MEMRISTOR OSCILLATORS

Tutoria ls and Revie ws International Journal of Bifurcation and Chaos, Vol. 18, No. 11 (2008) 3183-3206

© World Scientific Publishing Company

MEMRISTOR OSCILLATORS MAKOT O ITOH Dep artm ent of Information and Communication Engineering,

Fukuo ka Institute of Technology,

Fukuoka 811-0295, J apan

LEON O. CHUA Departm ent of El ectri cal Engineering and Compu ter S cien ces,

University of California, B erkeley,

B erkeley, CA 94120, US A

Received July 15, 2008; Revised Septemb er 18, 2008

The memristor has a ttracted phenomen al worldwide attent ion since its de but on 1 May 2008 issu e of Nature in view of its many po te nt ial applications, e.g. super- dense nonv ola tile com­ puter me mory and neural synapses. T he Hewlett-Packa rd memristor is a passive nonlin ear two­ t erminal circ uit element t ha t maintain s a fun cti on al rela t ionsh ip between t he time int egrals of current and voltage, res pective ly, viz. charge an d fl ux. In thi s paper, we derive seve ra l nonl inear oscillators from Chua's oscillators by rep lac ing Chua's d iodes with mem ristors.

K eywords: Memristor ; memrist ive devices; memristive systems; charge; flux; Chua's oscilla t or ; Chu a 's dio de; learning; ne ur ons; syn apses; Hodgkin-Huxley; nerve membrane model.

1. Memristors

'.

In a seminal paper [Strukov et al., 2008] which appeared on 1 May 2008 issu e of Nature, a team led by R. Stanley W illiams from t he Hewlet t­ P ackard Company anno unced t he fab ricat ion of a nanometer-size solid-state two-termina l dev ice called the m em ri stor, a contraction for memory resis tor, which was postulat ed in [Chua , 1971; Ch ua & Kang, 1976]. This passive elect ronic device

has generate d un precedented world wide interest .' because of its potent ial app lications [Tour & He,

2008; Johnson , 2008] in the next gener ation com­

puters and powerful brain-like "neur al" computers.

One imm ediat e applicat ion offers an enablin g low­ cost technology for n on-volatile mem ories/ where future computers would t urn on instant ly wit hou t t he usual "booting time", currently required in all pe rsonal com puters.

T he HP m em rist or shown in Fig. 1 is a pas­ sive two-terminal elect ronic device described by a nonlin ear constitutiv e relation v = M( q)i,

or

i = W (cp )v ,

(1)

between the device terminal voltage v and ter min al current i . The two nonlinear functions M (q) and W (cp), called the mem ristan ce and memductan ce, respectively, ar e defined by

M ( ) ~ dcp(q)

(2)

W ( ) ~ dq(cp)

(3)

q

dq '

an d

cp

dcp '

representing t he slope of a scalar function ip = cp( q) and q = q(cp), respect ively, called t he memristor con stit utive relat ion.

IMore tha n one million Google hits were registered as of J une 1, 2008.

2The Hewlett- Packard memristor is a t iny nano, passive, two-terminal device requiring no batteries. Memristors charact erized

by a nonmonotonic constit utive relat ion are called active memristors in this paper because t hey require a power supply.

3183

M. Itoh & L. O. Chua

3184

+

+

v

v

where a, b, c, d > O. Consequently, the memris­ tance M (q) and the memductance W (cp ) in Fi g. 2 are defined by

M (q) = d 1,

(7)

{C,

Iwl < 1, