Dynamics of Memristor Circuits We will consider in this paper the following broader generalization of an ideal memristor, which is called a memristive device in [Chua & Kang, 1976],

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International Journal of Bifurcation and Chaos, Vol. 24, No. 5 (2014) 1430015 (44 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S0218127414300158

Dynamics of Memristor Circuits

Makoto Itoh1-19-20-203, Arae, Jonan-ku, Fukuoka 814-0101, Japan

[email protected]

Leon O. ChuaDepartment of Electrical Engineering and Computer Sciences,

University of California, Berkeley, CA 94720, [email protected]

Received January 7, 2014

In this paper, we show that Hamilton’s equations can be recast into the equations of dissipa-tive memristor circuits. In these memristor circuits, the Hamiltonians can be obtained from theprinciples of conservation of “charge” and “flux”, or the principles of conservation of “energy”.Furthermore, the dynamics of memristor circuits can be recast into the dynamics of “idealmemristor” circuits. We also show that nonlinear capacitors are transformed into nonideal mem-ristors if an exponential coordinate transformation is applied. Furthermore, we show that thezero-crossing phenomenon does not occur in some memristor circuits because the trajectoriesdo not intersect the i = 0 axis. We next show that nonlinear circuits can be realized with fewerelements if we use memristors. For example, Van der Pol oscillator can be realized by only twoelements : an inductor and a memristor. Chua’s circuit can be realized by only three elements :an inductor, a capacitor, and a voltage-controlled memristor. Finally, we show an example oftwo-cell memristor CNNs. In this system, the neuron’s activity depends partly on the suppliedcurrents of the memristors.

Keywords : Hamiltonian; memristor; memristive device; dissipative circuit; conservation ofmomentum; conservation of charge and flux; conservation of energy; zero-crossing property;Newton’s equation; Hamilton’s equations; Chua’s circuit; CNN.

1. Introduction

In electrical circuits, Hamilton’s equations areformulated for lossless, nonlinear circuits [Chua &McPherson, 1974; Andronov et al., 1987; Bern-stein & Liberman, 1989]. It is generally believedthat dissipative systems do not have Hamilton’sequations because the trajectories are damped andhence not periodic. Recently, a dissipative memris-tor circuit was shown to have a Hamiltonian whosecontours are precisely the damped trajectories[Itoh & Chua, 2011]. However, the physical inter-pretation of Hamiltonians from memristor circuitsis not fully clarified.

In this paper, we show that Hamilton’s equa-tions can be recast into the equations of memristorcircuits. In these memristor circuits, the Hamilto-nian can be obtained from the principles of con-servation of “charge” and “flux”, or the principlesof conservation of “energy”. Similarly, in dis-sipative physical systems, Hamiltonians can beobtained from the principles of the conservationof momentum. Furthermore, the dynamics of 2-element “memristor” circuits can be recast intothat of 2-element “ideal memristor” circuits. Thesetransformed equations have the Hamiltonian as itsintegral. However, the time orientation of orbits

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M. Itoh & L. O. Chua

may not be preserved. We also show that nonlinearcapacitors are expressed as memristor if anexponential coordinate transformation is applied.Furthermore, although memristors possessed thezero-crossing property, that is, the terminal voltagev is zero whenever the terminal current i is zero,in some memristor circuits, the zero-crossing phe-nomenon does not occur because the trajectoriesdo not intersect the i = 0 axis. Finally, we showthat Van der Pol equation can be realized by twoelements: an inductor and a memristor. Chua’s cir-cuit [Madan, 1993] can be realized by only threeelements: an inductor, a capacitor, and a voltage-controlled memristor. It is well-known that Chua’scircuit with a typical double scroll attractor hasthree saddle-focuses as its equilibrium points. If thememristor exhibits the zero-crossing property, thenthe equilibrium points are transformed into saddle-centers from saddle-focuses. Furthermore, we showan example of two-cell memristor CNNs. In this sys-tem, the neuron’s activity partly depends on thesupplied currents of the memristors.

2. Memristors

Memristor is a 2-terminal electronic device, whichwas postulated in [Chua, 1971, 2012].

The memristor shown in Figs. 1 and 2 can bedescribed by a

(q, ϕ)-relation

ϕ = f(q) or q = g(ϕ), (1)

between the charge q and the flux ϕ. Its terminalvoltage v and the terminal current i are describedby an

(i, v)-relation

v = M(q)i or i = W (ϕ)v, (2)

where

v =dϕ

dtand i =

dq

dt. (3)

The two nonlinear functions M(q) and W (ϕ), calledthe memristance and memductance, respectively,are defined by

M(q)�=

df (q)dq

(4)

and

W (ϕ)�=

dg(ϕ)dϕ

, (5)

representing the slope of a scalar function ϕ = ϕ(q)and q = q(ϕ), respectively, called the memristorconstitutive relation.

The constitutive relation ϕ = f(q) cannot beuniquely determined from the terminal voltage vand the terminal current i of the memristor. Thatis, by integrating both sides of v = M(q)i, we obtainϕ = f(q) + D. Here D is a nonzero constant. Inother words, the memristance M(q) is well-definedat every operating point on the constitutive relationwhere ϕ = f(q) is differentiable. Hence, we concludeas follows [Itoh & Chua, 2013]:

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

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

• Memristance M(q) is uniquely defined viathe constitutive relation ϕ = f(q).

• Ohm’s law v = M(q)i is constrained on thespace of physical state ϕ = f(q).

• Memductance W (ϕ) is uniquely defined viathe constitutive relation q = g(ϕ).

• Ohm’s law i = W (ϕ)v is constrained on thespace of physical state q = g(ϕ).

A memristor characterized by a differentiableϕ − q (resp., q − ϕ) characteristic curve is pas-sive if, and only if, its small-signal memristanceM(q) (resp., small-signal memductance W (ϕ)) isnon-negative (see [Chua, 1971]). Since the instan-taneous power dissipated by a passive memristor isgiven by

P (t) = M(q(t)) i(t)2 ≥ 0 (6)

or

P (t) = W (ϕ(t)) v(t)2 ≥ 0, (7)

the energy flow into a passive memristor from timet0 to t satisfies ∫ t

t0

P (τ)dτ ≥ 0, (8)

for all t ≥ t0.

2.1. Generalization of idealmemristors

We will consider in this paper the following broadergeneralization of an ideal memristor, which is calleda memristive device in [Chua & Kang, 1976], or

now generically called a memristor [Chua, 2012],defined by

Generalized memristors:

dx

dt= f(x, i),

v = M(x)i,

(9)

where v and i denote the terminal voltage and cur-rent of the memristor, x is some internal physicalstate variable, and f(x, i) and M(x) are continuousscalar functions.

2.2. Nonvolatility property

Memristors exhibit the

Zero-crossing property:

v = M(x)i = 0 for i = 0. (10)

That is, the voltage v is zero whenever the current iis zero. This zero-crossing property is manifested inthe form of a Lissajous figure which always passesthrough the origin when i = 0.

The ideal memristor is endowed with the non-volatility property. When we switch off the power ofa charge-controlled memristor at t = t0, such that,i(t) = 0 for t > t0, then v(t) = 0 for t > t0. But thecharge q(t) satisfies

dq

dt= 0, (11)

for t > t0. Hence, q(t) = q(t0) for t ≥ t0. That is,the memristor did not lose the value of q when ibecame zero.

We next study the nonvolatility property ofa subclass of nonideal memristor (9) defined byf(x, i) = f(i)g(x). When we switch off the powerof the memristor at t = t0, that is, i(t) = 0 fort > t0, then v(t) = 0 for t > t0. The physical statex(t) satisfies

dx

dt= f(0)g(x), (12)

for t > t0. Consider the following three cases:

• Choose f(0) = 0. Then, Eq. (12) is given by

dx

dt= 0. (13)

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In this case, x(t) = x(t0) for t ≥ t0. That is,the memristor did not lose the value of x when ibecame zero.

• Choose f(0) = 1 and g(x) = −x3. Then Eq. (12)is given by

dx

dt= −x3. (14)

Thus, the physical state x(t) satisfies

limt→∞x(t) = 0. (15)

That is, the memristor did not hold the value ofx when i became zero. In this case, the memristoris “volatile”.

• Choose f(0) = 1 and g(x) = 1−x3. Then Eq. (12)is given by

dx

dt= 1 − x3. (16)

In this case, the physical state x(t) satisfies

limt→∞x(t) = sgn(x(t0))

�=

1 if x(t0) > 0,

0 if x(t0) = 0,

−1 if x(t0) < 0.

(17)

That is, the memristor did not lose the sign of xwhen i became zero.

Therefore, the nonvolatility property of the mem-ristor depends on the function f(0)g(x). In latersections, we discuss the case where f(i) has a sin-gularity at i = 0.

3. Newton’s Equations

Analogous electrical and mechanical systems havedifferential equations of the same form. Let us beginfrom Newton’s laws of motion, which are given bythree physical laws:

(I) First law: An object either is at rest ormoves at a constant velocity, unless anexternal force is applied to it.

(II) Second law: The relationship betweenan object’s mass m, its acceleration a,and the applied force F is F = ma.

(III) Third law: For every action there is anequal and opposite reaction.

The second law states that the rate of change ofmomentum is proportional to the imposed force,that is,

dp

dt=

d(mv)dt

= ma = F, (18)

where we assume that m does not change with time.Here, the momentum is traditionally represented bythe letter p, which is the product of two quantities,the mass m and velocity v:

p = mv, (19)

and the acceleration a is defined as the instanta-neous change in velocity v

a =dv

dt. (20)

According to Newton’s principle of determinacy[Arnold, 1978], all motions of a system are uniquelydetermined by their initial positions x(t0) and ini-tial velocities x(t0). They determine the accelera-tion, that is, there is a function G such that

Newton’s equation:

x = G(x, x, t), (21)

where

x(t0) =dx

dt

∣∣∣∣t=t0

, x(t0) =d2x

dt2

∣∣∣∣t=t0

. (22)

Equation (21) is the basis of classical mechanics,and it is called Newton’s equation [Arnold, 1978].We next show two examples of Newton’s equation.

3.1. Motion in a potential field

Choose

G(x, x, t) = −(

1m

)∂U(x)

∂x, (23)

where x denotes a position of a point of mass m.Then Eq. (21) can be described by

Newton’s equation in the potential field� �

mx = −∂U(x)∂x

, (24)

� �which states the motion of a point in a potentialfield with potential energy U(x).

