  C1 = , C2 = 1, L1 =  10 14.87  (141) 1    a = −1.27, b = −0.68, Rx = , 20 

(a)

(b)

Fig. 37. Synchronized attractors from two coupled Chua’s circuits. Initial condition: v1 (0) = 0, v2 (0) = 0.1, i3 (0) = 0.1, 1 1 1 , C2 = 1, L1 = 14.87 , a = −1.27, b = −0.68, Rx = 20 . va (0) = 0.2, vb (0) = 0.2, ic (0) = 0.2. Circuit parameters: C1 = 10 (a) Chua’s circuit (A) and (b) Chua’s circuit (B). 1330001-30

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Fig. 38. Synchronized trajectories from the two Chua’s circuits in Fig. 36 and described by Eq. (137). Lissajous curves on the (v1 , va )-plane, (v2 , vb )-plane, and (i3 , ic )-plane are illustrated from left to right. Observe that the Lissajous curves are 45◦ lines.

and v1 (0) = 0,

v2 (0) = 0.1,

va (0) = 0.2,

vb (0) = 0.2,

i3 (0) = 0.1,

ic (0) = 0.2,

(142)

respectively.

We can also synchronize the dual circuits shown in Fig. 39. In this circuit, a Chua’s circuit and a memristor Chua’s circuit are coupled by a differentiator and a resistor. The dynamics of this circuit is given by

Coupled dual circuits

  v2 − v1 dv1  = − f (v1 ), C1   dt R      v2 − v1 dv2 = −i3 − , C2  dt R       di3   = v2 , L  dt C1

dϕa dt

C2

dϕb dt

L

dqc dt

Chua’s circuit

  ϕb − ϕa  = − f (ϕa ) + K,   R      ϕb − ϕa = −qc − ,  R         = ϕb , 

(143)

Memristor Chua’s circuit

where

  t  dv1 (τ )   − v (τ ) a   dτ  K=  dτ  −∞ Rx     v1 (t) − v1 (−∞) − ϕa (t)  ,  =  Rx 

    ϕa va (t)dt,    −∞    t  vb (t)dt, ϕb   −∞    t     ic (t)dt.  qc 



(144)

and

t

−∞

1330001-31

(145)

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Fig. 39. Schematic diagram for synchronizing two dual circuits. A Chua’s circuit (left) and a memristor Chua’s circuit (right) are coupled by a differentiator and a resistor Rx (top).

Observe that Eqs. (137) and (143) are identical if v1 (−∞) = 0. Hence, if Rx is sufficiently small, and if v1 (−∞) = 0, the two dual circuits would synchronize as t → ∞. We show the chaotic trajectories and Lissajous curves of the synchronized attractors in Figs. 40 and 41. Observe that the two circuits are synchronized, and the Lissajous curves are 45◦ lines, since the two variables have the same value as t → ∞. The circuit parameters and the initial conditions are given by

1 C1 = , 10 a = −1.27,

C2 = 1,

1 , L1 = 14.87

b = −0.68,

     

and v1 (0) = 0, ϕa (0) = 0.2,

(a)

v2 (0) = 0.1, ϕb (0) = 0.2,

(146)

1    Rx = , 20 

i3 (0) = 0.1, qc (0) = 0.2,

(147)

(b)

Fig. 40. Synchronized attractors from (a) a Chua’s circuit and (b) a memristor Chua’s circuit. Initial condition: v1 (0) = 1 1 , C2 = 1, L1 = 14.87 ,a = 0, v2 (0) = 0.1, i3 (0) = 0.1, ϕa (0) = 0.2, ϕb (0) = 0.2, qc (0) = 0.2. Circuit parameters: C1 = 10 1 −1.27, b = −0.68, Rx = 20 . 1330001-32

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Fig. 41. Synchronized trajectories of Chua’s circuit and memristor Chua’s circuit in Eq. (143). Lissajous curves on the (v1 , ϕa )-plane, (v2 , ϕb )-plane, and (i3 , qc )-plane are illustrated from left to right. Observe that the Lissajous curves are 45◦ lines.

respectively. In this case, we have substituted v1 (−∞) = 0 in Eqs. (143)–(144). Note that in this case, v1 (t) and va (t) in Eq. (143) do not synchronize as shown in Fig. 42. Finally, we show the schematic diagram for synchronizing dual circuits in Figs. 43 and 44. The dynamics of the circuit in Fig. 43 is given by Coupled memristor Chua’s circuits

 dϕ1 ϕ2 − ϕ1 = − g(ϕ1 ),    dt R     dϕ2 ϕ2 − ϕ1  = −q3 − , C2 dt R       dq3    L = ϕ2 , dt

C1

Memristor Chua’s circuit (A)

 ϕb − ϕa ϕ1 − ϕa  dϕa  = − g(ϕa ) + , C1   dt R Rx      ϕb − ϕa dϕb = −qc − , C2  dt R       dqc   = ϕb , L  dt

(148)

Memristor Chua’s circuit (B)

where ϕ1

−∞

ϕ2 q3

v1 (t)dt, v2 (t)dt,

ϕb

i3 (t)dt,

qc

1330001-33

t

−∞

va (t)dt,

t

−∞

t

−∞

ϕa

t

−∞

t

vb (t)dt

t

−∞

ic (t)dt.

(149)

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Fig. 42.

Lissajous curve on the (v1 , va )-plane. Observe that the voltages v1 and va are not synchronized.

Fig. 43. Schematic diagram for synchronizing two memristor Chua’s circuits. Two memristor Chua’s circuits are coupled by a voltage follower and a resistor Rx (top).

Fig. 44. Schematic diagram for synchronizing two dual circuits. A Chua’s circuit (left) and a memristor Chua’s circuit (right) are coupled by an integrator and a resistor Rx (top). 1330001-34

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The dynamics of the circuit in Fig. 44 is given by Coupled dual circuits


  v2 − v1 dv1  = − f (v1 ) + K, C1   dt R      v2 − v1 dv2 = −i3 − , C2  dt R       di3   L = v2 ,  dt C1

dϕa dt

C2

dϕb dt

L

dqc dt

  ϕb − ϕa  = − f (ϕa ),   R      ϕb − ϕa = −qc − ,  R         = ϕb , 

Chua’s circuit

(150)

Memristor Chua’s circuit

where K= =

t −∞

va (τ ) dt − v1 (t) Rx

ϕa (t) − v1 (t) . Rx

(151)

Observe that Eqs. (148) and (150) are identical to Eq. (137). Hence, if Rx is sufficiently small, and if the initial conditions of the two subsystems are sufficiently close, they would synchronize as t → ∞.

