Dynamics of Memristor Circuits

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

Dynamics of Memristor Circuits Makoto Itoh 1-19-20-203, Arae, Jonan-ku, Fukuoka 814-0101, Japan [email protected]

Int. J. Bifurcation Chaos 2014.24. Downloaded from www.worldscientific.com by CITY UNIVERSITY OF HONG KONG on 05/28/14. For personal use only.

Leon O. Chua Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720, USA [email protected] Received January 7, 2014 In this paper, we show that Hamilton’s equations can be recast into the equations of dissipative memristor circuits. In these memristor circuits, the Hamiltonians can be obtained from the principles 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 “ideal memristor ” circuits. We also show that nonlinear capacitors are transformed into nonideal memristors if an exponential coordinate transformation is applied. Furthermore, we show that the zero-crossing phenomenon does not occur in some memristor circuits because the trajectories do not intersect the i = 0 axis. We next show that nonlinear circuits can be realized with fewer elements if we use memristors. For example, Van der Pol oscillator can be realized by only two elements: 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 of two-cell memristor CNNs. In this system, the neuron’s activity depends partly on the supplied currents of the memristors. Keywords: Hamiltonian; memristor; memristive device; dissipative circuit; conservation of momentum; 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 are formulated for lossless, nonlinear circuits [Chua & McPherson, 1974; Andronov et al., 1987; Bernstein & Liberman, 1989]. It is generally believed that dissipative systems do not have Hamilton’s equations because the trajectories are damped and hence not periodic. Recently, a dissipative memristor circuit was shown to have a Hamiltonian whose contours are precisely the damped trajectories [Itoh & Chua, 2011]. However, the physical interpretation of Hamiltonians from memristor circuits is not fully clarified.

In this paper, we show that Hamilton’s equations can be recast into the equations of memristor circuits. In these memristor circuits, the Hamiltonian can be obtained from the principles of conservation of “charge” and “flux”, or the principles of conservation of “energy”. Similarly, in dissipative physical systems, Hamiltonians can be obtained from the principles of the conservation of momentum. Furthermore, the dynamics of 2element “memristor ” circuits can be recast into that of 2-element “ideal memristor ” circuits. These transformed equations have the Hamiltonian as its integral. However, the time orientation of orbits

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M. Itoh & L. O. Chua

may not be preserved. We also show that nonlinear capacitors are expressed as memristor if an exponential coordinate transformation is applied. Furthermore, although memristors possessed the zero-crossing property, that is, the terminal voltage v is zero whenever the terminal current i is zero, in some memristor circuits, the zero-crossing phenomenon does not occur because the trajectories do not intersect the i = 0 axis. Finally, we show that Van der Pol equation can be realized by two elements: an inductor and a memristor. Chua’s circuit [Madan, 1993] can be realized by only three elements: an inductor, a capacitor, and a voltagecontrolled memristor. It is well-known that Chua’s circuit with a typical double scroll attractor has three saddle-focuses as its equilibrium points. If the memristor exhibits the zero-crossing property, then the equilibrium points are transformed into saddlecenters from saddle-focuses. Furthermore, we show an example of two-cell memristor CNNs. In this system, the neuron’s activity partly depends on the supplied currents of the memristors.

2. Memristors Memristor is a 2-terminal electronic device, which was postulated in [Chua, 1971, 2012]. The memristor shown in Figs. 1 and 2 can be described by a

(q, ϕ)-relation ϕ = f (q) or

q = g(ϕ),

(1)

between the charge q and the flux ϕ. Its terminal voltage v and the terminal current i are described by an (i, v)-relation v = M (q)i or

i = W (ϕ)v,

(2)

where v=

dϕ dt

and

i=

dq . dt

(3)

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

df (q) dq

(4)

dg(ϕ) , dϕ

(5)

M (q) = and W (ϕ) =

representing the slope of a scalar function ϕ = ϕ(q) and q = q(ϕ), respectively, called the memristor constitutive relation. The constitutive relation ϕ = f (q) cannot be uniquely determined from the terminal voltage v and the terminal current i of the memristor. That is, by integrating both sides of v = M (q)i, we obtain ϕ = f (q) + D. Here D is a nonzero constant. In other words, the memristance M (q) is well-defined at every operating point on the constitutive relation where ϕ = f (q) is differentiable. Hence, we conclude as follows [Itoh & Chua, 2013]:

Fig. 1. The constitutive relation of a monotone-increasing piecewise-linear memristor: charge-controlled memristor (left) and flux-controlled memristor (right). 1430015-2

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Dynamics of Memristor Circuits

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

 dx  = f (x, i),  dt   v = M (x)i,


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).

• 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).

(9)

where v and i denote the terminal voltage and current of the memristor, x is some internal physical state variable, and f (x, i) and M (x) are continuous scalar functions.

2.2. Nonvolatility property Memristors exhibit the Zero-crossing property: v = M (x)i = 0

• 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). A memristor characterized by a differentiable ϕ − q (resp., q − ϕ) characteristic curve is passive if, and only if, its small-signal memristance M (q) (resp., small-signal memductance W (ϕ)) is non-negative (see [Chua, 1971]). Since the instantaneous power dissipated by a passive memristor is given by P (t) = M (q(t))

i(t)2 ≥ 0

(6)

P (t) = W (ϕ(t))

v(t)2 ≥ 0,

(7)

or the energy flow into a passive memristor from time t0 to t satisfies t P (τ )dτ ≥ 0, (8) t0

for all t ≥ t0 .

for i = 0. (10)

That is, the voltage v is zero whenever the current i is zero. This zero-crossing property is manifested in the form of a Lissajous figure which always passes through the origin when i = 0. The ideal memristor is endowed with the nonvolatility property. When we switch off the power of a charge-controlled memristor at t = t0 , such that, i(t) = 0 for t > t0 , then v(t) = 0 for t > t0 . But the charge q(t) satisfies dq = 0, dt

(11)

for t > t0 . Hence, q(t) = q(t0 ) for t ≥ t0 . That is, the memristor did not lose the value of q when i became zero. We next study the nonvolatility property of a subclass of nonideal memristor (9) defined by f (x, i) = f (i)g(x). When we switch off the power of the memristor at t = t0 , that is, i(t) = 0 for t > t0 , then v(t) = 0 for t > t0 . The physical state x(t) satisfies dx = f (0)g(x), (12) dt for t > t0 . Consider the following three cases:

2.1. Generalization of ideal memristors We will consider in this paper the following broader generalization of an ideal memristor, which is called a memristive device in [Chua & Kang, 1976], or

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

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dx = 0. dt

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

(14)

d(mv) dp = = ma = F, (18) dt dt where we assume that m does not change with time. Here, the momentum is traditionally represented by the letter p, which is the product of two quantities, the mass m and velocity v:

(15)

p = mv,

Thus, the physical state x(t) satisfies lim x(t) = 0.


t→∞

That is, the memristor did not hold the value of x when i became zero. In this case, the memristor is “volatile”. • Choose f (0) = 1 and g(x) = 1−x3 . Then Eq. (12) is given by dx = 1 − x3 . dt

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

(16)

In this case, the physical state x(t) satisfies  1 if x(t0 ) > 0,   if x(t0 ) = 0, lim x(t) = sgn(x(t0 )) = 0 t→∞   −1 if x(t0 ) < 0.

(19)

and the acceleration a is defined as the instantaneous change in velocity v dv . (20) dt According to Newton’s principle of determinacy [Arnold, 1978], all motions of a system are uniquely determined by their initial positions x(t0 ) and initial velocities x(t ˙ 0 ). They determine the acceleration, that is, there is a function G such that a=

Newton’s equation: x ¨ = G(x, x, ˙ t),

(17)

where

That is, the memristor did not lose the sign of x when i became zero. Therefore, the nonvolatility property of the memristor depends on the function f (0)g(x). In later sections, we discuss the case where f (i) has a singularity at i = 0.

dx , x(t ˙ 0) = dt t=t0

(21)

d2 x x ¨(t0 ) = 2 . dt 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 3. Newton’s Equations

Choose

Analogous electrical and mechanical systems have differential equations of the same form. Let us begin from Newton’s laws of motion, which are given by three physical laws: (I) First law: An object either is at rest or moves at a constant velocity, unless an external force is applied to it. (II) Second law: The relationship between an object’s mass m, its acceleration a, and the applied force F is F = ma. (III) Third law: For every action there is an equal and opposite reaction.