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For example, in the case of “free fall”, that is,an object that is falling under the sole influence ofgravity, potential energy U(x) is given by

U(x) = gx, (25)

where g denotes the acceleration of gravity,9.81 m/s2 (meters per second per second). Then,Newton’s equation can be expressed as

mx = −∂U(x)∂x

= −g. (26)

3.2. Nonlinear dissipative system

Choose

G(x, x, t) = −k(x)xm

, (27)

where k(x) is a scalar function of x. Then Eq. (21)can be described by

Newton’s equation for a dissipative system� �

mx = −k(x)x, (28)� �which is a nonlinear dissipative system, where thedamping (friction) coefficient is a function of theposition x [Jeltsema & Doria-Cerezo, 2012]. If weintegrate both sides of Eq. (28) with respect to thetime variable t, that is,∫

mx dt = −∫

k(x)x dt, (29)

we would obtain

mx = −K(x) + D, (30)

where D is a constant of integration, and K(x) isdefined by

K(x)�=∫

k(x)x dt =∫

k(x)dx, (31)

which denotes the time integral of the dampingforce F . Here the symbol

∫denotes an indefinite

integral. Furthermore, Eq. (30) can be recast into

p + K(x) = D, (32)

where p is the momentum, which is defined byp = mx. The constant of integration D is deter-mined by the initial condition (x(0), p(0)), that is,

D = p(0) + K(x(0)). (33)

Equation (32) stipulates the law of conservation ofmomentum. Thus, we get the following theorem:

Theorem 1. Newton’s equation (28) can betransformed into an equation stating the lawof conservation of momentum.

In the case of linear dissipative systems, theresisting force F is proportional to velocity x ofthe mass. If the damping force F is expressed asF = −cx, that is, F is linearly dependent uponvelocity x, then we would obtain

mx = −cx, (34)

where c is the damping coefficient, which hasthe unit of “newton seconds per meter” (Ns/m).Furthermore, Eq. (32) can be expressed as

p + cx = D, (35)

where D = p(0) + cx(0) and (x(0), p(0)) is theinitial condition.

4. Classical Physical Systems

Hamiltonian mechanics is developed as a reformula-tion of classical mechanics. In Hamiltonian mechan-ics, a classical physical system is described by a setof canonical coordinates (x, p). Here, x is the posi-tion coordinate and p is the momentum. The timeevolution of the system is uniquely defined by the

Hamilton’s equations:

x =∂H(x, p)

∂p,

p = −∂H(x, p)∂x

,

(36)

where H(x, p) is the Hamiltonian, which corre-sponds to the total energy of the system. FromEq. (36), we obtain

dH(x, p)dt

=∂H(x, p)

∂xx +

∂H(x, p)∂p

p

=∂H(x, p)

∂x

∂H(x, p)∂p

− ∂H(x, p)∂p

∂H(x, p)∂x

= 0.

(37)

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It follows that Eq. (36) has a solution H(x, p) = H0

(H0 is any constant). The total mechanical energyin a system (i.e. the sum of the potential andkinetic energies) remains constant, which stipulatesthe principle of conservation of mechanical energy.Furthermore, the divergence of Eq. (36) is equal to0. Thus, the Hamilton’s equations (36) are conser-vative. We next show two examples of Hamilton’sequations.

4.1. Conservative systems

Consider the Newton’s equation defined by

Conservative system:

x = − 1m

∂U(x)∂x

, (38)

where U(x) denotes potential energy. If we multiplyboth sides of Eq. (38) by mx, we obtain

d

dt

(m

2x2)

= −dU(x)dt

. (39)

Hence, Eq. (38) has an integral

mx2

2+ U(x) =

p2

2m+ U(x) = D, (40)

where p�= mx and D is a constant of integration.

If we define the

Hamiltonian of conservative systems� �

H(x, p) =p2

2m+ U(x), (41)

� �then we would obtain the

Hamilton’s equations of conservative systems� �

x =∂H(x, p)

∂p=

p

m,

p = −∂H(x, p)∂x

= −∂U(x)∂x

.

(42)

� �The Hamiltonian (41) is the sum of the kineticenergy EK and the potential energy EP , which arerespectively defined by

Kinetic energy EK and the potential energy EP� �

EK =p2

2m,

EP = U(x).

(43)

� �Since Eq. (42) has a solution H(x, p) = H0 (H0

is any constant), the total mechanical energy in aHamiltonian system remains constant. It stipulatesthe principle of conservation of mechanical energy.The system is conservative since the divergence ofEq. (42) is equal to 0.

4.2. Dissipative systems

Consider the Newton’s equation defined by the

Dissipative system:

x = −k(x)m

x. (44)

Equation (44) has an integral

p + K(x) = D, (45)

where p = mx, D is a constant of integration, andK(x) is defined by

K(x)�=∫

k(x)dx. (46)

From the derivative of Eq. (45), we can obtain

x =p

m,

p = −k(x)m

p.

(47)

If k(x) > 0, the system is dissipative, since the diver-gence of Eq. (47) is less than 0.

Equation (47) can be expressed as

dp

dx= −

(k(x)m

p

)( p

m

) = −k(q), (48)

where p �= 0. Therefore, we can define a differential1-form

dH�= dp + k(x)dx = 0. (49)

From Eq. (49), we can obtain the HamiltonianH(x, p)

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Hamiltonian of dissipative systems� �

H(x, p) = p + K(x). (50)� �Hamilton’s equations are given by1

Hamilton’s equations of dissipative systems� �

dx

dτ=

∂H(x, p)∂p

= 1,

dp

dτ= −∂H(x, p)

∂x= −k(x),

(51)

� �where dτ = pdt. Hamilton’s equations (51) have asolution

H(x, p) = p + K(x)

= H0, (52)

where H0 is any constant. Thus, we get the follow-ing theorem:

Theorem 2. Hamilton’s equations (51) havea solution H(x, p) = p + K(x) = H0, whereH0 is any constant. Hence, it stipulates the“law of conservation of momentum” with aforce F = −k(x)x.

The two systems (47) and (51) are in one-to-one correspondence except at the singularity p = 0,although the time scaling dτ = pdt may not pre-serve the time orientation of orbits. That is, thetime orientation of orbits is not preserved in theregion p < 0. Note that x-axis is a continuous set ofequilibrium points of Eq. (47), since (x, p) = (0, 0)at p = 0. However, Hamilton’s equations (51) donot have an equilibrium point since x = 1 �= 0.Furthermore, Eq. (51) is conservative, even thoughEq. (47) is not conservative (dissipative).

The trajectories of Eqs. (47) and (51) are shownin Figs. 3 and 4. The following parameters are usedin our computer simulations:

(a) m = 1, K(x) = x3/3, k(x) = x2,(b) m = 1, K(x) = x2/2, k(x) = x.

(a) (b)

Fig. 3. Trajectories of Eq. (47) with (a) K(x) = x3/3 and (b) K(x) = x2/2. Arrows on the curves denote time orienta-tion of the orbits. Note that the x-axis is a continuous set of equilibrium points, that is, (x, p) = (0, 0) at p = 0. (a) Twotrajectories of Eq. (47) with K(x) = x3/3 and m = 1. They tend to the origin as t → ∞. Two initial conditions of theabove trajectories are given by (x(0), p(0)) = (2,−8/3), (−2, 8/3). (b) Three trajectories of Eq. (47) with K(x) = x2/2and m = 1. Trajectories starting from the first, second, and fourth quadrants tend to the x-axis as t → ∞. A trajec-tory starting from the third quadrant goes to infinity as t → ∞. Initial conditions of these three trajectories are given by(x(0), p(0)) = (−√

2.2,−0.1), (−√1.8, 0.1), (

√8,−3).

1After time scaling by dτ = pdt, Eq. (47) can be directly recast into Eq. (51) where p �= 0. For more details, see [Andronovet al., 1987; Nemytskii & Stepanov, 1989; Itoh & Chua, 2011]. Note that Eq. (51) can be expressed as Eq. (48).

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(a) (b)

Fig. 4. Trajectories of Hamilton’s equations (51) for (a) K(x) = x3/3 and (b) K(x) = x2/2. Blue arrows on the curves denotetime orientation of the orbits. Observe that the orbits of two systems (47) and (51) are in one-to-one correspondence exceptat the singularity p = 0, although the time scaling dτ = pdt may not preserve the time orientation of orbits. Furthermore,Eq. (51) does not have an equilibrium point, since x = 1 �= 0. (a) A trajectory starting from the second quadrant goes toinfinity as t → ∞. Initial condition of this trajectory is given by (x(0), p(0)) = (−2, 8/3), which is located on the contour ofH = 0 with K(x) = x3/3, m = 1. (b) A trajectory starting from the third quadrant goes to infinity as t → ∞. Initial conditionof this trajectory is given by (x(0), p(0)) = (−2.5,−2.125), which is located on the contour of H = 1 with K(x) = x2/2, m = 1.

Observe the following property of Hamilton’sequations (51):

• Time orientation of orbits is not preserved.• No equilibrium point exists on the (x, p)-plane.

4.3. Coordinate and time-scaletransformation

Let us recast Eq. (51) into a new Hamiltonian sys-tem, whose Hamiltonian has both a kinetic energyand a potential energy, by using coordinate andtime-scale transformation.

Case 1. p > 0

If we set p = s2

2m for p ≥ 0, we would obtain fromEq. (51)

dp

dτ=( s

m

) ds

dτ

= −k(x), (53)

where s = ±√2mp, which does not map the orbits

in a one-to-one manner. Here, p is defined to be thepseudo kinetic energy of the new state variable s.From Eqs. (51) and (53), we obtain

dx

dτ= 1,

( s

m

) ds

dτ= −k(x).

(54)

After time scaling by2

dτ =(m

s

)dτ, (55)

Eq. (54) assumes the equivalent form

dx

dτ=

s

m,

ds

dτ= −k(x),

(56)

where s �= 0. Equation (56) can be recast into

dx

dτ=

∂H(x, s)∂s

=s

m,

ds

dτ= −∂H(x, s)

∂x= −k(x),

(57)

2This time scaling maps orbits between systems (54) and (56) in a one-to-one manner except at the singularity s = 0, althoughit may not preserve the time orientation of orbits.

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where the Hamiltonian H(x, s) is given by

H(x, s) =s2

2m+ K(x). (58)

The Hamiltonian (58) is the sum of the pseudokinetic energy EK and the pseudo potential energyEP , which are respectively defined by

EK =s2

2m,

EP = K(x).

(59)

Observe that the pseudo kinetic energy EK , thepseudo potential energy EP , and the HamiltonianH do not represent real physical energy.

Compare the following relationship between thedissipative system and conservative system:

Newton’s equation� �

Dissipative system

x = − 1m

(∂K(x)

∂x

)x = −k(x)

mx

Conservative system

x = − 1m

∂U(x)∂x� �

Hamilton’s equations� �

Dissipative system

dq

dτ=

∂H(x, s)∂s

=s

m,

ds

dτ= −∂H(x, s)

∂q= −∂K(x)

∂x.

Conservative system

dx

dt=

∂H(x, p)∂p

=p

m,

dp

dt= −∂H(x, p)

∂x= −∂U(x)

∂x.

� �

Hamiltonian� �

Dissipative system

H(x, s) =s2

2m+ K(x)

Conservative system

H(x, p) =p2

2m+ U(x)

� �Observe that if K(x) = U(x), then these twoHamiltonians are equivalent, although their New-ton’s equations are distinct from each other. Notethat the time scaling (55) does not preserve the timeorientation of orbits in the region s < 0.

Case 2. p < 0

If we set p = − s2

2m for p ≤ 0, we would obtain fromEq. (51)

dp

dτ= −

( s

m

) ds

dτ= −k(x), (60)

where s = ±√−2mp, which does not map the orbitsin a one-to-one manner. From Eqs. (51) and (60),we obtain

dx

dτ= 1,

( s

m

) ds

dτ= k(x).