11. Identical Dynamics Consider a nonlinear circuit which consists of capacitors, inductors, and resistors. Replace all nonlinear elements with their corresponding memory elements as shown in Fig. 45. Then, we get a new memoryelement circuit. The relationship between these two circuits is described below:

Fig. 45. Dual circuits. The dynamics of these two circuits are identical. The two L-C-R circuits (colored moccasin) are identical, and contain linear inductors, linear capacitors, and linear resistors. The memory element circuit (b) is obtained by replacing all nonlinear elements (light-blue) in the circuit (a) with memory elements (yellow): meminductor, memcapacitor, and memristor (from the top to the bottom).

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M. Itoh & L. O. Chua Table 1. Relationship of corresponding circuit variables and equations between the nonlinear circuit and the memory element circuit.

⇐⇒

nonlinear circuit Basic circuit variables


(i, v)

(152)

memory element circuit Basic circuit variables

⇐⇒

(q, ϕ)

(153)

Kirchhoff circuit laws

Kirchhoff circuit laws

• Kirchhoff current law The algebraic sum of all currents im flowing into any node is zero: im = 0. (154) m

• Kirchhoff voltage law The algebraic sum of all voltages vn around any closed circuit is zero: vn = 0. (155) n

Nonlinear elements

• Kirchhoff charge law The algebraic sum of all charges qm flowing into any node is zero: qm = 0. (156) ⇐⇒

m

• Kirchhoff flux law The algebraic sum of all fluxes ϕn around any closed circuit is zero: ϕn = 0. (157) n

Nonlinear memory elements

• Capacitor

• Memcapacitor C(v)

dv =i dt

(158)

• Inductor di L(i) = v dt

dϕ =q dt

(161)

dq =ϕ dt

(162)

C(ϕ) ⇐⇒

• Meminductor

(159)

L(q)

• Resistor

• Memristor

v = f (i),

or

i = g(v).

(160)

ϕ = f (q),

or

q = g(ϕ). (163)

Linear elements

Linear elements

• Capacitor

• Capacitor C

dv =i dt

• Inductor L

C

(164) ⇐⇒

di =v dt

(165)

v = Ri,

or

(166)

i = Gv.

dq =ϕ dt • Memristor (resistor) ϕ = Rq,

1330001-36

(167)

• Inductor L

• Resistor

dϕ =q dt

or

q = Gϕ.

(168)

(169)

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dual equations. We next show an example of dual equations.


Fig. 46. Dual circuits. The dynamics of these three-element nonlinear circuits are identical. The memory-element circuit (b) is obtained by replacing all classical nonlinear elements (light-blue) in the circuit (a) with corresponding memory elements (yellow).

Note that a linear memristor is just a linear resistor. Observe that if we replace the state variables (i, v) in the left column with the state variables (q, ϕ), then we get the equations in the right column. Conversely, if we replace the state variables (q, ϕ) in the right column with the state variables (i, v), then we get the equations in the left column. It follows that the state variables in the left column are matched with the corresponding state variables in the right column. We refer to this situation as a one-to-one correspondence, and denote this relation by the symbol ⇐⇒. Thus, we can easily prove the following theorem.

Theorem 6. Consider a circuit consisting of

capacitors, inductors, and resistors. Replace all nonlinear elements in this circuit with their dual type memory elements. Then the dynamics of this new memory-element circuit for the state variables (q, ϕ) is identical to that of the original nonlinear circuit for the state variables (i, v). That is, they are duals. Furthermore, the dynamics of memory-element circuits can be simulated by using nonlinear circuits.

Recall that two systems are said to be duals of each other if we can exhibit some kind of one-toone correspondence between the variables of the two systems. Two equations which differ only in symbols but are otherwise identical are said to be

Example 4. Consider the three-element nonlinear circuits in Fig. 46. Applying Kirchhoff laws to the nonlinear circuit in Fig. 46(a), we obtain a set of equations which define the relation among two fundamental circuit variables, namely, the current and the voltage:   di  L(i) = v − f (i),   dt (170)  v  dv  = −i − ,  C(v)  dt R

where L(i), C(v), and R denote a nonlinear inductance, a nonlinear capacitance, and a linear resistance, respectively, and f (i) denotes the v–i curve of the current-controlled Chua’s diode. Applying Kirchhoff laws to the circuit in Fig. 46(b), we obtain a set of equations which define the relation among two fundamental circuit variables, namely, the charge and the flux:   dq  L(q) = ϕ − f (q),   dt (171)  ϕ  dϕ  = −q − ,  C(ϕ)  dt R where L(q), C(ϕ) and R denote the meminductance of a meminductor, the memcapacitance of a memcapacitor, and a linear resistance, respectively, and f (q) denotes the q–ϕ curve of a charge-controlled memristor. These two equations differ only in symbols, but are otherwise identical. Thus, they are dual equations.

12. 2-Terminal Circuit Elements A comprehensive family of fundamental 2-terminal or 1-port circuit elements is defined in [Chua, 2003, 2012]. In this section, we characterize 2-terminal circuit elements from the view point of their one-toone correspondences. Corresponding to the pair of terminal voltage and current (i(t), v(t)), we can define an associative pair of (i(k) , v (k) ), where