G(x, x, ˙ t) = −

1 m

∂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

m¨ x=−

∂U (x) , ∂x

(24)

which states the motion of a point in a potential field with potential energy U (x). 1430015-4

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

Equation (32) stipulates the law of conservation of momentum. Thus, we get the following theorem: Theorem 1. Newton’s equation (28) can be

(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 ∂U (x) = −g. m¨ x=− ∂x

(26)

3.2. Nonlinear dissipative system

transformed into an equation stating the law of conservation of momentum. In the case of linear dissipative systems, the resisting force F is proportional to velocity x˙ of the mass. If the damping force F is expressed as F = −cx, ˙ that is, F is linearly dependent upon velocity x, ˙ then we would obtain m¨ x = −cx, ˙


Choose G(x, x, ˙ t) = −

k(x)x˙ , m

(34)

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

(27)

p + cx = D,

where k(x) is a scalar function of x. Then Eq. (21) can be described by Newton’s equation for a dissipative system

(35)

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

4. Classical Physical Systems

Hamiltonian mechanics is developed as a reformulation of classical mechanics. In Hamiltonian mechan ics, a classical physical system is described by a set which is a nonlinear dissipative system, where the of canonical coordinates (x, p). Here, x is the posidamping (friction) coefficient is a function of the tion coordinate and p is the momentum. The time position x [Jeltsema & Doria-Cerezo, 2012]. If we evolution of the system is uniquely defined by the integrate both sides of Eq. (28) with respect to the time variable t, that is, Hamilton’s equations:  m¨ x dt = − k(x)x˙ dt, (29) ∂H(x, p)   ,  x˙ =   ∂p (36) we would obtain  ∂H(x, p)   , p˙ = −  mx˙ = −K(x) + D, (30) ∂x where D is a constant of integration, and K(x) is defined by where H(x, p) is the Hamiltonian, which corre sponds to the total energy of the system. From (31) K(x) = k(x)x˙ dt = k(x)dx, Eq. (36), we obtain  which denotes the time integral of the damping ∂H(x, p) ∂H(x, p)  dH(x, p)  force F . Here the symbol denotes an indefinite = x˙ + p˙     dt ∂x ∂p  integral. Furthermore, Eq. (30) can be recast into      ∂H(x, p) ∂H(x, p)  p + K(x) = D, (32) = ∂x ∂p (37)  where p is the momentum, which is defined by   ∂H(x, p) ∂H(x, p)   p = mx. ˙ The constant of integration D is deter −    ∂p ∂x mined by the initial condition (x(0), p(0)), that is,     = 0. D = p(0) + K(x(0)). (33) m¨ x = −k(x)x, ˙

(28)

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It follows that Eq. (36) has a solution H(x, p) = H0 (H0 is any constant). The total mechanical energy in a system (i.e. the sum of the potential and kinetic energies) remains constant, which stipulates the principle of conservation of mechanical energy. Furthermore, the divergence of Eq. (36) is equal to 0. Thus, the Hamilton’s equations (36) are conservative. We next show two examples of Hamilton’s equations.

4.1. Conservative systems


Consider the Newton’s equation defined by Conservative system: x ¨=−

1 ∂U (x) , m ∂x

Kinetic energy EK and the potential energy EP

EK =

p2 , 2m

(43)

EP = U (x).

Since Eq. (42) has a solution H(x, p) = H0 (H0 is any constant), the total mechanical energy in a Hamiltonian system remains constant. It stipulates the principle of conservation of mechanical energy. The system is conservative since the divergence of Eq. (42) is equal to 0.

4.2. Dissipative systems Consider the Newton’s equation defined by the

(38)

Dissipative system: where U (x) denotes potential energy. If we multiply both sides of Eq. (38) by mx, ˙ we obtain dU (x) d m 2 x˙ = − . dt 2 dt

(39)

x ¨=−

(44)

Equation (44) has an integral p + K(x) = D,

Hence, Eq. (38) has an integral p2 mx˙ 2 + U (x) = + U (x) = D, 2 2m

k(x) x. ˙ m

(40)

(45)

where p = mx, ˙ D is a constant of integration, and K(x) is defined by K(x) = k(x)dx. (46)

where p = mx˙ and D is a constant of integration. If we define the

From the derivative of Eq. (45), we can obtain  p   x˙ = ,   m

Hamiltonian of conservative systems

H(x, p) =

p2 + U (x), 2m

(41)

k(x)    p. p˙ = − m

(47)

If k(x) > 0, the system is dissipative, since the divergence of Eq. (47) is less than 0. Equation (47) can be expressed as Hamilton’s equations of conservative systems

k(x)  p  p ∂H(x, p)  dp m  = , x˙ = 

= −k(q), (48) = −  p ∂p m dx (42) m  ∂U (x)  ∂H(x, p)   =− . p˙ = − where p = 0. Therefore, we can define a differential ∂x ∂x 1-form

then we would obtain the

The Hamiltonian (41) is the sum of the kinetic energy EK and the potential energy EP , which are respectively defined by

dH = dp + k(x)dx = 0.

(49)

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

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Hamiltonian of dissipative systems

Theorem 2. Hamilton’s equations (51) have

Hamilton’s equations are given by1

a solution H(x, p) = p + K(x) = H0 , where H0 is any constant. Hence, it stipulates the “law of conservation of momentum” with a force F = −k(x)x. ˙

Hamilton’s equations of dissipative systems

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

(50)

The two systems (47) and (51) are in one-toone correspondence except at the singularity p = 0, ∂H(x, p) dx although the time scaling dτ = pdt may not pre= = 1, dτ ∂p serve the time orientation of orbits. That is, the (51) time orientation of orbits is not preserved in the   ∂H(x, p) dp  region p < 0. Note that x-axis is a continuous set of =− = −k(x),  dτ ∂x equilibrium points of Eq. (47), since (x, ˙ p) ˙ = (0, 0) at p = 0. However, Hamilton’s equations (51) do not have an equilibrium point since x˙ = 1 = 0. where dτ = pdt. Hamilton’s equations (51) have a Furthermore, Eq. (51) is conservative, even though solution Eq. (47) is not conservative (dissipative). The trajectories of Eqs. (47) and (51) are shown H(x, p) = p + K(x) in Figs. 3 and 4. The following parameters are used (52) = H0 , in our computer simulations:


     

where H0 is any constant. Thus, we get the following theorem:

(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 orientation 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) Two trajectories of Eq. (47) with K(x) = x3 /3 and m = 1. They tend to the origin as t → ∞. Two initial conditions of the above trajectories are given by (x(0), p(0)) = (2, −8/3), (−2, 8/3). (b) Three trajectories of Eq. (47) with K(x) = x2 /2 and m = 1. Trajectories starting from the first, second, and fourth quadrants tend to the x-axis as t → ∞. A trajectory starting from goes√to infinity as t → ∞. Initial conditions of these three trajectories are given by √ the third quadrant √ (x(0), p(0)) = (− 2.2, −0.1), (− 1.8, 0.1), ( 8, −3). 1

After time scaling by dτ = pdt, Eq. (47) can be directly recast into Eq. (51) where p = 0. For more details, see [Andronov et al., 1987; Nemytskii & Stepanov, 1989; Itoh & Chua, 2011]. Note that Eq. (51) can be expressed as Eq. (48). 1430015-7

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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 denote time orientation of the orbits. Observe that the orbits of 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 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 to infinity as t → ∞. Initial condition of this trajectory is given by (x(0), p(0)) = (−2, 8/3), which is located on the contour of H = 0 with K(x) = x3 /3, m = 1. (b) A trajectory starting from the third quadrant goes to infinity as t → ∞. Initial condition of 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’s equations (51):

dx = 1, dτ

After time scaling by2

Case 1. p > 0

If we set p = Eq. (51)

s2 2m

for p ≥ 0, we would obtain from

m

dτ, s Eq. (54) assumes the equivalent form  s dx   = ,   d˜ τ m  ds   = −k(x), d˜ τ d˜ τ=

Let us recast Eq. (51) into a new Hamiltonian system, whose Hamiltonian has both a kinetic energy and a potential energy, by using coordinate and time-scale transformation.

(54)

s ds    = −k(x). m dτ

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

4.3. Coordinate and time-scale transformation

    

(55)

(56)

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

dp s ds = dτ m 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 the pseudo kinetic energy of the new state variable s. From Eqs. (51) and (53), we obtain 2

˜ ∂ H(x, s) s dx = = , d˜ τ ∂s m

     

  ˜  ∂ H(x, s) ds  =− = −k(x), d˜ τ ∂x

(57)

This time scaling maps orbits between systems (54) and (56) in a one-to-one manner except at the singularity s = 0, although it may not preserve the time orientation of orbits. 1430015-8

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Hamiltonian

˜ where the Hamiltonian H(x, s) is given by s2 ˜ H(x, s) = + K(x). 2m

Dissipative system s2 ˜ + K(x) H(x, s) = 2m

(58)

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

EK =

s2 2m

   , 


  EP = K(x).