(61)

After time scaling by

dτ = −(m

s

)dτ, (62)

Eq. (61) assumes the equivalent form

dx

dτ= − s

m,

ds

dτ= −k(x),

(63)

where s �= 0. Equation (63) can be recast into

dx

dτ=

∂H(x, s)∂s

= − s

m,

ds

dτ= −∂H(x, s)

∂x= −k(x),

(64)

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where

H(x, s) = − s2

2m+ K(x). (65)

Here, the Hamiltonian (65) is the sum of the pseudokinetic energy EK and the pseudo potential energyEP , which are respectively defined by

EK = − s2

2m,

EP = K(x).

(66)

The Hamiltonian (65) has a negative pseudo kineticenergy. If we apply the reverse time scaling t = −τ ,

(a) (b)

(c)

Fig. 5. Trajectories of new Hamilton’s equations (51), (57) and (64) with K(x) = x3/3 and m = 1. Arrows on the curvesdenote time orientation of the orbits. The origin is an equilibrium point of Eqs. (57) and (64), that is, (x, s) = (0, 0). Allinitial conditions are located on the contour of H = 0. Observe that Hamilton’s equations (51), (57) and (64) do not preservethe time orientation of the orbits of Eq. (47) [see Fig. 3(a)]. (a) Trajectories of Eq. (57) with initial conditions (x(0), s(0)) =`−2.5,

q−2(−2.5)3

3

´ ≈ (−2.5, 3.23) and (x(0), s(0)) =`−0.2,−

q−2(−0.2)3

3

´ ≈ (−0.2,−0.073). (b) Trajectories of Eq. (64)

with initial conditions (x(0), s(0)) =`2.5,

q2(2.5)3

3

´ ≈ (2.5, 3.23) and (x(0), s(0)) =`0.2,−

q−2(−0.2)3

3

´ ≈ (0.2,−0.073).

(c) Trajectory of Eq. (51) with initial condition (x(0), p(0)) = (−2, 8/3). It is divided into two portions in blue and red, whichcorrespond to the red and blue trajectories in Fig. 3(a).

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(a) (b)

(c)

Fig. 6. Trajectories of new Hamilton’s equations (51), (57) and (64) with K(x) = x2/2 and m = 1. Arrows on the curvesdenote time orientation of the orbits. All initial conditions are located on the contour of H = 1. Observe that Hamilton’sequations (51), (57) and (64) do not preserve the time orientation of the orbits of Eq. (47) [see Fig. 3(b)]. (a) Trajectory ofEq. (57) with initial condition (x(0), s(0)) = (1, 1). (b) Trajectories of Eq. (64) with initial conditions (x(0), s(0)) = (3,

√7)

and (x(0), s(0)) = (−3,−√7). (c) Trajectory of Eq. (51) with initial condition (x(0), p(0)) = (−2.5,−2.125). It is divided into

three portions in blue, red, and purple, which correspond to the blue, red, and purple trajectories in Fig. 3(b).

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(a) (b)

(c) (d)

(e) (f)

Fig. 7. Three-dimensional plot and contour plot of Hamiltonians (50), (58) and (65). (a)–(b) Hamiltonian (50) for m = 1,K(x) = x3/3. (c)–(d) Hamiltonian (58) for m = 1, K(x) = x3/3. (e)–(f) Hamiltonian (65) for m = 1, K(x) = x3/3. Redthick curves denote the contours with the elevation H = 0. Arrows on the curves denote time orientation of the orbits. Twotrajectories in Fig. 7(d) are mapped into a blue portion of the trajectory in Fig. 7(b) by the function p = s2/2. Similarly, twotrajectories in Fig. 7(f) are mapped into a red portion of the trajectory in Fig. 7(b) by the function p = −s2/2. However, thetime orientation of the orbits is not preserved.

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(a) (b)

(c) (d)

(e) (f)

Fig. 8. Three-dimensional plot and contour plot of Hamiltonians (50), (58) and (65). (a)–(b) Hamiltonian (50) for m = 1,K(x) = x2/2. (c)–(d) Hamiltonian (58) for m = 1, K(x) = x2/2. (e)–(f) Hamiltonian (65) for m = 1, K(x) = x2/2. Blackthick curves denote the contours with the elevation H = 1. Arrows on the curves denote time orientation of the orbits. A redclosed trajectory in Fig. 8(d) is mapped into a red portion of the trajectory in Fig. 8(b) by the function p = s2/2. Similarly, ablue trajectory in the second and third quadrants of Fig. 8(f) are mapped into a blue portion of the trajectory in Fig. 8(b) bythe function p = −s2/2. A purple trajectory in the first and fourth quadrants of Fig. 8(f) are mapped into a purple portionof the trajectory in Fig. 8(b) by the function p = −s2/2. However, the time orientation of the orbits is not preserved.

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then the Hamilton’s equations (64) can have a posi-tive pseudo kinetic energy. Note that the time scal-ing (62) does not preserve the time orientation oforbits in the region s < 0. Compare the follow-ing relationship between the dissipative system andconservative system:


Dissipative system

dx

dτ=

∂H(x, s)∂s

= − s

m,

ds

dτ= −∂H(x, s)

∂q= −∂K(x)

∂x.

Conservative system

dx

dt=

∂H(x, p)∂p

=p

m,

dp

dt= −∂H(x, p)

∂x= −∂U(x)

∂x.

� �

Hamiltonian� �

Dissipative system

H(x, s) = − s2

2m+ K(x)

Conservative system

H(x, p) =p2

2m+ U(x)

� �The trajectories of Hamilton’s equations (51), (57)and (64) are shown in Figs. 5 and 6. The followingparameters are used in our computer simulations:

(a) m = 1, K(x) = x3/3, k(x) = x2,(b) m = 1, K(x) = x2/2, k(x) = x.

Observe the following relationship of Fig. 5:

• Two trajectories in Fig. 5(a) are mapped into ablue portion of the trajectory in Fig. 5(c) by thefunction p = s2/2.

• Two trajectories in Fig. 5(b) are mapped into ared portion of the trajectory in Fig. 5(c) by thefunction p = −s2/2.

• Time scaling does not preserve the time orienta-tion of orbits.

Observe the following relationship of Fig. 6:

• A closed orbit in Fig. 6(a) is mapped into a redportion of the trajectory in Fig. 6(c) by the func-tion p = s2/2.

• A trajectory moving in the second and thirdquadrants of Fig. 6(b) is mapped into a blue por-tion of the trajectory in Fig. 6(b) by the functionp = −s2/2.

• A trajectory moving in the first and fourth quad-rants of Fig. 6(b) is mapped into a purple por-tion of the trajectory in Fig. 6(c) by the functionp = −s2/2.

• Time scaling does not preserve the time orienta-tion of orbits.

The Hamiltonians (50), (58) and (65) and their con-tour plots are shown in Fig. 7 with K(x) = x3/3and m = 1. Thick red curves denote the con-tours with the elevation H = 0. We also show theHamiltonians (50), (58) and (65), and their con-tours with K(x) = x2/2 and m = 1 in Fig. 8. Thickblack curves denote the contours with the elevationH = 1. Observe relationships among the trajecto-ries of Eqs. (51), (57) and (64), which are denotedby thick curves.

5. Hamiltonians of ElectronicCircuits

The dynamics of physical systems can be simulatedby corresponding dual electronic circuits. Let usconsider first 2-element “lossless” and “dissipative”nonlinear circuits whose dynamics can be describedby conservative Hamilton’s equations.

5.1. Lossless LC-circuits

Consider the circuit in Fig. 9, which consists of alinear inductor and a nonlinear charge-controlledcapacitor. The dynamics of this circuit is describedby a set of differential equations:

Dynamics of a lossless LC-circuit:

dq

dt= i,

Ldi

dt= −g(q),

(67)

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Fig. 9. A 2-element Hamiltonian circuit, which consists ofa linear inductor and a nonlinear charge-controlled capacitorwith the characteristic curve v = g(q).

where the symbols q and i denote the charge andthe current of the nonlinear capacitor with the char-acteristic curve v = g(q), and L denotes the induc-tance of the inductor. This circuit is conservative,that is, lossless, since the divergence of Eq. (67) isequal to 0. A linear inductor is described by [Chua,1969]

ϕ = Li, (68)

where the flux ϕ(t) of the inductor is defined by

ϕ(t) = ϕL(t)�=∫

vL(τ)dτ

=∫

Ldi(τ)dτ

dτ = Li(t), (69)

that is, ϕ is an indefinite integral of vL.3 If we sub-stitute i = ϕ

L into Eq. (67), we would obtain

Dynamics of a lossless LC-circuit:

dq

dt=

ϕ

L,

dϕ

dt= −g(q).

(70)

Equation (70) can be recast into Hamilton’sequations:

Hamilton’s equations of the lossless circuit� �

dq

dt=

∂H(q, ϕ)∂ϕ

=ϕ

L,

dϕ

dt= −∂H(q, ϕ)

∂q

= −∂G(q)∂q

= −g(q),

(71)

� �where the Hamiltonian of Eq. (71) is given by

Hamiltonian of the lossless circuit� �

H(q, ϕ) =ϕ2

2L+ G(q). (72)

� �Here, G(q) is defined by

G(q)�=∫

g(q)dq. (73)

Let WL and WC be the stored energy in theabove inductor and capacitor during the time inter-val (t0, t1). Then we obtain

WL =∫ t1

t0

v(−i)dt

=∫ t1

t0

(L

di

dt

)idt

=∫ t1

t0

dϕ

dt

(ϕ

L

)dt

=∫ t1

t0

(ϕ

L

)dϕ

=ϕ2

2L

∣∣∣∣t1

t0

= EL

∣∣∣∣t1

t0

(74)

and

WC =∫ t1

t0

vidt

=∫ t1

t0

g(q)idt

3Note that vL + v = 0, that is, vL = −v. Thus, the flux of the capacitor is defined by

ϕC(t)�=

Zv(τ )dτ = −

ZvL(τ )dτ = −ϕL(t).

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=∫ t1

t0

g(q)dq

dtdt

=∫ t1

t0

g(q)dq

= G(q)∣∣t1t0

= EC

∣∣t1t0

, (75)

where EL and EC are defined by the

Energy stored in the lossless circuit� �

EL�=

ϕ2

2L,

EC�= G(q).

(76)

� �Equation (71) has a solution H(q, ϕ) = H0 (H0 isany constant), that is,

EL + EC = H0. (77)

It stipulates the “conservation law of energy” inlossless electrical circuits. Note that EC correspondsto the potential energy stored in the capacitor, andEL corresponds to the kinetic energy in classicalmechanics, since EL can be described as

EL =ϕ2

2L=

p2

2m, (78)

if we define

p�= ϕ,

m�= L.

(79)

Furthermore, from the derivative of Eq. (67),we can obtain the Newton’s equation

Newton’s equation of the lossless circuit� �

q = − 1L

∂G(q)∂q

= −g(q)L

. (80)

� �Compare Eq. (71) [resp., (80)] with Eq. (42) [resp.,(38)]. Observe that if x = q, p = ϕ, m = L, andU(·) = G(·), then they are equivalent.