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(k)

i

  dk i(t)    ,   dtk      i(t),   t   i(τ )dτ,    −∞    t τ|k|      · · ·  −∞ −∞

if k = 1, 2, . . . ∞ if k = 0 if k = −1 τ2 −∞

(172)

i(τ1 )dτ1 dτ2 · · · dτ|k| , if k = −2, −3, . . . − ∞


and

v

(k)

  dk v(t)    ,   dtk        v(t), t   v(τ )dτ,    −∞     t τ|k|     · · ·  −∞

−∞

if k = 1, 2, . . . ∞ if k = 0 if k = −1 τ2 −∞

(173)

v(τ1 )dτ1 dτ2 · · · dτ|k| , if k = −2, −3, . . . − ∞

where |k| denotes an integer. Similarly, we can define an associative pair of (q (k) , ϕ(k) ), where

q (k)

  dk q(t)    ,   dtk        q(t), t   q(τ )dτ,    −∞     t τ|k|     ···  −∞

−∞

if k = 1, 2, . . . ∞ if k = 0 if k = −1 τ2

−∞

(174)

q(τ1 )dτ1 dτ2 · · · dτ|k| , if k = −2, −3, . . . − ∞

and

ϕ(k)

 k    d ϕ(t) ,   k    dt      ϕ(t), t   ϕ(τ )dτ,    −∞    τ|k|    t   ···  −∞

−∞

if k = 1, 2, . . . ∞ if k = 0 if k = −1 τ2 −∞

ϕ(τ1 )dτ1 dτ2 · · · dτ|k| , if k = −2, −3, . . . − ∞

1330001-38

(175)

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or

We can now define 2-terminal elements by replacing the pair of variables (i, v) in Eq. (160) and (q, ϕ) in Eq. (163) with the associated pair of (i(m) , v (n) ) or (q (m) , ϕ(n) ), namely, (i, v) =⇒ (i(m) , v (n) ),

(q, ϕ) =⇒ (q (m) , ϕ(n) ),

where m and n are any positive integer, any negative integer, or zero. From Eqs. (160), (163), (166), and (169), we get the relationship between the following corresponding pairs of variables:

(176)

⇐⇒

Basic circuit variables: (i, v) Characteristic curve on (i(m) , v (n) )-plane


• Nonlinear element v

(n)

(m)

= f (i

(177)

Basic circuit variables: (q, ϕ) Characteristic curve on (q (m) , ϕ(n) )-plane

• Nonlinear element (m)

), or i

= g(v

(n)

). (178)

ϕ(n) = f (q (m) ), or q (m) = g(ϕ(n) ). (180)

⇐⇒

• Linear element

• Linear element

v (n) = Ri(m) , or i(m) = Gv (n) .

ϕ(n) = Rq (m) , or q (m) = Gϕ(n) .

(179)

(181)

Furthermore, we can assume the following Kirchhoff circuit laws for the mth order state variables: Kirchhoff circuit laws for (i(m) , v (n) )

• Kirchhoff current law The algebraic sum of all mth order element currents ij (m) flowing into any node is zero: ij (m) = 0. (182) j

⇐⇒

• Kirchhoff voltage law The algebraic sum of all nth order element voltages vk (n) around any closed circuit is zero: vk (n) = 0. (183)

k

Kirchhoff circuit laws for (q (m) , ϕ(n) )

• Kirchhoff charge law The algebraic sum of all mth order element charges qj (m) flowing into any node is zero: qj (m) = 0. (184) j

• Kirchhoff flux law The algebraic sum of all nth order element fluxes ϕk (n) around any closed circuit is zero: ϕk (n) = 0. (185)

k

Replacing the pair of variables (i, v) in Eqs. (158) and (159) with the associated pair (i(m) , v (n) ) or (q (m) , ϕ(n) ), we obtain the following four equations: • Capacitor type element dv (n) = i(m) dt • Inductor type element C(v (n) )

L(i(m) )

di(m) = v (n) dt

• Capacitor type element dϕ(n) = q (m) dt • Inductor type element C(ϕ(n) )

(186) ⇐⇒

L(q (m) )

(187)

1330001-39

dq (m) = ϕ(n) dt

(188)

(189)

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Since i(m) = q (m+1) and v (n) = ϕ(n+1) , the above are “dual” relationships. For example, the constitutive relation of the nonlinear resistors v = f (i) and

i = g(v),

(190)

q (1) = g(ϕ(1) ),

(191)

By integrating both sides of Eq. (197) with respect to time t, we obtain

C(v −∞

can be written as ϕ(1) = f (q (1) ) and

q = g(ϕ),

(192)


i(−1) = g(v (−1) ),

(193)

respectively. Note that we can derive Eqs. (186)–(189) from Eqs. (178)–(181). For example, by differentiating both sides of the equation: i(m−1) = h(v (n) ),

(194)

dv (n) dτ = ) dτ

t

i(m) dτ.

(198)

−∞

i(m−1) = h(v (n) ) + D,

(199)

D = i(m−1) (0) − h(v (n) (0)).

(200)

where

can be written as v (−1) = f (i(−1) ) and

(n)

Solving Eq. (198) with the initial condition at time t = 0, we obtain

respectively. Similarly, the constitutive relation of the nonlinear memristors ϕ = f (q) and

t

Hence, the constitutive relation i(m−1) = h(v (n) ) (n) cannot be uniquely obtained from C(v (n) ) dvdt = (n) i(m) . That is, C(v (n) ) dvdt = i(m) is well-defined at every point on the constitutive relation i(m−1) = h(v (n) ). Thus, we get the following theorem:

with respect to time t, we get

  di(m−1)  (m)  =i ,    dt      (n) (n) (n) dh(v ) dv dh(v ) =   dt dt dv (n)      (n)   (n) dv   , = C(v ) dt

Theorem 7. Equations (186)–(189) are welldefined at every point on the constitutive relation (178)–(181).

(195) Example 5. Consider the nonlinear capacitor whose q–v characteristic is given by

where h(v (n) ) is an indefinite integral of C(v (n) ), h(v (n) ) C(v (n) )dv (n) . (196)

q=

C(v (n) )

dt We also note the relation

= i(m) .

q= (197)

i(−1) =

Eq. (194)

C(v (n) )

(ϕ(1) )3 , 3

(202)

since v = ϕ(1) . We can also recast Eq. (201) into an (i, v)-space representation:

i(m−1) = h(v (n) )

(201)

We can recast Eq. (201) into a (q, ϕ)-space representation:

Hence, we obtain dv (n)

v3 . 3


(203)

since q = i(−1) . Equations (201)–(203) are identical, but expressed in different symbols. Differentiating both sides of Eq. (201) with respect to time t, we obtain

dv (n) = i(m) dt

Eq. (197)

v3 , 3

i = C(v)

dv , dt

(204)

where C(v) = v 2 . We note the following relation: 1330001-40

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is called the impedance and the symbol L denotes the Laplace transform operation (see Appendix B). Since it is often convenient to apply a sinusoidal signal (of the frequency ω) as a port voltage, it suffices to set s = jω, and the resulting Z(s) is given by

v3 q= 3 Eq. (201)


i = C(v)