Dissipative system

1 ∂K(x) k(x) x ¨=− x˙ = − x˙ m ∂x m Conservative system 1 ∂U (x) m ∂x

Dissipative system      

 ˜  ∂ H(x, s) ∂K(x)  ds  =− =− . d˜ τ ∂q ∂x Conservative system ∂H(x, p) p dx = = , dt ∂p m

Case 2. p < 0 2

After time scaling by d˜ τ =−

m

dτ, s Eq. (61) assumes the equivalent form  s  dx  =− ,   d˜ τ m    ds  = −k(x),  d˜ τ

(62)

(63)

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

    

˜ ∂ H(x, s) s dx = =− , d˜ τ ∂s m

     

  ˜  ∂ H(x, s) ds  =− = −k(x), d˜ τ ∂x

∂H(x, p) ∂U (x)  dp   =− =− . dt ∂x ∂x

p2 + U (x) 2m

s If we set p = − 2m for p ≤ 0, we would obtain from Eq. (51)

s ds dp =− = −k(x), (60) dτ m dτ √ where s = ± −2mp, which does not map the orbits in a one-to-one manner. From Eqs. (51) and (60), we obtain   dx  = 1,    dτ (61)

s ds    = k(x).  m dτ

Hamilton’s equations

˜ dq ∂ H(x, s) s = = , d˜ τ ∂s m

H(x, p) =

Observe that if K(x) = U (x), then these two Hamiltonians are equivalent, although their Newton’s equations are distinct from each other. Note that the time scaling (55) does not preserve the time orientation of orbits in the region s < 0.

(59)

Newton’s equation

Conservative system

Observe that the pseudo kinetic energy EK , the pseudo potential energy EP , and the Hamiltonian ˜ do not represent real physical energy. H Compare the following relationship between the dissipative system and conservative system:

x ¨=−

1430015-9

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where s2 ˜ + K(x). H(x, s) = − 2m

 2  s  , E˜K = − 2m   E˜P = K(x). 

(65)

Here, the Hamiltonian (65) is the sum of the pseudo kinetic energy E˜K and the pseudo potential energy E˜P , which are respectively defined by


(a)

(66)

The Hamiltonian (65) has a negative pseudo kinetic energy. If we apply the reverse time scaling tˆ = −˜ τ, (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 curves denote time orientation of the orbits. The origin is an equilibrium point of Eqs. (57) and (64), that is, (x, ˙ s) ˙ = (0, 0). All initial conditions are located on the contour of H = 0. Observe that Hamilton’s equations (51), (57) and (64) do not preserve the time of the orbits of Eq. (47) [see Fig. 3(a)]. (a) q Trajectories of Eq. (57) with initial conditions (x(0), s(0)) = q orientation ` ` −2(−2.5)3 ´ −2(−0.2)3 ´ −2.5, ≈ (−2.5, 3.23) and (x(0), s(0)) = −0.2, − ≈ (−0.2, −0.073). (b) Trajectories of Eq. (64) 3 3 q q ` ` 2(2.5)3 ´ −2(−0.2)3 ´ ≈ (2.5, 3.23) and (x(0), s(0)) = 0.2, − ≈ (0.2, −0.073). with initial conditions (x(0), s(0)) = 2.5, 3 3 (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, which correspond to the red and blue trajectories in Fig. 3(a). 1430015-10

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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 curves denote time orientation of the orbits. All initial conditions are located on the contour of H = 1. Observe that Hamilton’s equations (51), (57) and (64) do not preserve the time orientation of the orbits of Eq. (47) [see Fig. 3(b)]. (a) Trajectory √ of Eq. (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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M. Itoh & L. O. Chua (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. Red thick curves denote the contours with the elevation H = 0. Arrows on the curves denote time orientation of the orbits. Two trajectories in Fig. 7(d) are mapped into a blue portion of the trajectory in Fig. 7(b) by the function p = s2 /2. Similarly, two trajectories in Fig. 7(f) are mapped into a red portion of the trajectory in Fig. 7(b) by the function p = −s2 /2. However, the time orientation of the orbits is not preserved. 1430015-12

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Dynamics of Memristor Circuits (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. Black thick curves denote the contours with the elevation H = 1. Arrows on the curves denote time orientation of the orbits. A red closed 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, a blue trajectory in the second and third quadrants of Fig. 8(f) are mapped into a blue portion of the trajectory in Fig. 8(b) by the function p = −s2 /2. A purple trajectory in the first and fourth quadrants of Fig. 8(f) are mapped into a purple portion of the trajectory in Fig. 8(b) by the function p = −s2 /2. However, the time orientation of the orbits is not preserved. 1430015-13

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then the Hamilton’s equations (64) can have a positive pseudo kinetic energy. Note that the time scaling (62) does not preserve the time orientation of orbits in the region s < 0. Compare the following relationship between the dissipative system and conservative system: Hamilton’s equations

Dissipative system


˜ dx ∂ H(x, s) s = =− , d˜ τ ∂s m

     

 ˜  ∂ H(x, s) ∂K(x)  ds  =− =− . d˜ τ ∂q ∂x Conservative system ∂H(x, p) p dx = = , dt ∂p m

     

 ∂H(x, p) ∂U (x)  dp  =− =− .  dt ∂x ∂x Hamiltonian

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

5. Hamiltonians of Electronic Circuits

s2 ˜ + K(x) H(x, s) = − 2m

The dynamics of physical systems can be simulated by corresponding dual electronic circuits. Let us consider first 2-element “lossless” and “dissipative” nonlinear circuits whose dynamics can be described by conservative Hamilton’s equations.

Conservative system

p2 + U (x) 2m

Observe the following relationship of Fig. 6:

• A closed orbit in Fig. 6(a) is mapped into a red portion of the trajectory in Fig. 6(c) by the function p = s2 /2. • A trajectory moving in the second and third quadrants of Fig. 6(b) is mapped into a blue portion of the trajectory in Fig. 6(b) by the function p = −s2 /2. • A trajectory moving in the first and fourth quadrants of Fig. 6(b) is mapped into a purple portion of the trajectory in Fig. 6(c) by the function p = −s2 /2. • Time scaling does not preserve the time orientation of orbits.

Dissipative system

H(x, p) =

• Time scaling does not preserve the time orientation of orbits.

The trajectories of Hamilton’s equations (51), (57) and (64) are shown in Figs. 5 and 6. The following parameters 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 a blue portion of the trajectory in Fig. 5(c) by the function p = s2 /2. • Two trajectories in Fig. 5(b) are mapped into a red portion of the trajectory in Fig. 5(c) by the function p = −s2 /2.

5.1. Lossless LC -circuits Consider the circuit in Fig. 9, which consists of a linear inductor and a nonlinear charge-controlled capacitor. The dynamics of this circuit is described by a set of differential equations: Dynamics of a lossless LC-circuit:   dq   = i,   dt   di  L = −g(q),  dt

1430015-14

(67)

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Hamilton’s equations of the lossless circuit

∂H(q, ϕ) ϕ dq = = , dt ∂ϕ L dϕ ∂H(q, ϕ) =− dt ∂q

          

       ∂G(q)  = −g(q), =−  ∂q

(71)


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

where the symbols q and i denote the charge and the current of the nonlinear capacitor with the characteristic curve v = g(q), and L denotes the inductance of the inductor. This circuit is conservative, that is, lossless, since the divergence of Eq. (67) is equal 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 =

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

H(q, ϕ) =

Here, G(q) is defined by G(q) = g(q)dq.

t0

t1

= di(τ ) dτ = Li(t), L dτ

Dynamics of a lossless LC-circuit:   ϕ dq   = ,   dt L   dϕ  = −g(q).  dt

t0

(69)

t1

= t0

t1

= t0

di L dt

(73)

idt

dϕ ϕ dt dt L

ϕ L

dϕ

t t1 ϕ2 1 = = EL 2L t0 t0 (70)

(72)

Let WL and WC be the stored energy in the above inductor and capacitor during the time interval (t0 , t1 ). Then we obtain t1 v(−i)dt WL =

that is, ϕ is an indefinite integral of vL .3 If we subϕ into Eq. (67), we would obtain stitute i = L

and

WC =

Note that vL + v = 0, that is, vL = −v. Thus, the flux of the capacitor is defined by Z Z ϕC (t) = v(τ )dτ = − vL (τ )dτ = −ϕL (t). 1430015-15

t1

vidt t0

Equation (70) can be recast into Hamilton’s equations: 3

ϕ2 + G(q). 2L

t1

g(q)idt

= t0

(74)

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t1

=

g(q) t0

dq dt dt


t1

g(q)dq

= t0

t t = G(q) t10 = EC t10 ,

(75)

where EL and EC are defined by the Energy stored in the lossless circuit

ϕ2 , EL = 2L


    

   EC = G(q).

(76)

5.2. Exponential coordinate transformation

Equation (71) has a solution H(q, ϕ) = H0 (H0 is any constant), that is, EL + EC = H0 .