Thus, we get the following well-known theorem:

Theorem 3. Hamilton’s equations (71) havea solution H(q, ϕ) = H0 (H0 is any constant).The function H(q, ϕ) considered to be thesum of the magnetic energy EL stored in theinductor (i.e. the kinetic energy of the induc-tor) and the electric energy EC stored in thecapacitor (i.e. the potential energy of thecapacitor). Furthermore, the total energy ina lossless circuit remains constant. It stip-ulates the “conservation law of energy” inlossless electrical circuits. The Hamiltonian ofEq. (71) can be obtained from these laws.

5.2. Exponential coordinatetransformation

We will now show that Eq. (70) can be recast intothe dynamics of a memristor circuit. Substituting

Exponential coordinate transformation� �

ϕ = L ln|j|, (81)� �

into Eq. (67)4 and replacing q with w, we obtain

Dynamics of a memristor circuit� �

dw

dt= ln|j|,

Ldj

dt= −g(w)j,

(82)

� �which indicates the dynamics of the circuit whichconsists of a linear inductor and a memristordescribed by

Characteristic of the memristor� �

dw

dt= ln|j|,

v = g(w)j,

(83)

� �where v and j denote the terminal voltage andcurrent of the memristor, and the physical statevariable w is the memristor charge.

4Note that j can be expressed as j = ei. Hence, we call this transformation an exponential coordinate transformation.

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It follows from Eq. (72) that Eq. (82) has anintegral

I(w, j) =L(ln|j|)2

2+ G(w) = I0, (84)

where I0 is an arbitrary constant and G(q) is definedby

G(w)�=∫

g(w)dw. (85)

Hence, the nonlinear capacitor in Fig. 9 is trans-formed into a memristor [Chua, 2012] for allnonzero memristor current j, if an exponential coor-dinate transformation is applied to the flux ϕ.Observe that Eq. (83) defines a nonideal memris-tor. Thus, we get the following theorem:

Theorem 4. A nonlinear capacitor with thecharacteristic curve v = g(q) is transformedinto a memristor if an exponential coordinatetransformation is applied to the LC-circuitequation (70).

Note that if |g(w)| < ∞, then the memristor(83) apparently satisfies the zero-crossing property,that is,

v = l(w, j)i = g(w)j = 0 for j = 0. (86)

However, we obtain from Eq. (82)

dj

dt→ 0 and

dw

dt→ −∞ for j → 0, (87)

which implies that the trajectory cannot cross the v-axis on the (j, v)-plane. That is, j(t) cannot be zero.Furthermore, Eq. (82) and its integral I(w, j) havea singularity at j = 0. Therefore, the zero-crossingphenomenon cannot be observed in the form of aLissajous figure, which always passes through theorigin when j = 0. In other words, the system (82)must oscillate either in the upper half plane j > 0,or the lower half plane j < 0 without violating thezero-crossing property.

Example 1. Consider the LC-circuit in Fig. 9. Ifthe characteristic curve of the charge controlledcapacitor is given by

v = g(q) = q3, (88)

then the dynamics of the LC-circuit is given by

dq

dt= i,

Ldi

dt= −q3,

(89)

or

dq

dt=

ϕ

L,

dϕ

dt= −q3,

(90)

where the flux ϕ(t) is defined by

ϕ(t) = ϕL(t)�=∫

vL(τ)dτ

=∫

Ldi(τ)dτ

dτ = Li(t), (91)

that is, ϕ is an indefinite integral of vL.5

Equation (90) can be recast into Hamilton’sequations:

dq

dt=

∂H(q, ϕ)∂ϕ

=ϕ

L,

dϕ

dt= −∂H(q, ϕ)

∂q= −q3,

(92)

where the Hamiltonian is defined by

H(q, ϕ) =ϕ2

2L+

q4

4. (93)

Contour lines of Eq. (93) consist of closed curves.Applying the exponential coordinate transforma-tion ϕ = L ln|j| into Eq. (90) and replacing qwith w, we obtain the dynamics of a memristorcircuit

dw

dt= ln|j|,

Ldj

dt= −w3j.

(94)

5Note that vL + v = 0, that is, vL = −v. Thus, the flux of the capacitor is defined by

ϕC(t)�=

Zv(τ )dτ = −


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(a) (b)

Fig. 10. Trajectories (closed orbits) of Eqs. (90) and (94) with L = 1. (a) A closed orbit of Eq. (90) with initial condition(q(0), ϕ(0)) = (1, 1). (b) Two closed orbits of Eq. (94) with initial condition (w(0), j(0)) = (1, e), (1,−e), where e denotesNapier’s constant (e ≈ 2.71828). These two orbits are mapped into the same closed orbit in Fig. 10(a) by the function(q, i) = (w, ln|j|).

The characteristic of the memristor is given by

dw

dt= ln|j|,

v = w3j.

(95)

Here, v and j denote the terminal voltage and cur-rent of the memristor, and w is a physical statevariable. Equation (94) has an integral

I(w, j) =L(ln|j|)2

2+

w4

4= I0, (96)

where I0 is an arbitrary constant. Note that theexponential coordinate transformation i = ln|j| isnot in one-to-one correspondence, since j = ±1 aremapped to the same i = 0.

Furthermore, if |w| < ∞, the memristor appar-ently satisfies the zero-crossing property, that is,v = w3j → 0 for j → 0. However, Eq. (94) andits integral I(w, j) have a singularity at j = 0,and dj

dt → 0 for j → 0. That is, j(t) cannot bezero. Hence, the zero-crossing property cannot beobserved in the form of a pinched hysteresis loop inthis circuit because the periodic current waveformis not bipolar. In other words, the system (94) doesnot violate the zero-crossing property.

We show the trajectories (closed orbits) ofEqs. (90) and (94) with L = 1 in Fig. 10. Two closedorbits in Fig. 10(b) are mapped into a closed orbit

in Fig. 10(a) by the function (q, ϕ) = (w, ln |j|).We next show a hysteresis loop of the memristor inFig. 11. The zero-crossing property is not applica-ble in this case. We finally show the HamiltonianH(q, ϕ) and the integral I(w, j) in Fig. 12.

5.3. Dissipative memristor circuits

Consider the circuit in Fig. 13, which consists ofa linear inductor and a memristor. The terminal

Fig. 11. Lissajous trajectories of the memristor (95) on the(j, v)-plane (L = 1). Since the trajectories of the memristordid not pass through the v-axis, the zero-crossing property isnot applicable in this example.

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(a) (b)

(c) (d)

Fig. 12. (a)–(b) Hamiltonian H(q, ϕ) and contours of Eq. (93). (c)–(d) Integral I(w, j) and contours of Eq. (96). Blackthick curves denote the contours of H(q,ϕ) = 0.75 (top) and the contours of I(w,ϕ) = 0.75 (bottom). Initial conditionsgiven in Fig. 10 are located on these contours. The two closed orbits in Figs. 12(c)–12(d) are mapped into a closed orbit inFigs. 12(a)–12(b) by the function (q, ϕ) = (w, ln|j|).

voltage v and current i of the memristor are givenby

v = M(q)i. (97)

The dynamics of this circuit is given by a set ofdifferential equations:

Dynamics of a memristor circuit

dq

dt= i,

Ldi

dt= −M(q)i,

(98)Fig. 13. A 2-element memristor circuit, which consists of alinear inductor and a charge-controlled memristor. The ter-minal voltage v and current i of the memristor are given byv = M(q)i.

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where the symbols q and i denote the charge andthe current of the memristor, and L denotes theinductance of the inductor. If M(q) > 0, the cir-cuit is dissipative, since the divergence of Eq. (98)is less than 0. Furthermore, the q-axis (i.e. i = 0)is a continuous set of equilibrium points, that is,(dq

dt ,didt

)= (0, 0) at i = 0.

If we define ϕ = Lq, we would obtain fromEq. (98)

dq

dt=

ϕ

L,

dϕ

dt= −M(q)

ϕ

L.

(99)

Here, the flux ϕ(t) is defined by

ϕ(t) = ϕL(t)�=∫

vL(τ)dτ

=∫

Ldi(τ)dτ

dτ = Li(t), (100)

that is, ϕ is an indefinite integral of vL.6

Equation (99) can be expressed as

dϕ

dq= −

(M(q)

ϕ

L

)ϕ

L

= −M(q). (101)

Therefore, we can define a differential 1-form

dH�= dϕ + M(q)dq = 0. (102)

From Eq. (102), we can define the HamiltonianH(q, i)

Hamiltonians of the memristor circuit� �

H(q, ϕ) = ϕ + f(q), (103)� �where f(q) is given by

f(q)�=∫

M(q)dq + D0, (104)

where D0 is a constant of integration and ϕM

denotes the flux of the memristor. Thus, we obtain

the following rescaled7

Hamilton’s equations of the memristor circuit� �

dq

dτ=

∂H(q, ϕ)∂i

= 1,

dϕ

dτ= −∂H(q, ϕ)

∂q= −M(q),

(105)

� �where dτ =

(ϕL

)dt (i �= 0). Note that Eq. (105)

may not preserve the time orientation of orbits ofEq. (98) in the region i < 0.

5.4. Principle of conservationof flux

Hamilton’s equations (105) have a solution

H(q, ϕ) = ϕ + f(q) = H0, (106)

where H0 is any constant. It is equivalent to

ϕL + ϕM = H1, (107)

where ϕL and ϕM denote the flux of the inductorand the memristor, respectively. Here, ϕL = ϕ andϕM is given by

ϕM (t) =∫ t

−∞v(τ)dτ

=∫ t

−∞M(q)

dq(τ)dτ

dτ

=∫ q(t)

q(−∞)M(q)dq

= f(q(t)) − f(q(−∞)), (108)

and H1 = H0 + f(q(−∞)).Hence, Eq. (106) or (107) can be interpreted

as an example of the following principles ofconservation of charge and flux [Chua, 1969]:

6Note that vL + v = 0, that is, vL = −v. Thus, the flux of the memristor is defined by

ϕM (t)�=

Zv(τ )dτ = −


7After time scaling by dτ = idt, Eq. (98) can be recast into Eq. (105) where i �= 0 [Itoh & Chua, 2011]. Note that Eq. (105)can be expressed as Eq. (101).

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• Charge and flux can neither be created nordestroyed. The quantity of charge and fluxis always conserved.

Thus, we get the following theorem:

Theorem 5. Hamilton’s equations (105) havea solution

H(q, ϕ) = ϕ + f(q) = H0,

where H0 is any constant. It can be inter-preted as the law of conservation of flux. Inother words, the Hamiltonian of Eq. (105) canbe obtained from this law.

If we define

p�= ϕ,

x�= q,

m�= L,

(109)

then Eq. (106) can be recast into

p + f(x) = mx + f(x) = H0, (110)

which can be interpreted as the law of conserva-tion of momentum in classical mechanics. CompareEq. (110) with Eq. (52). If f(x) = K(x), they areequivalent.

Furthermore, Eq. (98) can be expressed asNewton’s equation

Newton’s equation of the memristor circuit� �

q = −M(q)L

q. (111)

� �Compare Eq. (105) [resp., (111)] with Eq. (51)[resp., (44)]. Observe that if x = q, p = ϕ, L = m,and k(x) = M(x), then they are equivalent.

5.5. Coordinate and time-scaletransformation

Let us next recast Eq. (105) into a new Hamiltoniansystem, whose Hamiltonian is defined as the sum of

a “pseudo kinetic energy” and a “pseudo potentialenergy”.8

Case 1. i > 0

If we set ϕ = s2

2L and dτ =(

Ls

)dτ , Eq. (105) can be

recast into

dq

dτ=

∂H(q, s)∂s

=s

L,

ds

dτ= −∂H(q, s)

∂q

= −∂f(q)∂q

= −M(q),

(112)

where s �= 0 and H(q, s) is given by

H(q, s) =s2

2L+ f(q). (113)

Here, f(q) is given by

f(q)�=∫

M(q)dq + D0, (114)

where D0 is a constant of integration. Compare thefollowing relationship between the dissipative mem-ristor circuit and the lossless LC -circuit:

Newton’s equation� �

Dissipative memristor circuit

q = −M(q)L

q

Lossless LC-circuit

q = −g(q)L� �

8The “pseudo kinetic energy” and “pseudo potential energy” do not represent real physical energy as stated before. Wesometimes abuse our terminology here by using the same “kinetic energy” and “potential energy” for simplicity.

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dq

dτ=

∂H(q, s)∂s

=s

L,

ds

dτ= −∂H(q, s)

∂q= −∂f(q)

∂q.

Lossless LC-circuit

dq

dt=

∂H(q, ϕ)∂ϕ

=ϕ

L,

dϕ

dt= −∂H(q, ϕ)

∂q= −∂G(q)

∂q.

� �

Hamiltonian� �


H(q, s) =s2

2L+ f(q)

Lossless LC-circuit

H(q, ϕ) =ϕ2

2L+ G(q)

� �Observe that if f(q) = G(q), then the above twoHamiltonians are equivalent, although their New-ton’s equations are distinct from each other. Thus,the dynamics of 2-element “memristor circuits” canbe recast into the dynamics of 2-element LC cir-cuits. Note that the time scaling dτ =

(Ls

)dτ does

not preserve the time orientation of orbits in theregion s < 0.

Case 2. i < 0Similarly, if we set ϕ = − s2

2L and dτ =(

Ls

)dτ ,

Eq. (105) can be recast into

dq

dτ=

∂H(q, s)∂s

= − s

L,

ds

dτ=

∂H(q, s)∂q

= −M(q),

(115)

where s �= 0 and

H(q, s) = − s2

2L+ f(q). (116)

Note that the Hamiltonian (116) has a negativekinetic energy.9 Compare the following relationshipbetween the dissipative memristor circuit and thelossless LC-circuit:



dq

dτ=

∂H(q, s)∂s

= − s

L,

ds

dτ= −∂H(q, s)

∂q= −∂f(q)

∂q.

Lossless LC-circuit

dq

dt=

∂H(q, ϕ)∂ϕ

=ϕ

L,

dϕ

dt= −∂H(q, ϕ)

∂q= −∂G(q)

∂q.

� �

Hamiltonian� �


H(q, s) = − s2

2L+ f(q)

Lossless LC-circuit

H(q, ϕ) =ϕ2

2L+ G(q).

� �

6. Coordinate Transformationof Hamilton’s Equations

Let us recast Hamilton’s equations into dissipativememristor circuit equations via coordinate transfor-mations.

9If we apply the reverse time scaling t = −τ , then the Hamilton’s equations (115) can have a positive kinetic energy.

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Consider

Hamilton’s equations:

dx

dt=

∂H(x, p)∂p

,

dp

dt= −∂H(x, p)

∂x,

(117)

where

Hamiltonian:

H(x, p) =∫

f(p)dp +∫

M(x)dx. (118)

Here, f(·) and M(·) are scalar functions, andthe symbol

∫denotes an indefinite integral. The

Hamiltonian (118) is the sum of the kinetic energydefined by

F (p) =∫

f(p)dp, (119)

and the potential energy defined by

U(x) =∫

M(x)dx. (120)

Substituting Eq. (118) into Eq. (117), we obtain

Hamilton’s equations for Eq. (118)� �

dx

dt=

∂H(x, p)∂p

= f(p),

dp

dt= −∂H(x, p)

∂x= −M(x).

(121)

� �Equation (121) can be expressed as

dp

dx= −M(x)

f(p), (122)

where f(p) �= 0. From Eqs. (118) and (122), weobtain a differential 1-form

dH = f(p)dp + M(x)dx = 0. (123)

Substituting

p = ln N(i),

x = ln g(y),

}(124)

into Eq. (123), we obtain

dH =f(i)N(i)

di +M(y)g(y)

dy = 0 (125)

and

H(y, i) =∫

f(i)N(i)

di +∫

M(y)g(y)

dy, (126)

where

f(i)�= f(ln N(i)),

M(y)�= M(ln g(y)).

(127)

Note that if N(i) = i+1 and g(y) = i+1, Eq. (124)can be written as

i = ep − 1,

y = ex − 1.

}(128)

Such exponential coordinate transformation is usedin the Toda Lattice CNN [Chua et al., 1995]. TheHamiltonian (126) is the sum of the kinetic energydefined by

Ek =∫

f(i)N(i)

di, (129)

and the potential energy defined by

EP =∫

M(y)g(y)

dy. (130)

That is,

H(y, i) = Ek + EP . (131)

From Eq. (125), we obtain

di

dy= −M(y)N(i)

f(i)g(y)= −

(M(y)g(y)

)(

f(i)N(i)

) . (132)

Equation (132) can be expressed as10

New Hamilton’s equations� �

dy

dτ=

∂H(y, i)∂i

=f(i)N(i)

,

di

dτ= −∂H(y, i)

∂y= −M(y)

g(y),

(133)

� �10This equation does not alter the trajectories of Eq. (132) except at the singularity (y, i) where N(i)g(y) = 0.

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whereNew Hamiltonian� �

H(y, i) =∫

f(i)N(i)

di +∫

M(y)g(y)

dy. (134)

� �Equation (133) has a solution H(y, i) = H0 (H0 isany constant), and the system is conservative sincethe divergence of the vector field is equal to 0.

Furthermore, Eq. (132) can be recast into

Dissipative equations� �

dy

dτ= f(i)g(y),

di

dτ= −M(y)N(i).

(135)

� �This kind of equation appears in memristor circuits(we will discuss this equation in the next subsec-tion).

Equation (135) has a solution H(y, i) = H0 (H0

is any constant), since

dH(y, i)dτ

=f(i)N(i)

di

dτ+

M(y)g(y)

dy

dτ

= − f(i)N(i)

M(y)N(i) +M(y)g(y)

f(i)g(y)

= 0. (136)

We note the followings:

• If g(y) = 1 and M(x) > 0, the divergence ofthe vector field is less than 0. Hence, Eq. (135)becomes dissipative.

• Equation (135) may not preserve the time orien-tation of orbits of Eq. (133).

• After time scaling by dt = N(i)g(y)dτ , Eq. (135)can be recast into Eq. (133).

Thus, we get the following theorem:

Theorem 6. Hamilton’s equations (121) canbe transformed into the dissipative memristorcircuit equations (135) by the “exponentialcoordinate transformation (124)”. However,the time orientation of orbits may not bepreserved.

7. 2-Element Memristor Circuits

Let us recast the dynamics of 2-element memris-tor circuits into the dynamics of “ideal memristor”circuits.

7.1. Dynamics of 2-elementmemristor circuits

Consider the 2-element memristor circuit in Fig. 19,which consists of a linear inductor L and a memris-tor. The memristor is described by

Characteristic of the memristor:

dx

dt= f(i)g(x),

v = M(x)i,

(137)

where v and i denote the terminal voltage and cur-rent of the memristor, x is a physical state vari-able, and f(·), g(·) and M(·) are continuous scalarfunctions. Therefore, the dynamics of this circuit isgiven by a set of differential equations:

Dynamics of 2-element memristor circuits

dx

dt= f(i)g(x),

Ldi

dt= −M(x)i.

(138)

Note that if

N(i) =i

L,

y = x,

M(y) = M(y),

f(i) = f(i),

(139)

then Eq. (135) is equivalent to Eq. (138).

7.2. Hamiltonians of 2-elementmemristor circuits

Let us recast the “nonconservative” system (138)into a “conservative” Hamiltonian system.11 From

11For more details, see [Itoh & Chua, 2011].

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Eq. (138), we can obtain

di

dx= − M(x)i

f(i)g(x)L= −

(M(x)g(x)

)(

Lf(i)i

) . (140)

Hence, Eq. (140) can be recast into Hamilton’sequations

Hamilton’s equations of memristor circuits� �

dx

dτ=

∂H(x, i)∂i

=Lf(i)

i,

di

dτ= −∂H(x, i)

∂y= −M(x)

g(x),

(141)

� �where g(x)i �= 0 and

Hamiltonian of memristor circuits� �

H(x, i) =∫

Lf(i)i

di +∫

M(x)g(x)

dx, (142)

� �which have a solution H(x, i) = H0 (H0 is any con-stant) [Itoh & Chua, 2011]. Note that Eq. (141)may not preserve the time orientation of orbits ofEq. (138).

The Hamiltonian H(x, i) considered to be thesum of the kinetic energy EK and the potentialenergy EP respectively defined by


EK�=∫

Lf(i)i

di,

EP�=∫

M(x)g(x)

dx,

(143)

� �that is,

H(x, i) = EK + EP = H0, (144)

(see Eqs. (129)–(131)).In the case of memristors, we can assume that

f(i) = i,

g(x) = 1,

x = q,

(145)

which defines an ideal memristor. It follows fromEqs. (143) that

EK =∫

Ldi = ϕL + DL,

EP =∫

M(q)dq = ϕM + DM ,

(146)

where q, ϕM and ϕL denote the charge of thememristor, the flux of the memristor, and the fluxof the inductor, respectively, and DL and DM

are arbitrary constants. Hence, Eq. (144) can bewritten as

H(q, i) = ϕL + ϕM = H0, (147)

which can be interpreted as the principles of con-servation of flux.

7.3. Coordinate transformation

In order to obtain a physical interpretation of theHamiltonian (142), let us recast the dynamics of theHamiltonian system (141) into the dynamics of a2-element “memristor” circuit. Define the new statevariables ξ and η by

dξ =di

i,

dη =dx

g(x).

(148)

By integrating both sides of Eq. (148), we obtain

ξ = ln|i| + D1,

η = G(x)�=∫

1g(x)

dx + D2,

(149)

where D1 and D2 are constants. If η = G(x) can besolved as a function of η at a driving point, that is,x = h(η), then we would have

i = deξ ,

x = h(η),

}(150)

where d = e−D1 and h(·) �= G−1(·).

From Eqs. (140), (141) and (148), we obtain

di

dx=

idξ

g(x)dη= − M(x)i

f(i)g(x)L, (151)

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which can be recast into

dξ

dη= −M(x)

Lf(i), (152)

where g(x) �= 0. Substituting Eq. (148) into (142),we obtain

H(x, i) =∫

Lf(i)dξ +∫

M(x)dη. (153)

Substituting Eq. (150) into Eqs. (152) and (153),we obtain

dξ

dη= − M(η)

Lf(ξ)(154)

and

H(η, ξ) =∫

Lf(ξ)dξ +∫

M(η)dη, (155)

where

H(η, ξ)�= H(h(η), deξ),

f(ξ)�= f(deξ),

M(η)�= M(h(η)).

(156)

From Eqs. (154) and (155), we obtain

dH(η, ξ)�= Lf(ξ)dξ + M(η)dη = 0. (157)

Define

dζ�= f(ξ)dξ, (158)

which can be defined as a function of i, that is,

dζ =f(i)

idi. (159)

Then we obtain from Eq. (157)

Ldζ + M(η)dη = 0. (160)

Multiplying both sides of Eq. (160) by ζ, we get

Lζdζ + ζM(η)dη = 0. (161)

From Eq. (161), we can define the differentialequation

dζ

dη= −

(ζM(η)

L

)

ζ, (162)

which can be recast into

Memristor circuit equations� �

dη

dτ= ζ,

Ldζ

dτ= −M(η)ζ.

(163)

� �The differential equation (163) is equivalent toEq. (98), which governs the dynamics of a 2-element“memristor” circuit. That is, the variables η and ζin Eq. (163) correspond to the charge q and thecurrent i of the memristor in Eq. (98), respectively.Furthermore, the Hamiltonian (155) can be recastinto

Hamiltonian of Eq. (163)� �

H(η, ζ)�=∫

Ldζ +∫

M(η)dη

= EK + EP , (164)� �

where


EK�=∫

Ldζ,

EP�=∫

M(η)dη.

(165)

� �Therefore, EK and EP correspond to the flux ofthe inductor and the flux of a “memristor”, respec-tively. Thus, we get the following theorem:

Theorem 7. Dynamics of 2-element memris-tor circuits can be recast into the dynamicsof 2-element “ideal memristor” circuits. Theprinciple of conservation of flux also holds forthe new (η, ζ)-coordinates.

Note that Eq. (163) has the Hamiltonian (164) as itsintegral. Furthermore, Eq. (163) may not preservethe time orientation of orbits of Eq. (138).

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Example 2. Choose

f(i) = α(i2 − 1),

g(x) =1x2

,

M(x) = x + β,

L = 1,

(166)

where α and β denote some constants. Then, thedynamics of the memristor circuit, that is, Eq. (138)

assumes the form


dx

dt=

α(i2 − 1)x2

,

di

dt= −(x + β)i.

(167)

� �

(a) (b)

(c)

Fig. 14. Trajectories of Eqs. (167), (168) and (170). (a) Trajectories of the memristor circuit equation (167) with initialcondition (x(0), i(0)) = (0.0857712, 1.77357), (−0.119501, 0.397994). They are not closed orbits, since

˛dxdt

˛ → ∞ as x → 0except at i = ±1. (b) A closed orbit of Hamilton’s equations (168) with initial condition (x(0), i(0)) = (0.565063, 1.70000).(c) Trajectory of the memristor circuit equation (170) with initial condition (η(0), ζ(0)) = (−2.20653, 0.0992411). Note thatη-axis is a continuous set of equilibrium points, since (η, ζ) = (0, 0) at ζ = 0. Initial conditions of Eqs. (167) and (168) arelocated on the contour H(x, i) = 1. Similarly, initial condition of Eq. (170) is located on the contour I(η, ζ) = 1.0. Note thatinitial conditions are rounded to six significant figures in our computer simulations.

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Hamilton’s equations (141) can be written as


dx

dτ= α

(i − 1

i

),

di

dτ= −(x + β)x2,

(168)

� �where dτ = g(x)idt, and the Hamiltonian is givenby

H(x, i) = α

(i2

2− ln|i|

)+

x4

4+ β

x3

3. (169)

Furthermore, Eq. (168) can be recast into

Memristor circuit equation� �

dη

dτ= ζ,

dζ

dτ= −((3η − 3η0)

13 + β)ζ.

(170)

� �

(a) (b)

(c) (d)

Fig. 15. Relationship between the trajectories of Eqs. (167) and (170). (a) Trajectory of Eq. (167). (b) Trajectory of Eq. (170).(c) Trajectory of Eq. (167) on the three-dimensional (x, i, t)-space. (d) Trajectory of Eq. (167) on the three-dimensional (η, ζ, t)-space, which is mapped by Eq. (171). The nonlinear functions (171) can map the orbit of Eq. (167) [Fig. 15(a)] into the orbit ofEq. (170) [Fig. 15(b)]. That is, a red (resp., blue) trajectory on the right (resp., left) half plane in Fig. 15(a) is mapped into ared (resp., blue) trajectory on the right (resp., left) half plane in Fig. 15(b). Note that the orbit in Fig. 15(a) cannot be mappedinto the orbit in Fig. 15(b) in a one-to-one manner [see their three-dimensional plots in Figs. 15(c)–15(d)]. Furthermore, thetime orientation of the orbits is not preserved. The two orbits in Fig. 15(a) is the projection of the orbits in Fig. 15(c) intothe (x, i)-plane. Similarly, the orbit in Fig. 15(b) is the projection of the orbit in Fig. 15(d) into the (η, ζ)-plane.

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Here, η and ζ are given by

η =∫

1g(x)

dx + η0 =∫

x2dx + η0

=x3

3+ η0,

ζ =∫

f(i)i

di + ζ0 =∫

α(i2 − 1)i

di + ζ0

= α

(i2

2− ln|i|

)+ ζ0,

(171)

where ζ0 and η0 are constants of integration.Equation (170) has an integral

I(η, ζ)�= ζ +

14(3η − 3η0)

43 + βη = I0, (172)

where I0 also denotes a constant of integration.The variables η and ζ in Eq. (170) correspond tothe charge and the current of the ideal memristor,respectively. In this case, i cannot be expressed as afunction of ζ, since i = ±1 are mapped to the sameζ = 1

2 + ζ0 by Eq. (171). Therefore, Eqs. (167),(168) and (170) are not in one-to-one corre-spondence.

We show the trajectories of Eqs. (167), (168)and (170) in Figs. 14 and 15. Observe the rela-tionship among their trajectories. We also show theHamiltonian (169) and the integral (172) in Fig. 16.The following parameters are used in our computer

(a) (b)

(c) (d)

Fig. 16. (a)–(b) Hamiltonian H(x, i) and contours of Eq. (169). (c)–(d) Integral I(η, ζ) and contours of Eq. (172). Blackthick curves denote the contours of H(x, i) = 1 (top) and the contours of I(η, ζ) = 1 (bottom). Initial conditions given inFig. 14 are located on these contours.

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simulations:

α = β = 1, η0 = ζ0 = 0.

8. 2N-Element Memristor Circuits

Consider next the N -particle Hamiltonian definedby the sum of the kinetic and potential energiesdenoted by T and V , respectively:

N -particle Hamiltonian:

H = T + V,

T =N∑

k=1

pk2

2mk,

V = V (x1, x2, . . . , xN ),

(173)

where xi is the position of ith particle whose massis mi, and pi is the momentum pi = mixi. The cor-responding Hamilton’s equations are given by:

N -particle Hamilton’s equations:

dxk

dt=

∂H

∂pk=

pk

mk,

dpk

dt= − ∂H

∂xk= − ∂V

∂xk.

(174)

Let us next recast Eq. (174) into the equationof a lossless LC-circuit. Substituting

mk = Lk,

pk = ϕk,

xk = qk,

(175)

into Eqs. (173) and (174), we obtain

Hamiltonian of 2N -element LC-circuits� �

H = T + V ,

T =N∑

k=1

ϕk2

2Lk,

V = V (q1, q2, . . . , qN ),

(176)

� �

and

2N -element LC-circuit equations� �

dqk

dt=

∂H

∂ϕk=

ϕk

Lk,

dϕk

dt= −∂H

∂qk= −Mk(q1, q2, . . . , qN ),

(177)

� �where

Mk(q1, q2, . . . , qN ) =∂V (q1, q2, . . . , qN )

∂qk. (178)

Equation (177) describes the dynamics of a 2N -element circuit, which consist of N linear inductorsand N nonlinear charge-controlled capacitors. Thekth capacitor is described by

Coupled capacitors� �

vk = Mk(q1, q2, . . . , qN ), (179)� �

where vk and qk denote the terminal voltage andthe charge of the kth capacitors, whose states arecoupled among the N capacitors. The kth linearinductor is defined by

Lkdqk

dt= ϕk, (180)

where Lk denotes the inductance. Hence, we obtainthe following well-known theorem:

Theorem 8. The N -particle Hamilton’s equa-tions (174) can be transformed into the 2N -element LC-circuit equations (177) via thecoordinate transformation (175).

Equation (177) has a solution

H =N∑

k=1

ϕk2

2Lk+ V (q1, q2, . . . , qN ) = H0, (181)

where H0 is any constant. The function H is thesum of the energy EL stored in inductors andthe potential energy EC stored in the capacitors,

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where

EL�=

N∑k=1

ϕk2

2Lk,

EC�= V (q1, q2, . . . , qN ).

(182)

The total energy in a circuit remains constant sinceEq. (177) has a solution H = H0 (H0 is any con-stant). It implies the conservation law of energy inelectrical circuits. Furthermore, the circuit is con-servative, since the divergence of Eq. (177) is equalto 0.

8.1. Coordinate transformation

In this section, we transform the 2N -elementLC-circuit equations (177) into memristor circuitequations via the exponential coordinate transfor-mation. Substituting

Exponential coordinate transformation� �

ϕk = Lk ln|jk|, (jk �= 0)

qk = wk,

}(183)

� �into Eq. (177), we obtain the dynamics of a 2N -element memristor circuit, which consists of N lin-ear inductors and N memristors,


dwk

dt= ln|jk|,

Lkdjk

dt= −Mk(w1, w2, . . . , wN )jk.

(184)

� �Here, the kth memristor is described by

Coupled memristors� �

dwk

dt= ln|jk|,

vk = Mk(w1, w2, . . . , wN )jk,

(185)

� �where wk, vk, and jk denote the physical state,the terminal voltage, and the terminal currentof the kth memristor, whose states are coupled

among the N memristors. Their memristance isdescribed by

Mk(w1, w2, . . . , wN ) =∂V (w1, w2, . . . , wN )

∂wk.

(186)

Equation (184) can be realized by the circuit ofFig. 17. Furthermore, Eq. (184) has a solution

N∑k=1

Lk(ln|jk|)22

+ V (w1, w2, . . . , wN ) = H1,

(187)

where H1 is any constant. It can be interpreted asthe conservation law of energy in electrical circuits.If Mk(w1, w2, . . . , wN ) > 0 (k = 1, 2, . . . , N), thenthe divergence of Eq. (184) is less than 0, and thecircuit would become dissipative. Hence, we obtainthe following two theorems:

Theorem 9. The 2N -element LC-circuitequations (177) can be transformed into mem-ristor circuit equations (184) by the exponen-tial coordinate transformation (183).

Theorem 10. The nonlinear capacitors (179)are recast into memristors if an exponentialcoordinate transformation is applied. Con-versely, the memristors defined in Eq. (185)are expressed as capacitors if a logarithmiccoordinate transformation is applied.

Note that if Mk(w1, w2, . . . , wN ) < ∞, then thememristor (185) apparently satisfies

vk = 0 for ik = 0. (188)

However, Eq. (184) and its integral (187) have a sin-gularity at jk = 0. Hence, jk(t) cannot be zero, andthe memristor zero-crossing phenomenon cannot beobserved in this system.

Example 3. Choose

T (p1, p2) =p1

2 + p22

2,

V (x1, x2) =x1

2 + x22

2+ x1

2x2 − 13x2

3,

mk = 1,

(189)

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Fig. 17. A 2N-element memristor Hamiltonian circuit, which consists of N inductors with the inductance Lk (k = 1, 2, . . . , N)and N memristors described by

vk = Mk(w1, w2, . . . , wN )jk,dwk

dt= ln|jk| (k = 1, 2, . . . , N),

where Mk(w1, w2, . . . , wN ) denotes the memristance of the kth memristor. Even though the memristors appear to be dis-connected, their dynamics are coupled via the memristance equation involving the same state variables (w1, w2, . . . , wN ).

(a) (b)

Fig. 18. Poincare sections of the Henon and Heiles memristor equation (191) starting from 50 random initial conditions withH = 0.11. (a) Points (w2(t), j2(t)) are successively plotted when w1(t) = 0. (b) Points (j2(t), v2(t)) are successively plottedwhen w1(t) = 0. In this figure, different color represents different trajectory. Observe that any trajectory cannot cross thev2-axis on the (j2, v2)-plane. That is, we cannot observe the zero-crossing phenomenon of the memristor.

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where k = 1, 2. Then Eq. (174) is recast as

Henon and Heiles equation� �

dx1

dt= p1,

dx2

dt= p2,

dp1

dt= −x1 − 2x1x2,

dp2

dt= −x2 − x1

2 + x22.

(190)

� �If we set pk = ln|jk| and xk = wk, then Eq. (190) isrecast into

Henon and Heiles memristor equation� �

dw1

dt= ln|j1|,

dw2

dt= ln|j2|,

dj1

dt= (−w1 − 2w1w2)j1,

dj2

dt= (−w2 − w1

2 + w22)j2.

(191)

� �Equation (190) is a nonlinear nonintegrable Hamil-tonian equation, called Henon and Heiles equation[Henon & Heiles, 1964]. An example of a Henon andHeiles memristor equation, which exhibits chaoticmotion, is given in [Itoh & Chua, 2011]. We show thePoincare sections of Eq. (191) in Fig. 18. Observethat trajectories cannot cross the v2-axis (i.e. i2 =0) on the (j2, v2)-plane. That is, we cannot observethe zero-crossing phenomenon of the memristor.Roughly speaking, Eq. (191) forms attractors inthe j2 > 0 half plane, thereby precluding the zero-crossing phenomenon.

9. Generalized Memristor Circuits

A generalized current-controlled memristor [Chua,2012] is described by

Generalized current-controlled memristor

dw

dt= k(w, i),

v = l(w, i)i,

(192)

where v and i denote the terminal voltage and cur-rent of the memristor, w is the physical state vari-able, k and l are scalar functions [Chua & Kang,1976]. If we use this kind of memristors, nonlinearcircuits may be realized by fewer elements. Notethat the generalized memristor can exhibit the

Zero-crossing property:

v = l(w, i)i = 0 for i = 0. (193)

That is, the voltage v is zero whenever the currenti is zero. Furthermore, if i(t) = 0 for t ≥ t0, thenthe physical state w satisfies

dw

dt= k(w, 0), (194)

for t ≥ t0. In this case (i.e. i = 0), the memris-tor continues to evolve to an equilibrium state viaEq. (194).

9.1. 2-element Van der Pol circuit

Consider the 2-element memristor circuit in Fig. 19,which consists of a linear inductor L and a mem-ristor. Assume that the memristor is described byEq. (192). Choose

k(w, i) = ln|i|,l(w, i) = −µ(1 − w2)ln|i| + w,

}(195)

where µ is a parameter. Then the current-controlledmemristor is described by

Current-controlled memristor� �

dw

dt= ln|i|,

v = {−µ(1 − w2)ln|i| + w}i.

(196)

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Fig. 19. A 2-element memristor circuit, which consists of alinear inductor and a current-controlled memristor. Van derPol oscillator can be realized by this circuit.

The dynamics of the 2-element circuit is given by

Dynamics of the 2-element circuit� �

dw

dt= k(w, i) = ln|i|,

Ldi

dt= −v = −l(w, i)i

= {µ(1 − w2)ln|i| − w}i.

(197)

� �If |w| < ∞, the memristor (196) apparently satisfiesthe zero-crossing property, that is,

v = l(w, i)i

= {−µ(1 − w2) ln |i| + w}i → 0, (198)

for i → 0. However, Eq. (197) has the singularityat i = 0. Thus, the current i(t) cannot be zero. Notrajectory intersects the v-axis (i.e. i = 0) on the

(i, v)-plane. In this case, the system (197) oscillatesin the i > 0 half plane, thereby precluding the zero-crossing phenomenon.

Differentiating the first equation with respectto time variable t, we obtain the Van der Polequation

Dynamics of the Van der Pol circuit

w − µ(1 − w2)w + w = 0, (199)

where we set L = 1. The two equivalent elec-tronic circuits are shown in Fig. 20. Trajectories ofEqs. (197) and (199) are shown in Fig. 21, whereµ = 1. The zero-crossing phenomenon cannot occuras shown in the enlargement in Fig. 22(b), wherethe trajectory does not intersect the i = 0 axis.

Choose next

k(w, i) = i − µ

(w3

3− w

),

l(w, i) =w|sgn(i)|

i,

(200)

where the sign function sgn(x) is defined asfollows12

sgn(x) =

−1 if x < 0,

0 if x = 0,

1 if x > 0.

(201)

Fig. 20. Two nonlinear circuits exhibiting identical dynamics for L = C = 1. The dynamics of the 2-element memristorcircuit (right) is identical to the dynamics of the Van der Pol circuit (left).

12A smooth approximation of the sign function is given by sgn(x) ≈ tanh(kx) for k � 1. A piecewise-linear approximation ofthe sign function is given by sgn(x) ≈ 0.5k

`˛x + 1

k

˛ − ˛x − 1

k

˛´for k � 1.

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(a) (b)

Fig. 21. Trajectories of a 2-element memristor circuit (left) and Van der Pol equation (right). (a) Trajectory of the 2-elementmemristor circuit (197) with initial condition (w(0), w(0)) = (0.1,−0.1), that is, (w(0), i(0)) = (0.1, e−0.1). (b) Trajectory ofEq. (199) (Van der Pol equation) with initial condition (w(0), w(0)) = (0.1,−0.1). Observe that our computer simulations arethe same since Eqs. (197) and (199) have equivalent dynamics.

(a) (b)

Fig. 22. (a) Global image of a hysteresis loop for the memristor (196). (b) Hysteresis loop in the neighborhood of the origin:(i, v) = (0, 0). Observe that the Lissajous trajectory cannot pass through the v-axis as shown in the enlargement near i = 0 inFig. 22(b). The hysteresis loop in this case [Fig. 22(a)] lies in the second and third quadrants because the memristor is active.

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Then the current-controlled memristor is describedby

Current-controlled memristor� �

dw

dt= i − µ

(w3

3− w

),

v = w|sgn(i)|.

(202)

� �The memristor (200) satisfies the zero-crossingproperty. If we choose L = 1, then the dynamicsof the 2-element circuit in Fig. 19 is given by

Dynamics of the 2-element circuit� �

dw

dt= i − µ

(w3

3− w

),

di

dt= −w|sgn(i)|.

(203)

� �If i �= 0, we would obtain from Eq. (203)

dw

dt= i − µ

(w3

3− w

),

di

dt= −w.

(204)

Equation (204) is equivalent to the Van der Polequation.

Since sgn(x) ≈ tanh(kx) for k � 1, Eq. (203)can be approximated by

Smooth approximation of sgn(x)� �

dw

dt= i − µ

(w3

3− w

),

di

dt= −w|tanh(ki)|,

(205)

� �where k � 1 and

l(w, i) = w|tanh(ki)|. (206)

In the case of a piecewise-linear approximation,we can obtain

Piecewise-linear approximation of sgn(x)� �

dw

dt= i − µ

(w3

3− w

),

di

dt= −0.5k

∣∣∣∣∣∣∣∣i +

1k

∣∣∣∣−∣∣∣∣i − 1

k

∣∣∣∣∣∣∣∣w,

(207)

� �where k � 1 and

l(w, i) = 0.5k∣∣∣∣∣∣∣∣x +

1k

∣∣∣∣−∣∣∣∣x − 1

k

∣∣∣∣∣∣∣∣w.

(208)

If i = 0, then Eqs. (205) and (207) can be writtenas

dw

dt= −µ

(w3

3− w

),

di

dt= 0.

(209)

Hence, Eqs. (205) and (207) cannot have a closedorbit since the w-axis (i = 0) is an invariant set,and three equilibrium points are located on the w-axis. Note that there must be at least one equilib-rium point inside a closed orbit, and the closed orbitintersects with the w-axis.

We show the trajectories of Eqs. (205) and(207) with k = 10 and µ = 1 in Figs. 23 and24, respectively. Observe that Eqs. (205) and (207)do not have a closed orbit, and the trajectoriestend to the origin on the (i, v)-plane, which showsthe zero-crossing property, that is, v = 0 wheni = 0.

9.2. 3-element Chua’s circuit

Consider the 3-element memristor circuit in Fig. 25,which consists of a linear inductor L, a linear capac-itor C, and a voltage-controlled memristor. Thevoltage-controlled memristor is described by

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(a) (b)

Fig. 23. Trajectories of the 2-element memristor circuit (205) with k = 10. (a) Trajectories on the (w, i)-plane. (b) Tra-jectories on the (i, v)-plane. Initial condition: (w(0), i(0)) = (2.5,−1.0), (−2.5, 1.0). Observe that the trajectories go to theequilibrium point on the w-axis on the (w, i)-plane. They tend to the origin on the (i, v)-plane, which shows the zero-crossingproperty.

(a) (b)

Fig. 24. Trajectories of the 2-element memristor circuit (207) with k = 10. (a) Trajectories on the (w, i)-plane. (b) Trajectorieson the (i, v)-plane. Initial condition: (w(0), i(0)) = (2.5,−1.0), (−2.5, 1.0). Observe that the trajectories go to the equilibriumpoint on the w-axis on the (w, i)-plane. They also tend to the origin on the (i, v)-plane, which shows the zero-crossing property.

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Fig. 25. A 3-element memristor circuit, which consists of alinear inductor, a linear capacitor and a voltage-controlledmemristor.

Voltage-controlled memristor

dw

dt= k(w, v),

i = l(w, v)v,

(210)

where v and i denote the terminal voltage and cur-rent of the memristor, w is the physical state vari-able, and k and l are scalar functions.

The dynamics of the 3-element circuit in Fig. 25is given by

Dynamics of the 3-element circuit:

dw

dt= k(w, v),

Cdv

dt= iL − l(w, v)v,

LdiLdt

= −v.

(211)

Choose

k(w, v) =

v − w

R− f(w)

C1,

l(w, v) =(

v − w

Rv

),

(212)

where f is a scalar function of w. In this case, thezero-crossing property is not satisfied, that is,

i �= 0 for v = 0. (213)

From Eqs. (211) and (212), we obtain

C1dw

dt=

v − w

R− f(w),

C2dv

dt= iL − v − w

R,

LdiLdt

= −v,

(214)

where C2 = C.Equation (214) is equivalent to the dynamics of

Chua’s circuit [Madan, 1993]

Chua’s circuit:

C1dv1

dt=

v2 − v1

R− f(v1),

C2dv2

dt= iL − v2 − v1

R,

LdiLdt

= −v2,

(215)

where f(w) is given by

f(w) = bw + 0.5(a − b)(|w + 1| − |w − 1|).(216)

Here, a and b are some constants, v1 and v2 denotethe voltage across the capacitors C1 and C2, andiL denotes the current through the inductor L. Weshow two dynamically equivalent electronic circuitsin Fig. 26.

Choose next

k(w, v) =

v − w

R− f(w)

C1,

l(w, v) =(

v − w

Rv

)|sgn(v)|,

(217)

where f is given by Eq. (216). Note that this mem-ristor satisfies the zero-crossing property:

i = l(w, v)v = 0 for v = 0. (218)

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Fig. 26. Two identical nonlinear circuits exhibiting identical dynamics. The dynamics of the 3-element circuit (right) isidentical to the dynamics of Chua’s circuit with five elements (left), if the voltage-controlled memristor does not have thezero-crossing property.

Then, the dynamics of the 3-element circuit is givenby

C1dw

dt=

v − w

R− f(w),

Cdv

dt= iL −

(v − w

R

)|sgn(v)|,

LdiLdt

= −v,

(219)

which is identical to the dynamics of Chua’s circuitexcept at v = 0.

Since sgn(x) ≈ tanh(kx) for k � 1, we canapproximate Eq. (219) by

Smooth approximation of sgn(x)� �

dw

dt=

v − w

R− f(w)

C1,

Cdv

dt= iL −

(v − w

R

)|tanh(kv)|,

LdiLdt

= −v,

(220)

� �where k � 1.

If we approximate sgn(x) by a piecewise-linearfunction,

g(x) = 0.5k∣∣∣∣∣∣∣∣v +

1k

∣∣∣∣−∣∣∣∣v − 1

k

∣∣∣∣∣∣∣∣, (221)

we can obtain

Piecewise-linear approximation of sgn(x)� �

dw

dt=

v − w

R− f(w)

C1,

Cdv

dt= iL − 0.5k

∣∣∣∣∣∣∣∣v +

1k

∣∣∣∣−∣∣∣∣v − 1

k

∣∣∣∣∣∣∣∣(

v − w

R

),

LdiLdt

= −v,

(222)� �where k � 1.

We show the trajectories and the pinched hys-teresis loops of Eqs. (220) and (222) in Fig. 27.The zero-crossing phenomenon can be observed inthe form of a Lissajous figure which always passesthrough the origin when v = 0. However, when kis sufficiently large, we could not find a chaotic ora periodic oscillation in these systems as shown inFig. 28. The zero-crossing property may suppressthe oscillation for large k. We used the followingparameters in our computer simulations:

C1 =110

, C2 = C = 1, L =1

14.87,

R = 1, a = −1.27, b = −0.68.

(223)

We finally study the property of equilibriumpoints of Eqs. (214), (220) and (222). Let ej (j =1, 2, 3) be the equilibrium points of Eqs. (220)and (222). Then, they are given by

(w, v, iL) = (wj , 0, 0), (224)

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(a) (b)

(c) (d)

Fig. 27. Pinched hysteresis loops on the (v, i)-plane (left) and their corresponding trajectories on the (w, v, iL)-space (right).Observe that the Lissajous trajectories pass through the origin when v = 0 (left). The pinched hysteresis loops lie in all quad-rants because the memristor is active. Observe also that the trajectories on the (w, v, iL)-space are slightly different (right),since they are not obtained from the same equations. (a) Pinched hysteresis loop for Eq. (220) for k = 4.3. (b) Trajectory ofEq. (220) for k = 4.3. (c) Pinched hysteresis loop for Eq. (222) for k = 2.8. (d) Trajectory of Eq. (222) for k = 2.8. In ourcomputer simulations, initial condition is identical: (w(0), v(0), iL(0)) = (0.5, 0.5, 0.5).

where wj (j = 1, 2, 3) is a solution of the equation

w + f(w) = 0, (225)

which can be written as

w1 = 0, w2,3 ≈ ±1.84375. (226)

The eigenvalues of ej for Eqs. (220) and (222) aregiven by

e1 : λ1 = 2.7, λ2,3 ≈ ±i3.856

e2,3 : µ1 = −3.2, µ2,3 ≈ ±i3.856(227)

They are saddle-centers.

We next obtain the equilibrium points rj (j =1, 2, 3) of Eq. (214). They are given by

(w, v, iL) = (wj , 0,−wj). (228)

Their eigenvalues are given by

r1 : λ1 ≈ 3.848, λ2,3 ≈ −1.074 ± i3.046

r2,3 : µ1 ≈ −4.660, µ2,3 ≈ 0.2298 ± i3.187

(229)

which are saddle-focuses. Hence, if the memris-tor exhibits the zero-crossing property, then the

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(a) (b)

(c) (d)

Fig. 28. Pinched hysteresis loops of memristor (left) for large parameter k (k = 30), and their corresponding trajectories onthe (w, v, iL)-space (right). Observe that Lissajous trajectories pass through the origin when v = 0 (left). They finally tend tothe origin as t → ∞. Their corresponding trajectories on the (w, v, iL)-space tend to an equilibrium point. (a) Pinched hystere-sis loop for Eq. (220) for k = 30. (b) Trajectory of Eq. (220) for k = 30. (c) Pinched hysteresis loop for Eq. (222) for k = 30. (d)Trajectory of Eq. (222) for k = 30. In our computer simulations, initial condition is identical: (w(0), v(0), iL(0)) = (0.5, 0.5, 0.5).

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equilibrium points are transformed into saddle-centers from saddle-focuses.

9.3. Two-cell memristor CNNs

We show an example of memristor CNNs. Definetwo coupled memristors:

Two-cell memristor CNN:

dw1

dt= −w1 + a00f(w1)g(i1) + a01f(w2)g(i2),

v1 = f(w1)g(i1),

dw2

dt= −w2 + a10f(w2)g(i2) + a00f(w1)g(i1),

v2 = f(w2)g(i2),

(230)

where w1 and w2 are the physical state variables,f(wj) is a scalar function, and ajk are some con-stants. If we choose

yj�= f(wj) =

12(|wj + 1| − |wj − 1|),

g(ij) = 1,

(231)

then we would obtain the dynamics of the two-cellCNNs without input and threshold templates [Chua,1998; Zhou & Nosseck, 1991; Civalleri & Gilli, 1993;Itoh & Chua, 2004]. Here, yj denotes the outputof the jth cell. Equation (230) can be realized bytwo coupled memristor and two current sources asshown in Fig. 29.

The zero-crossing phenomenon does not occurin this two-cell CNNs. However, if we choose

f(wj) =12(|wj + 1| − |wj − 1|),

g(ij) = |tanh(ij)|,

(232)

then vj is given by

vj = f(wj)|tanh(ij)|, (j = 1, 2) (233)

and it satisfies the zero-crossing property, that is,the voltage vj is zero whenever the supplied currentij is zero.

Fig. 29. A two-cell memristor CNN circuit, which consistsof two current sources and two coupled memristors describedby

dw1

dt= −w1 + a00f(w1)g(i1) + a01f(w2)g(i2),

v1 = f(w1)g(i1),

dw2

dt= −w2 + a10f(w2)g(i2) + a00f(w1)g(i1),

v2 = f(w2)g(i2),

where w1 and w2 are the physical state variables, f(wj) isa scalar function, and ajk are some constants. Even thoughthe memristors appear to be disconnected, their dynamicsare coupled via the memristance equation involving the samestate variables (w1, w2).

If we set |i1| 1 and i2 � 1, then v1 =f(w1)|tanh(i1)| ≈ 0 and v2 = f(w2)|tanh(i2)| ≈f(w2). Hence, we would obtain

dw1

dt≈ −w1 + a01f(w2),

v1 ≈ 0,

dw2

dt≈ −w2 + a10f(w2),

v2 = f(w2),

(234)

which shows that the state w2 does not depend onthe other state w1. We can reduce the influence ofw1 by supplying the current |i1| 1.

We show the trajectories of Eq. (230) with(i1, i2) = (10, 10) and with (i1, i2) = (0.1, 10) inFig. 30. The following parameters are used in ourcomputer simulations [Itoh & Chua, 2004]:

(a00, a01, a10, a11) = (1.1, 0.5,−0.2, 1.1)

Observe that if we choose (i1, i2) = (10, 10), there isa stable limit cycle. Thus, two neurons are excited.However, if we choose (i1, i2) = (0.1, 10), then a

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(a) (b)

(c) (d)

Fig. 30. Trajectories of the two-cell memristor CNNs (230). (a)–(b) A trajectory on the (w1, w2)-plane tends to a sta-ble limit cycle when i1 = i2 = 10 � 1. Similarly, a trajectory on the (v1, v2)-plane also tends to a stable limit cycle.Initial condition used in our computer simulations: (w1(0), w2(0)) = (0.1, 0.1). In this case, two neurons are excited, thatis, the two cells are oscillating. (c)–(d) Trajectories on the (w1, w2)-plane tend to one of two equilibrium points wheni1 = 0.1 � 1 and i2 = 10 � 1. Similarly, trajectories on the (v1, v2)-plane also tend to one of two equilibrium points.Those two equilibrium points are located at (w1, w2) ≈ (0.5616, 1.089), (−0.5616,−1.089) on the (w1, w2)-plane, which cor-respond to (v1, v2) ≈ (0.05597, 1), (−0.05597,−1) on the (v1, v2)-plane. Initial condition used in our computer simulations:(w1(0), w2(0)) = (2,−2), (−2, 2). In this case, two neurons quiet down, that is, two cells cease to oscillate.

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limit cycle disappears, but the trajectory tends toone of two stable equilibrium points (since the originis a saddle point). Thus, two neurons quiet down.We conclude that the neuron’s activity dependsin part on the supplied currents (i1, i2) of thememristors.

10. Conclusion

We have shown that Hamilton’s equations can berecast into the equation of dissipative memristorcircuits. In dissipative memristor circuits, Hamilto-nians can be obtained from the principles of conser-vation of “charge” and “flux”, or the principles ofconservation of “energy”. Nonlinear capacitors areexpressed as memristor if an exponential coordinatetransformation is applied. We have also shown thatsome nonlinear circuits can be realized with fewerelements if we use memristors. Furthermore, thezero-crossing phenomenon does not occur in somememristor circuits.

Acknowledgment

This work is supported in part by an AFOSR Grantno. FA 9550-13-1-0136, and by an EC Marie CurieFellowship.

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Documents

Dynamics of Memristor Circuits We will consider in this paper the following broader generalization of an ideal memristor, which is called a memristive device in [Chua & Kang, 1976],