Z(jω) −

(213)

which is called a Frequency-Dependent Capacitor [Chua, 2003]. If n = 3 and m = 0 for Eq. (209), we would obtain

dv dt

Eq. (204)


1 , (jω)Cω 2

In order to obtain Eq. (201) from Eq. (204), we have to solve a set of equations   dq  = i,   dt (205)   dv  = i, C(v)  dt

i . C

v (3) =

(214)

The Laplace transform of Eq. (214) under zero initial conditions is given by s3 L{v(t)} =

L{i(t)} , C

(215)

which can be recast into

under the initial condition v(0) = v0 ,

q(0) = q0 .

v(t)3 + K, 3

Z(s)

(207)

where 3

v0 . (208) 3 If K = 0, or if all initial conditions are assumed to be zero, namely, q0 = v0 = 0, then Eqs. (201) and (204) are in one-to-one correspondence. K = q0 −

Example 6. Consider the linear element defined by

i(m) , (209) C where v and i are the terminal voltage and current, respectively. If n = 0 and m = −3, we would obtain i(−3) . (210) v= C The Laplace transform of Eq. (210) under zero initial conditions is given by L{v(t)} = Z(s)L{i(t)},

(211)

where 1 , C s3

1 . C s3

(217)

Hence, Eqs. (210) and (214) have the same Z(s). Furthermore, if n − m = 3, Eq. (209) also has the same Z(s) = C1s3 under zero initial conditions. Hence, there are many distinct 2-terminal circuit elements which have the same impedance under zero initial conditions about some operating points. By differentiating Eq. (210) three times with respect to time t, we get Eq. (214). However, we note the relation

v (n) =

Z(s)

(216)

where

Solving Eq. (205), we obtain q(t) =

L{v(t)} = Z(s)L{i(t)},

(206)

(212) 1330001-41

v=

i(−3) C

Eq. (210)

plus terms involving kj tj where kj are

v (3) =

arbitrary constants.

i C

Eq. (214)

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Let us define x(t) i(−3) (t). Then, Eq. (214) can be recast into v (3) (t) =

x(3) (t) . C

(218)

The Laplace transform of Eq. (214) under nonzero initial conditions is given by s3 L{v(t)} − s2 v(0) − sv (1) (0) − v (2) (0)


=

1 3 s L{x(t)} − s2 x(0) C − sx(1) (0) − x(2) (0) .

Hence, if ak are all zero, or if all initial conditions are assumed to be zero, then Eqs. (210) and (214) are in one-to-one correspondence. Example 7. Consider the linear element defined by

Eq. (179) v (n) = Ri(m) ,

(224)

where v and i are the terminal voltage and current, respectively. If n = 0 and m = 4, we would obtain the equation v = Ri(4) .

(219)

(225)

Replacing x(t) in Eq. (219) with i(−3) (t), we obtain

The Laplace transform of Eq. (225) under zero initial conditions is given by

s3 L{v(t)} − s2 v(0) − sv (1) (0) − v (2) (0)

L{v(t)} = Z(s)L{i(t)},

(226)

Z(s) Rs4 .

(227)

=

1 3 s L{i(−3) (t)} − s2 i(−3) (0) C − si(−2) (0) − i(−1) (0) .

where (220)

Dividing both sides of Eq. (220) by s3 , we obtain

Z(jω) R(jω)4 = Rω 4 ,

v(0) v (1) (0) v (2) − − 3 s s2 s 1 i(−3) (0) L{i(−3) (t)} − = C s

L{v(t)} −

−

i(−2) (0) s2

−

i(−1) (0)

If we apply a sinusoidal signal (with frequency ω) as the port voltage, then Z(s) is given by

which is a Frequency-Dependent Resistor.5 Furthermore, if n = −4 and m = 0, we would obtain the equation v (−4) = Ri.

s3

.

(221)

2

i(−3) (t) + ak tk , C

L{v(t)} = RL{i(t)}, s4

i(−3) (0) , a0 = v(0) − C a1 = v (1) (0) −

i(−2) (0) , C

L{v(t)} = Z(s)L{i(t)},

(231)

Z(s) Rs4 .

(232)

where

            

       (−1)  1 i (0)  (2) v (0) − . a2 =   2 C 5

(230)

which can be recast into (222)

k=0

where

(229)

The Laplace transform of Eq. (229) under zero initial conditions is given by

Taking the inverse Laplace transform of Eq. (221), we get v(t) =

(228)

(223)

Hence, Eqs. (225) and (229) have the same Z(s). Furthermore, if n − m = −4, then Eq. (224) has the same Z(s) = Rs4 under zero initial conditions. Hence, there are many distinct 2-terminal circuit elements which have the same impedance. By differentiating Eq. (229) four times with respect to time t, we get Eq. (225). However, we

Frequency-dependent linear elements are defined in [Chua, 2003]. 1330001-42

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Dividing both sides of Eq. (235) by s4 , we obtain

note the relation v (−4) = Ri

L{v (−4) (t)} −

Eq. (229)

v (−2) (0) v (−1) (0) − s3 s4 i(0) i(1) (0) − = R L{i(t)} − s s2 i(2) (0) i(3) (0) − − . s3 s4

plus terms involving kj tj where kj are

−

arbitrary constants.

v = Ri(4)


Eq. (225) Let us define x(t) v (−4) (t). Then, Eq. (225) can be recast into x(4) = Ri(4) .

v (−4) (0) v (−3) (0) − s s2

The inverse Laplace transform of Eq. (236) is given by

(233)

v (−4) (t) = Ri(t) +

The Laplace transform of Eq. (233) is given by

3

ak tk ,

where a0 = v (−4) (0) − R i(0),

− x(2) (0)s − x(3) (0) = R s4 L{i(t)} − i(0)s3 − i(1) (0)s2 − i(2) (0)s − i(3) (0) . (234)

a1 = v (−3) (0) − R i(1) (0),

          

1 (−2) v (0) − R i(2) (0) ,  2       1 (−1)  (3) (0) − R i (0) . a3 = v 3

a2 =

Replacing x(t) in Eq. (234) with v (−4) (t), we obtain s4 L{v (−4) (t)} − v (−4) (0)s3 − v (−3) (0)s2

(238)

Hence, if ak are all zero, or if all initial conditions are assumed to be zero, then Eqs. (225) and (229) are in one-to-one correspondence.