(77)

It stipulates the “conservation law of energy” in lossless electrical circuits. Note that EC corresponds to the potential energy stored in the capacitor, and EL corresponds to the kinetic energy in classical mechanics, since EL can be described as EL = if we define

p2 ϕ2 = , 2L 2m

(79)

Newton’s equation of the lossless circuit

ϕ = L ln|j|,

(81)

into Eq. (67)4 and replacing q with w, we obtain Dynamics of a memristor circuit

dw = ln|j|, dt

     (82)

which indicates the dynamics of the circuit which consists of a linear inductor and a memristor described by Characteristic of the memristor

  dw  = ln|j|,  dt   v = g(w)j,

(80)

Compare Eq. (71) [resp., (80)] with Eq. (42) [resp., (38)]. Observe that if x = q, p = ϕ, m = L, and U (·) = G(·), then they are equivalent. Thus, we get the following well-known theorem: 4

Exponential coordinate transformation

 dj  L = −g(w)j,  dt

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

g(q) 1 ∂G(q) =− . L ∂q L

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

(78)

 p = ϕ,   m = L.

q¨ = −

a solution H(q, ϕ) = H0 (H0 is any constant). The function H(q, ϕ) considered to be the sum of the magnetic energy EL stored in the inductor (i.e. the kinetic energy of the inductor ) and the electric energy EC stored in the capacitor (i.e. the potential energy of the capacitor). Furthermore, the total energy in a lossless circuit remains constant. It stipulates the “conservation law of energy” in lossless electrical circuits. The Hamiltonian of Eq. (71) can be obtained from these laws.

(83)

where v and j denote the terminal voltage and current of the memristor, and the physical state variable w is the memristor charge.

Note that j can be expressed as j = ei . Hence, we call this transformation an exponential coordinate transformation. 1430015-16

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It follows from Eq. (72) that Eq. (82) has an integral L(ln|j|)2 + G(w) = I0 , I(w, j) = 2

(84)


where I0 is an arbitrary constant and G(q) is defined by G(w) = g(w)dw. (85) Hence, the nonlinear capacitor in Fig. 9 is transformed into a memristor [Chua, 2012] for all nonzero memristor current j, if an exponential coordinate transformation is applied to the flux ϕ. Observe that Eq. (83) defines a nonideal memristor. Thus, we get the following theorem: Theorem 4. A nonlinear capacitor with the

characteristic curve v = g(q) is transformed into a memristor if an exponential coordinate transformation is applied to the LC-circuit equation (70).

v = g(q) = q 3 ,

(88)

then the dynamics of the LC-circuit is given by   dq  = i,    dt (89)   di  L = −q 3 ,  dt or ϕ dq = , dt L

     

  dϕ  = −q 3 ,  dt

(90)

where the flux ϕ(t) is defined by ϕ(t) = ϕL (t) = vL (τ )dτ =

L

di(τ ) dτ = Li(t), dτ

(91)

that is, ϕ is an indefinite integral of vL .5 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 → 0 and dt

dw → −∞ dt

for j → 0,

(87)

where the Hamiltonian is defined by

which implies that the trajectory cannot cross the vaxis on the (j, v)-plane. That is, j(t) cannot be zero. Furthermore, Eq. (82) and its integral I(w, j) have a singularity at j = 0. Therefore, the zero-crossing phenomenon cannot be observed in the form of a Lissajous figure, which always passes through the origin 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 the zero-crossing property. Example 1. Consider the LC-circuit in Fig. 9. If

the characteristic curve of the charge controlled capacitor is given by 5

Equation (90) can be recast into Hamilton’s equations:   ∂H(q, ϕ) ϕ dq   = = ,   dt ∂ϕ L (92)   ∂H(q, ϕ) dϕ  =− = −q 3 ,  dt ∂q

H(q, ϕ) =

q4 ϕ2 + . 2L 4

(93)

Contour lines of Eq. (93) consist of closed curves. Applying the exponential coordinate transformation ϕ = L ln|j| into Eq. (90) and replacing q with w, we obtain the dynamics of a memristor circuit   dw  = ln|j|,    dt (94)   dj  L = −w3 j.  dt

Note that vL + v = 0, that is, vL = −v. Thus, the flux of the capacitor is defined by Z Z ϕC (t) = v(τ )dτ = − vL (τ )dτ = −ϕL (t). 1430015-17

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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 denotes Napier’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  = ln|j|, dt (95)    v = w3 j.

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

Here, v and j denote the terminal voltage and current of the memristor, and w is a physical state variable. Equation (94) has an integral

5.3. Dissipative memristor circuits

I(w, j) =

L(ln|j|)2 w4 + = I0 , 2 4

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

(96)

where I0 is an arbitrary constant. Note that the exponential coordinate transformation i = ln|j| is not in one-to-one correspondence, since j = ±1 are mapped to the same i = 0. Furthermore, if |w| < ∞, the memristor apparently satisfies the zero-crossing property, that is, v = w3 j → 0 for j → 0. However, Eq. (94) and its integral I(w, j) have a singularity at j = 0, dj → 0 for j → 0. That is, j(t) cannot be and dt zero. Hence, the zero-crossing property cannot be observed in the form of a pinched hysteresis loop in this circuit because the periodic current waveform is not bipolar. In other words, the system (94) does not violate the zero-crossing property. We show the trajectories (closed orbits) of Eqs. (90) and (94) with L = 1 in Fig. 10. Two closed orbits in Fig. 10(b) are mapped into a closed orbit

Fig. 11. Lissajous trajectories of the memristor (95) on the (j, v)-plane (L = 1). Since the trajectories of the memristor did not pass through the v-axis, the zero-crossing property is not 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). Black thick curves denote the contours of H(q, ϕ) = 0.75 (top) and the contours of I(w, ϕ) = 0.75 (bottom). Initial conditions given in Fig. 10 are located on these contours. The two closed orbits in Figs. 12(c)–12(d) are mapped into a closed orbit in Figs. 12(a)–12(b) by the function (q, ϕ) = (w, ln|j|).

voltage v and current i of the memristor are given by v = M (q)i.

(97)

The dynamics of this circuit is given by a set of differential equations: Dynamics of a memristor circuit   dq   = i,   dt   di  L = −M (q)i,  dt

(98) Fig. 13. A 2-element memristor circuit, which consists of a linear inductor and a charge-controlled memristor. The terminal voltage v and current i of the memristor are given by v = M (q)i. 1430015-19

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where the symbols q and i denote the charge and the current of the memristor, and L denotes the inductance of the inductor. If M (q) > 0, the circuit is dissipative, since the divergence of Eq. (98) is less than 0. Furthermore, the q-axis (i.e. i = 0) is continuous set of equilibrium points, that is, dqa di dt , dt = (0, 0) at i = 0. If we define ϕ = Lq, ˙ we would obtain from Eq. (98)   ϕ dq   = ,   dt L (99)  ϕ  dϕ  = −M (q) .  dt L Here, the flux ϕ(t) is defined by ϕ(t) = ϕL (t) = vL (τ )dτ =

L

di(τ ) dτ = Li(t), dτ

Hamilton’s equations of the memristor circuit

∂H(q, ϕ) dq = = 1, dτ ∂i

where f (q) is given by f (q) = M (q)dq + D0 ,

H(q, ϕ) = ϕ + f (q) = H0 ,

(100)

ϕL + ϕM = H1 ,

(107)

where ϕL and ϕM denote the flux of the inductor and the memristor, respectively. Here, ϕL = ϕ and ϕM is given by ϕM (t) =

t

v(τ )dτ −∞

t

=

M (q) −∞

(103)

dq(τ ) dτ dτ

q(t)

M (q)dq

= q(−∞)

= f (q(t)) − f (q(−∞)), (104)

(106)

where H0 is any constant. It is equivalent to

(102)

where D0 is a constant of integration and ϕM denotes the flux of the memristor. Thus, we obtain 6

ϕ

Hamilton’s equations (105) have a solution

Hamiltonians of the memristor circuit

(105)

where dτ = L dt (i = 0). Note that Eq. (105) may not preserve the time orientation of orbits of Eq. (98) in the region i < 0.

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

H(q, ϕ) = ϕ + f (q),

    

 ∂H(q, ϕ) dϕ   =− = −M (q), dτ ∂q

5.4. Principle of conservation of flux

that is, ϕ is an indefinite integral of vL .6 Equation (99) can be expressed as

ϕ M (q) dϕ L = −M (q). (101) =− ϕ dq L Therefore, we can define a differential 1-form dH = dϕ + M (q)dq = 0.

the following rescaled7

(108)

and H1 = H0 + f (q(−∞)). Hence, Eq. (106) or (107) can be interpreted as an example of the following principles of conservation of charge and flux [Chua, 1969]:

Note that vL + v = 0, that is, vL = −v. Thus, the flux of the memristor is defined by Z Z ϕM (t) = v(τ )dτ = − vL (τ )dτ = −ϕL (t).

7

After 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). 1430015-20

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

• Charge and flux can neither be created nor destroyed. The quantity of charge and flux is always conserved.