(−1)

−v (0)s − v (0) 4 = R s L{i(t)} − i(0)s3 − i(1) (0)s2 − i(2) (0)s − i(3) (0) . (235)

(a)

(237)

k=0

s4 L{x(t)} − x(0)s3 − x(1) (0)s2

(−2)

(236)

Example 8. Consider the circuits in Fig. 47, which consist of a linear resistor with the resistance of 4Ω

(b)

(c)

Fig. 47. Two-element circuits consisting of a 4Ω linear resistor connected to a 2-terminal (α, β) element (yellow). The v–i (4) (−4) curves of the 2-terminal elements (yellow) are given by (a) v1 = i1 , (b) v2 = i2 , and (c) the yellow element in this case is (−4)

a time-varing circuit element described by v3

= i3 − 1 − t. 1330001-43

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and a 2-terminal element. The 2-terminal elements in Figs. 47(a) and 47(b) have the same impedance Z(s) = s4 (see Eqs. (227) and (232) for R = 1). Consider the circuit in Fig. 47(a). The dynamics of this circuit is given by a set of equations v1 = i1 (4) , (239) v1 = −4i1 . From Eq. (239), we obtain i1 (4) + 4i1 = 0.

(240)


Let ξ(t) be a solution of the differential equation (240). Then, we obtain ξ(t) = (b1 et + b2 e−t ) cos(t) + (b3 et + b4 e−t ) sin(t),

Consider the circuit in Fig. 47(b). The dynamics of this circuit is given by a set of equations  i2 = v2 (−4) ,  (243) v2 i2 = − .   4 From Eq. (243), we obtain v2 = v2 (−4) . 4

v3 (−4) = i3 − (1 + t).

(248)

Differentiating Eq. (248) with respect to time t, we obtain v3 = i3 (4) ,

(249)

which is equivalent to Eq. (239a). The Laplace transform of Eq. (248) under zero initial conditions is given by L{v3 (t)} 1 1 + = L{i (t)} − . (250) 3 s4 s s2 Hence, Eq. (250) can be recast into L{v3 (t)} = Z(s)L{i3 (t)} − (s3 + s2 ),

(251)

where Z(s) = s4 . The dynamics of the circuit is given by a set of equations  i3 = v3 (−4) + 1 + t, (252) v3  i3 = − . 4 From Eq. (252), we obtain −

v3 = v3 (−4) + 1 + t. 4

(253)

Let us define w(t) v3 (−4) (t). Then, Eq. (253) can be recast into w(4) + 4w + 4 + 4t = 0.

(245)

Let η(t) be a solution of Eq. (245). Then, we obtain

(254)

Let ρ(t) be a solution of Eq. (254). Then, we obtain

η(t) = (c1 et + c2 e−t ) cos(t) + (c3 et + c4 e−t ) sin(t),

4

Consider next the time-varying circuit in Fig. 47(c). The v–i characteristic of the timevarying 2-terminal circuit element (yellow) is given by

(244)

Let us define y(t) v2 (−4) (t). Then, Eq. (244) can be recast into y (4) + 4y = 0.

3

(241)

where bk (k = 1, 2, 3, 4) are constants. Therefore, i1 (t) and v1 (t) can be respectively written as  i1 (t) = ξ(t)     t −t   = (b1 e + b2 e ) cos(t)     t −t + (b3 e + b4 e ) sin(t),  (242)  v1 (t) = ξ (4) (t) = −4ξ(t)      = −4(b1 et + b2 e−t ) cos(t)      t −t − 4(b3 e + b4 e ) sin(t).

−

where ck (k = 1, 2, 3, 4) are constants. Therefore, i2 (t) and v2 (t) can be respectively written as  i2 (t) = v2 (−4) (t) = y(t) = η(t)      t −t   = (c1 e + c2 e ) cos(t)     t −t + (c3 e + c4 e ) sin(t),  (247)  v2 (t) = y (4) (t) = η (4) (t) = −4η(t)     = −4(c1 et + c2 e−t ) cos(t)      t −t − 4(c e + c e ) sin(t). 

ρ(t) = (r1 et + r2 e−t ) cos(t) (246) 1330001-44

+ (r3 et + r4 e−t ) sin(t) − (1 + t). (255)

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Therefore, v3 (t) and i3 (t) can be respectively written as  v3 (t) = w(4) (t) = ρ(4) (t) = −4ρ(t) − 4 − 4t     t −t   = −4(r1 e + r2 e ) cos(t)     t −t  − 4(r3 e + r4 e ) sin(t), i3 (t) = v3 (−4) (t) + 1 + t = ρ(t) + 1 + t = (r1 et + r2 e−t ) cos(t) + (r3 et + r4 e−t ) sin(t),

           


(256) where rk (k = 1, 2, 3, 4) are constants. Note that v3 (−4) (t) in Eq. (248) contains a term − (1 + t), that is, v3 (−4) (t) = −(1 + t) + (r1 et + r2 e−t ) cos(t) + (r3 et + r4 e−t ) sin(t).

(257)

Observe that Eqs. (242), (247), and (256) are identical except for the coefficients bk , ck , and rk . Thus, it is difficult to distinguish between the three circuits in Fig. 47 using only the terminal voltage vj (t) and current ij (t). Furthermore, these 2terminal elements are locally-active (i.e. not passive), since the instantaneous power is given by  v1 (t) i1 (t) = −4ξ 2 (t) ≤ 0,   2 v2 (t) i2 (t) = −4η (t) ≤ 0, (258)   2 v3 (t) i3 (t) = −4(ρ(t) + 1 + t) ≤ 0. Note that if bk , ck , and rk are nonzero, then |vj (t)| → ∞,

|ij (t)| → ∞,

(259)

as t → ∞ or t → −∞ (j = 1, 2, 3). However, if the initial conditions at time t = 0, vi

(j)

(0) =

(j) v0 ,

(i = 1, 2, 3, j = 0, 1, 2, 3)

That is, any order of time-derivatives are identical. However, the integrals of v2 (t) and v3 (t) are not identical, that is,  v2 (−1) (t) = v3 (−1) (t),     (−2) (−2) (t) = v3 (t), v2 (262) v2 (−3) (t) = v3 (−3) (t),      (−4) (−4) v2 (t) = v3 (t). Furthermore, the integrals of v1 (t) may also be different from the integrals of v2 (t) and v3 (t), since they can be written as  t   (−1) (−1)  (t) = v1 (τ )dτ + v1 (0), v1    0     t    (−1) (−2) (−2) (t) = v1 (τ ) dτ + v1 (0), v1   0 (263) t    (−2) (−3) v1 (−3) (t) = v1 (τ ) dτ + v1 (0),    0     t    (−3) (−4) (−4) (t) = v1 (τ ) dτ + v1 (0),  v1 0

where v1 (t) is given by Eq. (242), and v1 (−k) (0) (k = 1, 2, 3, 4) are initial conditions at time t = 0, which should be specified.

13. Generalized 2-Terminal Circuit Elements The 2-terminal circuit elements (178) and (180) can be generalized [Chua, 2003; Itoh, 1997] as follows: Characteristic curve on (i† , v † )-space

(260)

are given, then bk , ck , and rk are uniquely determined, and they are identical. In this case, the derivatives of v1 (t), v2 (t) and v3 (t) satisfy  v1 (0) (t) = v2 (0) (t) = v3 (0) (t),     v1 (1) (t) = v2 (1) (t) = v3 (1) (t),      (2) (2) (2) v1 (t) = v2 (t) = v3 (t), (261) v1 (3) (t) = v2 (3) (t) = v3 (3) (t),      v1 (4) (t) = v2 (4) (t) = v3 (4) (t),     .. .. ..  . . .

f (i† , v † ) = 0,

where f is a scalar function of i† i(−M ) , i(−M +1) , . . . , i(−1) , i, i(1) , i(2) , . . . , i(N −1) , i(N ) , v † v (−M ) , v (−M +1) , . . . , v (−1) , v, v (1) , v (2) , . . . , v (N −1) , v (N ) ,

(264)

(265)

where M, N are positive integers. We can also define a generalized nonlinear element as follows:

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Characteristic curve on (q † , ϕ† )-space

g(q † , ϕ† ) = 0,

(266)

where g is a scalar function of q † q (−M ) , q (−M +1) , . . . , q (−1) , q, q (1) , q (2) , . . . , q (N −1) , q (N ) , ϕ† ϕ(−M ) , ϕ(−M +1) , . . . , ϕ(−1) , ϕ, ϕ(1) , ϕ(2) , . . . , ϕ(N ) , ϕ(N ) ,

(267)

where M and N are positive integers. Furthermore, Kirchhoff circuit laws for these nonlinear 2-terminal elements can be given by


Kirchhoff current and voltage laws

Kirchhoff charge and flux laws

• Kirchhoff current law The algebraic sum of all kth order currents in (k) flowing into any node is zero: in (k) = 0. (268)

• Kirchhoff charge law The algebraic sum of all kth order charges qn (k) flowing into any node is zero: qn (k) = 0. (270)

n

⇐⇒

• Kirchhoff voltage law The algebraic sum of all kth order element voltages vn (k) around any closed circuit is zero: vn (k) = 0. (269) n

n

• Kirchhoff flux law The algebraic sum of all kth order element fluxes ϕn (k) around any closed circuit is zero: ϕn (k) = 0. (271) n

Here, equation: k ∈ −M, −M + 1, . . . , −1, 0, 1, . . . , N − 1, N.

v (2) − (1 − v 2 )v (1) + v − A sin(ωt) = 0,

(272) We next show the examples of these generalized 2terminal circuit elements. Example 9.

Consider

the

nonlinear

element

defined by f (i, v, v (1) , v (2) ) v (2) − (1 − v 2 )v (1) + v − i = 0,

(275)

where i = A sin(ωt), and A, ω are constants. Example 10. Consider the nonlinear 2-terminal element defined by

g(q, ϕ, ϕ(1) , ϕ(2) ) ϕ(2) + kϕ(1) + ϕ3 − q − B = 0, (276)

(273) where k and B are constants, and where v and i are the terminal voltage and current, respectively. In this example, M = 0 and N = 2 in Eq. (264). If the terminal current i is zero for all t, i.e. the terminal is open-circuited, Eq. (273) reduces to the Van der Pol equation: v (2) − (1 − v 2 )v (1) + v = 0.

(274)

By connecting a current source A sin(ωt) to this element and applying Kirchhoff current law for k = 0 to this circuit, we obtain the forced Van der Pol

t

i(τ )dτ,

q(t) = −∞

t

ϕ(t) =

v(τ )dτ. (277) −∞

The symbols v and i denote the terminal voltage and current, respectively. If we connect the current source

−B sin(t), if t ≥ 0, s(t) = (278) 0, if t < 0,

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to this element, then q(t) is given by t s(τ )dtτ q(t) =

with respect to time t. The arbitrary constant K can be determined from the initial condition at time t = 0, that is,

−∞

v 3 (0) − v(0) + v (1) (0). (285) 3 Therefore, if K = 0, or if all initial conditions are assumed to be zero, then Eqs. (282) and (283) are in one-to-one correspondence.

=

B(cos(t) − 1), 0,

if t ≥ 0, if t < 0.

K = v (−1) (0) +

(279)

Substituting q(t) into Eq. (276), we obtain the differential equation of the Japanese Attractor [Abraham & Ueda, 2001] ϕ(2) + kϕ(1) + ϕ3 = B cos(t),

Example 12. Consider a linear 2-terminal circuit element defined by

(280)

f (i(1) , i(3) , i(5) , v)


for t ≥ 0.

m1 i(1) + m3 i(3) + m5 i(5) − v = 0, (286)

Consider the nonlinear 2-terminal element defined by Example 11.

g(i, v (−1) , v, v (1) ) v (−1) +

v3 − v + v (1) − i = 0, 3

Z(s)L{i(t)} − L{v(t)} = 0, (281)

where v and i are the terminal voltage and current, respectively. In this example, M = 1 and N = 1 in Eq. (264). If the terminal current i is equal to zero, then v3 − v + v (1) = 0. 3

v (−1) +

where m1 , m3 , m5 are constants. The Laplace transform of Eq. (286) under zero initial conditions is given by

(282)

where the symbol L denotes the Laplace transform operation, and Z(s) m1 s + m3 s3 + m5 s5 ,

where

d(v(−1) ) dt

(288)

is called the impedance. If we apply a sinusoidal signal (with frequency ω) across the input terminals and set s = jω, the resulting Z(s) is given by

Differentiating this equation with respect to time t, we obtain the Van der Pol equation v (2) − (1 − v 2 )v (1) + v = 0,

(287)

Z(jω) = m1 (jω) + m3 (jω)3 + m5 (jω)5 = jωL(ω),

(283)

(289)

where = v. We note the relation

v (−1) +

L(ω) = m1 − m3 ω 2 + m5 ω 4 ,

v3 − v + v (1) = 0 3

is equivalent to a Frequency-Dependent Inductance [Chua, 2003]. From the viewpoint of circuit synthesis, there are many 2-terminal elements which are distinct but have the same small-signal impedance Z(s). For example, consider the 2-terminal element defined by

Eq. (282)

plus terms containing arbitrary constants

g(i(−2) , i, i(2) , v (−3) )

v (2) − (1 − v 2 )v (1) + v = 0

m1 i(−2) + m3 i + m5 i(2) − v (−3) = 0. (291)

Eq. (283) Equation (283) can be obtained by differentiating both sides of v (−1) +

v3 − v + v (1) = K, 3

(290)

(284)

The Laplace transform of Eq. (291) under zero initial conditions is given by m 1 1 2 + m + m s L{i(t)} − 3 L{v(t)} = 0, 3 5 2 s s (292)

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which can be recast into 3

5

(m1 s + m3 s + m5 s )L{i(t)} − L{v(t)} = 0, (293) that is, Z(s)L{i(t)} − L{v(t)} = 0, s3

(294)

s5 .

Hence, the 2where Z(s) = m1 s + m3 + m5 terminal elements (286) and (291) have the same impedance Z(s). We can obtain Eq. (286) by differentiating Eq. (291) three times with respect to time t. However, we note the relation


m1 i(−2) + m3 i + m5 i(2) − v (−3) = 0

resistor oscillator, which differs only in the symbols chosen for the state variables. The former oscillator can be defined on the (q, ϕ)-state space, and the latter oscillator can be defined on the (i, v)-state space. They are duals. Furthermore, we have studied the chaotic synchronization of these dual oscillators. We also studied memory element circuits and 2-terminal circuit elements from the view point of duality and one-to-one correspondence. Finally, we have presented an example of a 2-terminal circuit element such that the terminal voltage and current signals, and their time derivatives of any order are identical, but their time integrals are different.

Eq. (291)

Acknowledgment

This paper is supported in part by grant no. FA 9550-10-1-0290.

plus terms involving kj tj where kj are arbitrary constants

References

m1 i(1) + m3 i(3) + m5 i(5) − v = 0 Eq. (286) Equation (286) can be obtained by differentiating m1 i(−2) + m3 i + m5 i(2) − v (−3) −

2

ak tk = 0,

k=0

(295) three times with respect to time t. The arbitrary constant ak can be determined from the initial condition at time t = 0:   a0 = m1 i(−2) (0) + m3 i(0)     (2) (−3)  (0),  + m5 i (0) − v    (−1) (1)  (0) + m3 i (0)  a1 = m1 i   (3) (−2) (296) + m5 i (0) − v (0),      1 (2)  a2 = m1 i(0) + m3 i (0)    2      (4) (−1) (0) , + m5 i (0) − v (see Appendix C for more details). Therefore, if ak are all zero, or if all initial conditions are assumed to be zero, then Eqs. (286) and (291) are in one-to-one correspondence.

14. Conclusion We have shown that the dynamics of any memristor oscillator is identical to that of a “dual ” nonlinear

Abraham, R. H. & Ueda, Y. (eds.) [2001] The Chaos Avant-Garde: Memoirs of the Early Days of Chaos Theory (World Scientific, Singapore). Bilotta, E., Bossio, E. & Pantano, P. [2010] “Chua’s circuit for students in junior and senior high school,” Int. J. Bifurcation and Chaos 20, 1–28. Chua, L. O. [1969] Introduction to Nonlinear Network Theory (McGraw-Hill, NY). Chua, L. O. [1971] “Memristor — The missing circuit element,” IEEE Trans. Circuit Theory CT-18, 507– 519. Chua, L. O. & Kang, S. M. [1976] “Memristive devices and systems,” Proc. IEEE 64, 209–223. Chua, L. O., Komuro, M. & Matsumoto, T. [1986] “The double scroll family,” IEEE Trans. Circuits Syst. 33, 1072–1118. Chua, L. O. & Lin, G. N. [1990] “Canonical realization of Chua’s circuit family,” IEEE Trans. Circuits Syst. 37, 885–902. Chua, L. O., Itoh, M., Kocarev, Lj. & Eckert, K. [1992] “Experimental chaos synchronization in Chua’s circuit,” Int. J. Bifurcation and Chaos 2, 705–708. Chua, L. O. [1993] “Global unfolding of Chua’s circuit,” IEICE Trans. Fundam. E76-A, 704–734. Chua, L. O. [2003] “Nonlinear circuit foundations for nanodevices, Part I: The four-element torus,” Proc. IEEE 91, 1830–1859. Chua, L. O. [2012] “The fourth element,” Proc. IEEE 100, 1920–1927. Gandhi, G., Cserey, G., Zbrozek, J. & Roska, T. [2009] “Anyone can build Chua’s circuit: Hands-onexperience with chaos theory for high school students,” Int. J. Bifurcation and Chaos 19, 1113–1125.

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Itoh, M. & Tomiyasu, R. [1989] “Instruments for displaying the Lorenz maps and the Poincaré maps on a synchroscope,” Electron. Commun. Jpn. 72, 86–94. Itoh, M. [1997] “Synthesis of topologically conjugate chaotic nonlinear circuits,” Int. J. Bifurcation and Chaos 7, 1195–1223. Itoh, M. & Chua, L. O. [2008] “Memristor oscillators,” Int. J. Bifurcation and Chaos 18, 3183–3206. Itoh, M. & Chua, L. O. [2011] “Memristor Hamiltonian circuits,” Int. J. Bifurcation and Chaos 21, 2395– 2425. Madan, R. N. [1993] Chua’s Circuit: A Paradigm for Chaos (World Scientific, Singapore). Muthuswamy, B. [2010] “Implementing memristor based chaotic circuits,” Int. J. Bifurcation and Chaos 20, 1335–1350. Muthuswamy, B. & Chua, L. O. [2010] “Simplest chaotic circuit,” Int. J. Bifurcation and Chaos 20, 1567–1580. Pecora, L. M. & Carroll, T. L. [1990] “Synchronization in chaotic systems,” Phys. Rev. Lett. 64, 821–824. Strukov, D. B., Snider, G. S., Stewart, G. R. & Williams, R. S. [2008] “The missing memristor found,” Nature 453, 80–83.

Appendices Appendix A The Ohm’s law for a charge-controlled memristor is given by v = M (q)i,

(A.1)

where v, i, and M (q) denote the terminal voltage, the terminal current, and the memristance, respectively. In order to obtain the constitutive relation from Eq. (A.1), we have to solve a set of equations  dϕ    = v,   dt   (A.2) dq  = i,   dt     v = M (q)i, under the initial condition at time t = 0 ϕ(0) = ϕ0 ,

q(0) = q0 ,

(A.3)

where q0 and ϕ0 are constants. From Eq. (A.2), we obtain dq dϕ = M (q) . (A.4) dt dt Integrating both sides of Eq. (A.4) with respect to time t, we obtain

t

dϕ(τ ) dτ = dτ

t

dq(τ ) dτ. dτ

(A.5)

ϕ(t) − ϕ(−∞) = F (q(t)) − F (q(−∞)),

(A.6)

−∞

M (q(τ )) −∞

Thus, we get

where F (q) is an indefinite integral of M (q), F (q) M (q)dq. (A.7) Substituting the initial condition for t = 0 into Eq. (A.6), we get ϕ0 − ϕ(−∞) = F (q0 ) − F (q(−∞)).

(A.8)

Subtracting Eq. (A.8) from Eq. (A.6), we obtain ϕ(t) = F (q(t)) + ϕ0 − F (q0 ).

(A.9)

Hence, arbitrary constants of integration are unavoidable, if ϕ0 = F (q0 ).

Appendix B The Laplace transform of a function f (t) is defined by ∞ f (t)e−st dt F (s), (B.1) L{f (t)} = 0

which transforms f (t) to a function F (s) with a complex argument s = σ + jω. By assumption, f (t) = 0 for −∞ < t ≤ 0.

(B.2)

If f (t) = 1 for t ≥ 0, then ∞ 1 −st ∞ 1 −st e dt = − e = , (B.3) L{1} = s s 0 0 where Re{s} = σ > 0 so that e−st → 0 as t → ∞. It can be shown that  ∞ 1   −st te dt = 2 ,  L{t} =   s 0 (B.4) ∞   2  t2 e−st dt = 3 . L{t2 } =  s 0 We can also define the inverse Laplace transform: given a function F (s), its inverse Laplace transform L−1 {F (s)} is a function f (t), t ≥ 0, such that F (s) = L{f (t)}. It follows from Eqs. (B.3) and (B.4) that

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L

−1

1 1 −1 = t, = 1, L s s2 1 1 −1 = 2. L 3 s 2t

(B.5)

In general, the inverse Laplace transform L−1 is given by the following complex integral


f (t) = L−1 {F (s)} σ+iT 1 lim est F (s)ds, = 2πi T →∞ σ−iT

(B.6)

where σ is any real number such that the contour path of integration is in the region of convergence of F (s). The important property of the Laplace transform under zero initial condition is as follows: t 1 f (τ )dτ = F (s), L s 0 t τn−1 τ1 ··· f (τ )dτ dτ1 · · · dτn−1 L 0

=

0

0

where the symbol L denotes the Laplace transform operation, and ak and bk are given by  b1 = m5 i(0),      (1)  b2 = m5 i (0),     (−2)  (0) + m3 i(0) a0 = m1 i     (2) (−3)  (0),  + m5 i (0) − v   (−1) (1) (C.3) (0) + m3 i (0) a1 = m1 i     + m5 i(3) (0) − v (−2) (0),       1  (2)  a2 = m1 i(0) + m3 i (0)    2     (4) (−1) (0) . + m5 i (0) − v Dividing both sides of Eq. (C.2) by s3 , we obtain m 1 1 2 + m + m s L{i(t)} − 3 L{v(t)} 3 5 2 s s a0 a1 2a2 − 2 − 3 = 0. (C.4) − b1 s − b2 − s s s Taking the inverse Laplace transform of Eq. (C.4), we get m1 i(−2) + m3 i + m5 i(2) − v (−3)

1 F (s), sn

− b1 δ(t) − b2 δ(1) (t) −

L{f (1) (t)} = sF (s), L{f

(n)

(t)} = s F (s),

where L{f (t)} = F (s) and n is a positive integer. The Laplace transforms of f (n) (t) and f (−n) (t) under nonzero initial condition are given respectively by

= s F (s) − s

δ(t)

f (0) − · · · − f

(n−1)

(0),

F (s) + sn−1 f (−n) (0) + · · · + f (−1) (0) , = sn where n is a positive integer.

Appendix C

dH(t) , dt

 0     1 H(t) =  2     1

L{f (−n) (t)}

(C.6)

(t < 0) (t = 0)

(C.7)

(t > 0).

Since δ(t) = δ(1) (t) = 0 for t = 0, Eq. (C.5) can be written as 2 ak tk = 0, m1 i(−2) + m3 i + m5 i(2) − v (−3) − k=0

The Laplace transform of (1)

m1 i

(3)

+ m3 i

(C.8) (5)

+ m5 i

− v = 0,

(C.1)

is given by (m1 s + m3 s3 + m5 s5 )L{i(t)} − L{v(t)} 4

(C.5)

where δ(t) is the delta function which can be viewed as the derivative of the Heaviside step function H(t),

where

L{f (n) (t)} n−1

ak tk = 0,

k=0

n

n

2

3

2

for t > 0. Furthermore, if ai are all zero, or if the initial condition is assumed to be zero, then we would obtain m1 i(−2) + m3 i + m5 i(2) − v (−3) = 0,

− b1 s − b2 s − a0 s − a1 s − 2a2 = 0, (C.2)

for t > 0.

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(C.9)