Case 1. i > 0

If we set ϕ = recast into

Thus, we get the following theorem:

s2 2L

and d˜ τ=

L s dτ , Eq. (105) can be


a solution

˜ s) ∂ H(q, s dq = = , d˜ τ ∂s L

H(q, ϕ) = ϕ + f (q) = H0 ,


where H0 is any constant. It can be interpreted as the law of conservation of flux. In other words, the Hamiltonian of Eq. (105) can be obtained from this law. If we define

 p = ϕ,     x = q,     m = L,

˜ s) ds ∂ H(q, =− d˜ τ ∂q

        ∂f (q)  = −M (q), =−  ∂q

(109)

2 ˜ s) = s + f (q). H(q, 2L

Newton’s equation of the memristor circuit

M (q) q. ˙ L

(113)

(110)

which can be interpreted as the law of conservation of momentum in classical mechanics. Compare Eq. (110) with Eq. (52). If f (x) = K(x), they are equivalent. Furthermore, Eq. (98) can be expressed as Newton’s equation

q¨ = −

(112)

˜ s) is given by where s = 0 and H(q,

then Eq. (106) can be recast into p + f (x) = mx˙ + f (x) = H0 ,

           

Here, f (q) is given by

f (q) =

M (q)dq + D0 ,

(114)

where D0 is a constant of integration. Compare the

following relationship between the dissipative memristor circuit and the lossless LC -circuit:

(111)

Newton’s equation

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.

Dissipative memristor circuit q¨ = −

5.5. Coordinate and time-scale transformation

M (q) q˙ L

Lossless LC-circuit

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

q¨ = −

8

g(q) L

The “pseudo kinetic energy” and “pseudo potential energy” do not represent real physical energy as stated before. We sometimes abuse our terminology here by using the same “kinetic energy” and “potential energy” for simplicity. 1430015-21

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Hamilton’s equations

where s = 0 and

Dissipative memristor circuit

2

˜ s) = − s + f (q). H(q, 2L

     

˜ s) ∂ H(q, s dq = = , d˜ τ ∂s L

 ˜ s)  ds ∂ H(q, ∂f (q)   =− =− . d˜ τ ∂q ∂q Lossless LC-circuit


Note that the Hamiltonian (116) has a negative kinetic energy.9 Compare the following relationship between the dissipative memristor circuit and the lossless LC-circuit: Hamilton’s equations

     

∂H(q, ϕ) ϕ dq = = , dt ∂ϕ L

Hamiltonian

˜ s) ∂ H(q, s dq = =− , d˜ τ ∂s L

Lossless LC-circuit

2 ˜ s) = s + f (q) H(q, 2L

∂H(q, ϕ) ϕ dq = = , dt ∂ϕ L

Lossless LC-circuit

2L

+ G(q)

2

s and d˜ τ = Similarly, if we set ϕ = − 2L Eq. (105) can be recast into

˜ s) ∂ H(q, s dq = =− , d˜ τ ∂s L

9

Hamiltonian

Dissipative memristor circuit 2

˜ s) = − s + f (q) H(q, 2L Lossless LC-circuit

L s dτ ,

     

  ˜ s)  ∂ H(q, ds  = = −M (q), d˜ τ ∂q

     

 dϕ ∂H(q, ϕ) ∂G(q)   =− =− . dt ∂q ∂q 

Observe that if f (q) = G(q), then the above two Hamiltonians are equivalent, although their Newton’s equations are distinct from each other. Thus, the dynamics of 2-element “memristor circuits” can be recast into the dynamics of 2-element LC circuits. Note that the time scaling d˜ τ = Ls dτ does not preserve the time orientation of orbits in the region s < 0. Case 2. i < 0

     

˜ s) ∂ H(q, ∂f (q)   ds  =− =− . d˜ τ ∂q ∂q 


H(q, ϕ) =


 ∂H(q, ϕ) ∂G(q)  dϕ  =− =− .  dt ∂q ∂q

ϕ2

(116)

(115)

H(q, ϕ) =

ϕ2 + G(q). 2L

6. Coordinate Transformation of Hamilton’s Equations Let us recast Hamilton’s equations into dissipative memristor circuit equations via coordinate transformations.

If we apply the reverse time scaling tˆ = −˜ τ , then the Hamilton’s equations (115) can have a positive kinetic energy. 1430015-22

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Consider

into Eq. (123), we obtain

Hamilton’s equations:

dH =

 ∂H(x, p)  dx  = ,    dt ∂p  ∂H(x, p)  dp  =− ,  dt ∂x

and (117)

H(y, i) =

f˜(i) = f (ln N (i)),

Here, f (·) and M (·) are scalar functions, and the symbol denotes an indefinite integral. The Hamiltonian (118) is the sum of the kinetic energy defined 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)

∂H(x, p) dx = = f (p), dt ∂p

Note that if N (i) = i + 1 and g(y) = i + 1, Eq. (124) can be written as i = ep − 1, (128) y = ex − 1. Such exponential coordinate transformation is used in the Toda Lattice CNN [Chua et al., 1995]. The Hamiltonian (126) is the sum of the kinetic energy defined by ˜ f (i) di, (129) Ek = N (i) and the potential energy defined by ˜ M (y) dy. EP = g(y)

That is,

     

H(y, i) = Ek + EP . From Eq. (125), we obtain

Equation (121) can be expressed as M (x) dp =− , dx f (p)

(122)

where f (p) = 0. From Eqs. (118) and (122), we obtain a differential 1-form dH = f (p)dp + M (x)dx = 0.

x = ln g(y),

(131)

˜ (y) M g(y) ˜ (y)N (i) di M . =− =− dy f˜(i)g(y) f˜(i) N (i)

(121)

p = ln N (i),

(130)

  ∂H(x, p) dp  =− = −M (x).  dt ∂x

Substituting

(126)

(127)

˜ (y) = M (ln g(y)). M

Hamiltonian: H(x, p) = f (p)dp + M (x)dx. (118) Int. J. Bifurcation Chaos 2014.24. Downloaded from www.worldscientific.com by CITY UNIVERSITY OF HONG KONG on 05/28/14. For personal use only.

˜ M (y) f˜(i) di + dy, N (i) g(y)

(125)

where

where

10

˜ (y) f˜(i) M di + dy = 0 N (i) g(y)

(132)

Equation (132) can be expressed as10 New Hamilton’s equations

∂H(y, i) f˜(i) dy = = , dτ ∂i N (i)

(123)

      

˜ (y)   ∂H(y, i) M di  =− =− ,  dτ ∂y g(y) 

(124)

This equation does not alter the trajectories of Eq. (132) except at the singularity (y, i) where N (i)g(y) = 0. 1430015-23

(133)

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where New Hamiltonian H(y, i) =

˜ M (y) f˜(i) di + dy. N (i) g(y)

(134)

Equation (133) has a solution H(y, i) = H0 (H0 is any constant), and the system is conservative since the divergence of the vector field is equal to 0. Furthermore, Eq. (132) can be recast into


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

7.1. Dynamics of 2-element memristor circuits

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

Dissipative equations

dy = f˜(i)g(y), d˜ τ

7. 2-Element Memristor Circuits

     

  di ˜  = −M(y)N (i).  d˜ τ

Characteristic of the memristor:   dx  = f (i)g(x), dt (137)    v = M (x)i,

(135)

This kind of equation appears in memristor circuits (we will discuss this equation in the next subsection). Equation (135) has a solution H(y, i) = H0 (H0 is any constant), since ˜ (y) dy f˜(i) di M dH(y, i) = + d˜ τ N (i) d˜ τ g(y) d˜ τ =−

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

= 0.

(136)

where v and i denote the terminal voltage and current of the memristor, x is a physical state variable, and f (·), g(·) and M (·) are continuous scalar functions. Therefore, the dynamics of this circuit is given by a set of differential equations: Dynamics of 2-element memristor circuits   dx  = f (i)g(x),   dt (138)   di  L = −M (x)i.  dt

We note the followings: • If g(y) = 1 and M (x) > 0, the divergence of the vector field is less than 0. Hence, Eq. (135) becomes dissipative. • Equation (135) may not preserve the time orientation of orbits of Eq. (133). • After time scaling by dt = N (i)g(y)d˜ τ , Eq. (135) can be recast into Eq. (133).

Note that if i N (i) = , L y = x,

  ˜ (y) = M (y),  M     ˜ f (i) = f (i), 

Thus, we get the following theorem:

11

         

Theorem 6. Hamilton’s equations (121) can

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

be transformed into the dissipative memristor circuit equations (135) by the “exponential coordinate transformation (124)”. However, the time orientation of orbits may not be preserved.

7.2. Hamiltonians of 2-element memristor circuits

(139)

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

For more details, see [Itoh & Chua, 2011]. 1430015-24

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Eq. (138), we can obtain

M (x) M (x)i di g(x) . =− = −

Lf (i) dx f (i)g(x)L i

(140)

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

where q, ϕM and ϕL denote the charge of the memristor, the flux of the memristor, and the flux of the inductor, respectively, and DL and DM are arbitrary constants. Hence, Eq. (144) can be written as

Hamilton’s equations of memristor circuits


∂H(x, i) Lf (i) dx = = , dτ ∂i i

     

 ∂H(x, i) M (x)  di  =− =− ,  dτ ∂y g(x)

which defines an ideal memristor. It follows from Eqs. (143) that     EK = Ldi = ϕL + DL ,   (146)    EP = M (q)dq = ϕM + DM , 

(141)

H(q, i) = ϕL + ϕM = H0 ,

(147)

which can be interpreted as the principles of conservation of flux.

where g(x)i = 0 and

7.3. Coordinate transformation

Hamiltonian of memristor circuits

In order to obtain a physical interpretation of the Hamiltonian (142), let us recast the dynamics of the M (x) Lf (i) di + dx, (142) H(x, i) = Hamiltonian system (141) into the dynamics of a i g(x) 2-element “memristor” circuit. Define the new state variables ξ and η by which have a solution H(x, i) = H0 (H0 is any con stant) [Itoh & Chua, 2011]. Note that Eq. (141) di   may not preserve the time orientation of orbits of dξ = ,    Eq. (138). i (148) The Hamiltonian H(x, i) considered to be the  dx   sum of the kinetic energy EK and the potential . dη =  g(x) energy EP respectively defined by


  Lf (i)  di,    i

 M (x)   dx,  g(x)

EK =

EP =

By integrating both sides of Eq. (148), we obtain  ξ = ln|i| + D1 ,   (149) 1  η = G(x) = dx + D2 , g(x)

(143)

where D1 and D2 are constants. If η = G(x) can be

solved as a function of η at a driving point, that is,

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, (145)   x = q,

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

1430015-25

idξ M (x)i di = =− , dx g(x)dη f (i)g(x)L

(151)

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which can be recast into

which can be recast into M (x) dξ =− , dη Lf (i)

(152)

Memristor circuit equations

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 ˆ (η) M dξ =− dη Lfˆ(ξ) Int. J. Bifurcation Chaos 2014.24. Downloaded from www.worldscientific.com by CITY UNIVERSITY OF HONG KONG on 05/28/14. For personal use only.

and

ˆ H(η, ξ) =

where

(154)

Lfˆ(ξ)dξ +

ˆ (η)dη, M

(155)

  dζ  ˆ  = −M(η)ζ. L  d˜ τ

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

H(η, ζ) =

(156)

ˆ (η)dη = 0. ˆ dH(η, ξ) = Lfˆ(ξ)dξ + M

Ldζ +

ˆ (η)dη M (164)

where (157)


EK = (158)

EP =

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

Ldζ,

(159)

     

  ˆ (η)dη.  M 

(165)

Then we obtain from Eq. (157)

Therefore, E K and E P correspond to the flux of the inductor and the flux of a “memristor ”, respectively. Thus, we get the following theorem:

(160)

Multiplying both sides of Eq. (160) by ζ, we get ˆ (η)dη = 0. Lζdζ + ζ M

= EK + EP ,

Define

ˆ (η)dη = 0. Ldζ + M

(163)

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

f (i) di. dζ = i

     

dη = ζ, d˜ τ

Hamiltonian of Eq. (163)

 ξ ˆ  H(η, ξ) = H(h(η), de ),    ξ ˆ f (ξ) = f (de ),      ˆ M (η) = M (h(η)).

dζ = fˆ(ξ)dξ,

(161)

From Eq. (161), we can define the differential equation ˆ ζ M(η) L dζ =− , (162) dη ζ

Theorem 7. Dynamics of 2-element memris-

tor circuits can be recast into the dynamics of 2-element “ideal memristor” circuits. The principle of conservation of flux also holds for the new (η, ζ)-coordinates.

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

1430015-26

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assumes the form

Example 2. Choose

 f (i) = α(i2 − 1),       1  g(x) = 2 , x     M (x) = x + β,     L = 1,


(166)

dx = dt

α(i2

  di  = −(x + β)i.  dt

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


 − 1)    ,  x2

(a)

(167)

(b)

(c) Fig. 14. Trajectories of Eqs. (167), (168) and (170). (a) Trajectories of the memristor circuit equation ˛ ˛(167) with initial ˛ condition (x(0), i(0)) = (0.0857712, 1.77357), (−0.119501, 0.397994). They are not closed orbits, since ˛ dx dt → ∞ as x → 0 except 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 that ˙ = (0, 0) at ζ = 0. Initial conditions of Eqs. (167) and (168) are η-axis is a continuous set of equilibrium points, since (η, ˙ ζ) located on the contour H(x, i) = 1. Similarly, initial condition of Eq. (170) is located on the contour I(η, ζ) = 1.0. Note that initial conditions are rounded to six significant figures in our computer simulations. 1430015-27

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Hamilton’s equations (141) can be written as Hamilton’s equations

  1 dx   =α i− ,   dτ i   di  = −(x + β)x2 ,  dτ

i2 x4 x3 H(x, i) = α − ln|i| + +β . 2 4 3 Furthermore, Eq. (168) can be recast into Memristor circuit equation

(168)

dη = ζ, d˜ τ

     

  1 dζ  = −((3η − 3η0 ) 3 + β)ζ.  d˜ τ

where dτ = g(x)idt, and the Hamiltonian is given by


(169)

(a)

(b)

(c)

(d)

(170)

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 of Eq. (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 a red (resp., blue) trajectory on the right (resp., left) half plane in Fig. 15(b). Note that the orbit in Fig. 15(a) cannot be mapped into the orbit in Fig. 15(b) in a one-to-one manner [see their three-dimensional plots in Figs. 15(c)–15(d)]. Furthermore, the time orientation of the orbits is not preserved. The two orbits in Fig. 15(a) is the projection of the orbits in Fig. 15(c) into the (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. 1430015-28

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Here, η and ζ are given by η=

1 dx + η0 = g(x)

x2 dx + η0

                

x3 + η0 , 3   α(i2 − 1) f (i)  di + ζ0 = di + ζ0  ζ=    i i    

2   i   − ln|i| + ζ0 , =α  2 =

(171)


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

4 1 I(η, ζ) = ζ + (3η − 3η0 ) 3 + βη = I0 , (172) 4 where I0 also denotes a constant of integration. The variables η and ζ in Eq. (170) correspond to the charge and the current of the ideal memristor, respectively. In this case, i cannot be expressed as a function of ζ, since i = ±1 are mapped to the same ζ = 12 + ζ0 by Eq. (171). Therefore, Eqs. (167), (168) and (170) are not in one-to-one correspondence. We show the trajectories of Eqs. (167), (168) and (170) in Figs. 14 and 15. Observe the relationship among their trajectories. We also show the Hamiltonian (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). Black thick curves denote the contours of H(x, i) = 1 (top) and the contours of I(η, ζ) = 1 (bottom). Initial conditions given in Fig. 14 are located on these contours. 1430015-29

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simulations:

and α = β = 1,

2N -element LC-circuit equations

η0 = ζ0 = 0.

     

8. 2N -Element Memristor Circuits

ϕk ∂H dqk = , = dt ∂ϕk Lk

Consider next the N -particle Hamiltonian defined by the sum of the kinetic and potential energies denoted by T and V , respectively:

  dϕk ∂H  =− = −Mk (q1 , q2 , . . . , qN ),  dt ∂qk (177)

N -particle Hamiltonian:

       

H = T + V, Int. J. Bifurcation Chaos 2014.24. Downloaded from www.worldscientific.com by CITY UNIVERSITY OF HONG KONG on 05/28/14. For personal use only.

N pk 2 , T = 2mk

   k=1     V = V (x1 , x2 , . . . , xN ),

where Mk (q1 , q2 , . . . , qN ) =

(173)

where xi is the position of ith particle whose mass is mi , and pi is the momentum pi = mi x˙ i . The corresponding Hamilton’s equations are given by: N -particle Hamilton’s equations:   ∂H pk dxk   = = ,   dt ∂pk mk  ∂H ∂V  dpk  =− =− .  dt ∂xk ∂xk

Coupled capacitors

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

(174)

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

(175)

Hamiltonian of 2N -element LC-circuits

N ϕk 2 , 2Lk k=1

V˜ = V (q1 , q2 , . . . , qN ),

dqk = ϕk , dt

(180)

where Lk denotes the inductance. Hence, we obtain the following well-known theorem: Theorem 8. The N -particle Hamilton’s equa-

Equation (177) has a solution

              

(179)

tions (174) can be transformed into the 2N element LC -circuit equations (177) via the coordinate transformation (175).

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

T˜ =

where vk and qk denote the terminal voltage and the charge of the kth capacitors, whose states are coupled among the N capacitors. The kth linear inductor is defined by Lk

˜ = T˜ + V˜ , H

(178)

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

 mk = Lk ,  pk = ϕk ,   xk = q k ,

∂ V˜ (q1 , q2 , . . . , qN ) . ∂qk

˜ = H (176)

N ϕk 2 + V˜ (q1 , q2 , . . . , qN ) = H0 , 2Lk

(181)

k=1

˜ is the where H0 is any constant. The function H sum of the energy EL stored in inductors and the potential energy EC stored in the capacitors, 1430015-30

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where

EL =

among the N memristors. Their memristance is described by

     

N ϕk 2 , 2Lk

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

(182)

k=1

    EC = V˜ (q1 , q2 , . . . , qN ).


The total energy in a circuit remains constant since ˜ = H0 (H0 is any conEq. (177) has a solution H stant). It implies the conservation law of energy in electrical circuits. Furthermore, the circuit is conservative, since the divergence of Eq. (177) is equal to 0.

8.1. Coordinate transformation In this section, we transform the 2N -element LC-circuit equations (177) into memristor circuit equations via the exponential coordinate transformation. Substituting Exponential coordinate transformation

ϕk = Lk ln|jk |,

(jk = 0)

Equation (184) can be realized by the circuit of Fig. 17. Furthermore, Eq. (184) has a solution N Lk (ln|jk |)2 k=1

The 2N -element LC -circuit equations (177) can be transformed into memristor circuit equations (184) by the exponential coordinate transformation (183).

Theorem 9.

(183)

Theorem 10. The nonlinear capacitors (179)

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

are recast into memristors if an exponential coordinate transformation is applied. Conversely, the memristors defined in Eq. (185) are expressed as capacitors if a logarithmic coordinate transformation is applied.


dwk = ln|jk |, dt

     

  djk  Lk = −Mk (w1 , w2 , . . . , wN )jk .  dt

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

(184)

vk = 0 for ik = 0.

Coupled memristors

dwk = ln|jk |, dt

   

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

(188)

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

Here, the kth memristor is described by

+ V˜ (w1 , w2 , . . . , wN ) = H1 ,

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

2

(187)

qk = wk ,

∂ V˜ (w1 , w2 , . . . , wN ) . ∂wk (186)

Example 3. Choose

(185)

p1 2 + p2 2 , T (p1 , p2 ) = 2

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

x1 2 + x2 2 1 + x1 2 x2 − x2 3 ,   2 3       mk = 1,

V (x1 , x2 ) =

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          (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 = ln|jk | dt

(k = 1, 2, . . . , N ),

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

(a)

(b)

Fig. 18. Poincaré sections of the Hénon and Heiles memristor equation (191) starting from 50 random initial conditions with ˜ = 0.11. (a) Points (w2 (t), j2 (t)) are successively plotted when w1 (t) = 0. (b) Points (j2 (t), v2 (t)) are successively plotted H when w1 (t) = 0. In this figure, different color represents different trajectory. Observe that any trajectory cannot cross the v2 -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 Hénon and Heiles equation

dx1 = p1 , dt dx2 = p2 , dt

               

            dp2  2 2  = −x2 − x1 + x2 . dt

where v and i denote the terminal voltage and current of the memristor, w is the physical state variable, k and l are scalar functions [Chua & Kang, 1976]. If we use this kind of memristors, nonlinear circuits may be realized by fewer elements. Note that the generalized memristor can exhibit the

(190)

dp1 = −x1 − 2x1 x2 , dt


Generalized current-controlled memristor   dw  = k(w, i),  dt (192)    v = l(w, i)i,

Zero-crossing property: v = l(w, i)i = 0

If we set pk = ln|jk | and xk = wk , then Eq. (190) is recast into Hénon and Heiles memristor equation

dw1 = ln|j1 |, dt dw2 = ln|j2 |, dt

(193)

That is, the voltage v is zero whenever the current

i is zero. Furthermore, if i(t) = 0 for t ≥ t0 , then the physical state w satisfies

               

            dj2  2 2  = (−w2 − w1 + w2 )j2 . dt

for i = 0.

dw = k(w, 0), dt

for t ≥ t0 . In this case (i.e. i = 0), the memristor continues to evolve to an equilibrium state via Eq. (194).

(191)

dj1 = (−w1 − 2w1 w2 )j1 , dt

(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-

Equation (190) is a nonlinear nonintegrable Hamiltonian equation, called Hénon and Heiles equation [Hénon & Heiles, 1964]. An example of a Hénon and Heiles memristor equation, which exhibits chaotic motion, is given in [Itoh & Chua, 2011]. We show the Poincaré sections of Eq. (191) in Fig. 18. Observe that trajectories cannot cross the v2 -axis (i.e. i2 = 0) on the (j2 , v2 )-plane. That is, we cannot observe the zero-crossing phenomenon of the memristor. Roughly speaking, Eq. (191) forms attractors in the j2 > 0 half plane, thereby precluding the zerocrossing phenomenon.

ristor. Assume that the memristor is described by Eq. (192). Choose

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

where µ is a parameter. Then the current-controlled memristor is described by Current-controlled memristor

9. Generalized Memristor Circuits

v = {−µ(1 −

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   

dw = ln|i|, dt

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

(195)

w2 )ln|i|

  + w}i.

(196)

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(i, v)-plane. In this case, the system (197) oscillates in the i > 0 half plane, thereby precluding the zerocrossing phenomenon. Differentiating the first equation with respect to time variable t, we obtain the Van der Pol equation Dynamics of the Van der Pol circuit w ¨ − µ(1 − w2 )w˙ + w = 0,

(199)


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

where we set L = 1. The two equivalent electronic circuits are shown in Fig. 20. Trajectories of The dynamics of the 2-element circuit is given by Eqs. (197) and (199) are shown in Fig. 21, where µ = 1. The zero-crossing phenomenon cannot occur Dynamics of the 2-element circuit as shown in the enlargement in Fig. 22(b), where  the trajectory does not intersect the i = 0 axis.  dw  Choose next  = k(w, i) = ln|i|,   dt    (197) 

3 di   L = −v = −l(w, i)i w     dt  − w , k(w, i) = i − µ    3    (200) 2 = {µ(1 − w )ln|i| − w}i.   w|sgn(i)|   , l(w, i) =  i If |w| < ∞, the memristor (196) apparently satisfies the zero-crossing property, that is, where the sign function sgn(x) is defined as v = l(w, i)i follows12 = {−µ(1 − w2 ) ln |i| + w}i → 0, (198)  −1 if x < 0,   for i → 0. However, Eq. (197) has the singularity if x = 0, sgn(x) = 0 (201) at i = 0. Thus, the current i(t) cannot be zero. No   1 if x > 0. trajectory intersects the v-axis (i.e. i = 0) on the

Fig. 20. Two nonlinear circuits exhibiting identical dynamics for L = C = 1. The dynamics of the 2-element memristor circuit (right) is identical to the dynamics of the Van der Pol circuit (left). 12

A smooth approximation of the sign function is given ≈ tanh(kx) for k 1. A piecewise-linear approximation of ˛ ˛ by sgn(x) ˛´ `˛ the sign function is given by sgn(x) ≈ 0.5k ˛x + k1 ˛ − ˛x − k1 ˛ for k 1. 1430015-34

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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-element memristor 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 of Eq. (199) (Van der Pol equation) with initial condition (w(0), w(0)) ˙ = (0.1, −0.1). Observe that our computer simulations are the 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 in Fig. 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 described by Current-controlled memristor

  w3  − w , 3    v = w|sgn(i)|.

dw =i−µ dt

Piecewise-linear approximation of sgn(x)

w3 −w , 3 1 di = −0.5k i + − i − dt k

dw =i−µ dt

(202)

The memristor (200) satisfies the zero-crossing property. If we choose L = 1, then the dynamics of the 2-element circuit in Fig. 19 is given by Int. J. Bifurcation Chaos 2014.24. Downloaded from www.worldscientific.com by CITY UNIVERSITY OF HONG KONG on 05/28/14. For personal use only.

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

Dynamics of the 2-element circuit

dw =i−µ dt

 1   w,   k (207)

where k 1 and l(w, i) = 0.5k x +

  − w ,   3    

      

w3

di = −w|sgn(i)|. dt

1 − x− k

1 w. k

(203)

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

as

If i = 0, we would obtain from Eq. (203) dw =i−µ dt

di = −w. dt

  w3   − w ,  3     

dw = −µ dt di = 0. dt

(204)

w3 3

    − w ,      

(209)

Hence, Eqs. (205) and (207) cannot have a closed orbit since the w-axis (i = 0) is an invariant set, and three equilibrium points are located on the waxis. Note that there must be at least one equilibrium point inside a closed orbit, and the closed orbit intersects with the w-axis. We show the trajectories of Eqs. (205) and (207) with k = 10 and µ = 1 in Figs. 23 and 24, respectively. Observe that Eqs. (205) and (207) do not have a closed orbit, and the trajectories tend to the origin on the (i, v)-plane, which shows (205) the zero-crossing property, that is, v = 0 when i = 0.

Equation (204) is equivalent to the Van der Pol equation. Since sgn(x) ≈ tanh(kx) for k 1, Eq. (203) can be approximated by Smooth approximation of sgn(x)

  w3  − w ,  3    di = −w|tanh(ki)|,  dt

dw =i−µ dt

where k 1 and

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

9.2. 3-element Chua’s circuit (206)

Consider the 3-element memristor circuit in Fig. 25, which consists of a linear inductor L, a linear capacitor C, and a voltage-controlled memristor. The voltage-controlled memristor is described by 1430015-36

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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) Trajectories 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 the equilibrium point on the w-axis on the (w, i)-plane. They tend to the origin on the (i, v)-plane, which shows the zero-crossing property.

(a)

(b)

Fig. 24. Trajectories of the 2-element memristor circuit (207) with k = 10. (a) Trajectories on the (w, i)-plane. (b) Trajectories 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 the equilibrium point 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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where f is a scalar function of w. In this case, the zero-crossing property is not satisfied, that is, i = 0


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

Voltage-controlled memristor   dw  = k(w, v),  dt   i = l(w, v)v,

From Eqs. (211) and (212), we obtain   v−w dw  = − f (w), C1   dt R      v−w dv = iL − , C2  dt R       diL   L = −v,  dt

(213)

(214)

where C2 = C. Equation (214) is equivalent to the dynamics of Chua’s circuit [Madan, 1993] Chua’s circuit:

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

(210)

where v and i denote the terminal voltage and current of the memristor, w is the physical state variable, and k and l are scalar functions. The dynamics of the 3-element circuit in Fig. 25 is given by Dynamics of the 3-element circuit:   dw   = k(w, v),   dt      dv (211) C = iL − l(w, v)v,  dt       diL   = −v. L  dt

(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 denote the voltage across the capacitors C1 and C2 , and iL denotes the current through the inductor L. We show two dynamically equivalent electronic circuits in Fig. 26. Choose next   v−w   − f (w)   R k(w, v) = ,   C1 

  v−w  |sgn(v)|, l(w, v) =   Rv

Choose  v−w  − f (w)     k(w, v) = R , C1 

   v−w   , l(w, v) = Rv

for v = 0.

(217)

(212) where f is given by Eq. (216). Note that this memristor satisfies the zero-crossing property: i = l(w, v)v = 0 for v = 0. 1430015-38

(218)

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

Then, the dynamics of the 3-element circuit is given by   dw v−w   = − f (w), C1    dt R   

 v−w dv = iL − |sgn(v)|, C dt R        diL   = −v, L dt

we can obtain Piecewise-linear approximation of sgn(x)

v−w  − f (w) R   ,   C1  

 v − w  1 dv 1 C = iL − 0.5k v + − v − , dt k k R       diL  L = −v, dt (222) dw = dt

(219)

which is identical to the dynamics of Chua’s circuit except at v = 0. Since sgn(x) ≈ tanh(kx) for k 1, we can approximate Eq. (219) by Smooth approximation of sgn(x)

dw dt C

dv dt

diL L dt

  v−w   − f (w)   R   , =   C1   

v−w = iL − |tanh(kv)|,   R         = −v, 

where k 1. We show the trajectories and the pinched hysteresis loops of Eqs. (220) and (222) in Fig. 27. The zero-crossing phenomenon can be observed in the form of a Lissajous figure which always passes through the origin when v = 0. However, when k is sufficiently large, we could not find a chaotic or a periodic oscillation in these systems as shown in Fig. 28. The zero-crossing property may suppress the oscillation for large k. We used the following (220) parameters in our computer simulations: C1 =

where k 1. If we approximate sgn(x) by a piecewise-linear function, g(x) = 0.5k v +

1 − v− k

1 , k

1 , 10

R = 1,

C2 = C = 1, a = −1.27,

L=

1 , 14.87

(223)

b = −0.68.

We finally study the property of equilibrium points 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

(221) 1430015-39

˜j , 0, 0), (w, v, iL ) = (w

(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 quadrants 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 of Eq. (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 our computer simulations, initial condition is identical: (w(0), v(0), iL (0)) = (0.5, 0.5, 0.5).

where w ˜j (j = 1, 2, 3) is a solution of the equation w + f (w) = 0,

(225)

We next obtain the equilibrium points rj (j = 1, 2, 3) of Eq. (214). They are given by ˜j , 0, −w ˜j ). (w, v, iL ) = (w

which can be written as w ˜1 = 0,

w ˜2,3 ≈ ±1.84375.

(226)

Their eigenvalues are given by

The eigenvalues of ej for Eqs. (220) and (222) are given by e1 : λ1 = 2.7,

λ2,3 ≈ ±i3.856

e2,3 : µ1 = −3.2,

µ2,3 ≈ ±i3.856

They are saddle-centers.

(228)

r1 : λ1 ≈ 3.848,

λ2,3 ≈ −1.074 ± i3.046

r2,3 : µ1 ≈ −4.660,

µ2,3 ≈ 0.2298 ± i3.187 (229)

(227)

which are saddle-focuses. Hence, if the memristor exhibits the zero-crossing property, then the 1430015-40

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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 on the (w, v, iL )-space (right). Observe that Lissajous trajectories pass through the origin when v = 0 (left). They finally tend to the origin as t → ∞. Their corresponding trajectories on the (w, v, iL )-space tend to an equilibrium point. (a) Pinched hysteresis 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 saddlecenters from saddle-focuses.

9.3. Two-cell memristor CNNs We show an example of memristor CNNs. Define two coupled memristors: Two-cell memristor CNN:


 dw1   = −w1 + a00 f (w1 )g(i1 ) + a01 f (w2 )g(i2 ),   dt       v1 = f (w1 )g(i1 ),

Fig. 29. A two-cell memristor CNN circuit, which consists of two current sources and two coupled memristors described by

  dw2  = −w2 + a10 f (w2 )g(i2 ) + a00 f (w1 )g(i1 ),   dt      v2 = f (w2 )g(i2 ),

dw1 = −w1 + a00 f (w1 )g(i1 ) + a01 f (w2 )g(i2 ), dt v1 = f (w1 )g(i1 ), dw2 = −w2 + a10 f (w2 )g(i2 ) + a00 f (w1 )g(i1 ), dt

(230)

v2 = f (w2 )g(i2 ),

where w1 and w2 are the physical state variables, f (wj ) is a scalar function, and ajk are some constants. If we choose   1  yj = f (wj ) = (|wj + 1| − |wj − 1|), 2    g(ij ) = 1, (231) then we would obtain the dynamics of the two-cell CNNs without input and threshold templates [Chua, 1998; Zhou & Nosseck, 1991; Civalleri & Gilli, 1993; Itoh & Chua, 2004]. Here, yj denotes the output of the jth cell. Equation (230) can be realized by two coupled memristor and two current sources as shown in Fig. 29. The zero-crossing phenomenon does not occur in this two-cell CNNs. However, if we choose   1  f (wj ) = (|wj + 1| − |wj − 1|), 2 (232)    g(i ) = |tanh(i )|, j

j

then vj is given by vj = f (wj )|tanh(ij )|,

(j = 1, 2)

where w1 and w2 are the physical state variables, f (wj ) is a scalar function, and ajk are some constants. Even though the memristors appear to be disconnected, their dynamics are coupled via the memristance equation involving the same state 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  ≈ −w1 + a01 f (w2 ),    dt       v1 ≈ 0, (234)   dw2  ≈ −w2 + a10 f (w2 ),   dt       v2 = f (w2 ), which shows that the state w2 does not depend on the other state w1 . We can reduce the influence of w1 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) in Fig. 30. The following parameters are used in our computer simulations [Itoh & Chua, 2004]: (a00 , a01 , a10 , a11 ) = (1.1, 0.5, −0.2, 1.1)

(233)

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

Observe that if we choose (i1 , i2 ) = (10, 10), there is a 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 stable 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, that is, the two cells are oscillating. (c)–(d) Trajectories on the (w1 , w2 )-plane tend to one of two equilibrium points when i1 = 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 correspond 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 to one of two stable equilibrium points (since the origin is a saddle point). Thus, two neurons quiet down. We conclude that the neuron’s activity depends in part on the supplied currents (i1 , i2 ) of the memristors.


10. Conclusion We have shown that Hamilton’s equations can be recast into the equation of dissipative memristor circuits. In dissipative memristor circuits, Hamiltonians can be obtained from the principles of conservation of “charge” and “flux”, or the principles of conservation of “energy”. Nonlinear capacitors are expressed as memristor if an exponential coordinate transformation is applied. We have also shown that some nonlinear circuits can be realized with fewer elements if we use memristors. Furthermore, the zero-crossing phenomenon does not occur in some memristor circuits.

Acknowledgment This work is supported in part by an AFOSR Grant no. FA 9550-13-1-0136, and by an EC Marie Curie Fellowship.

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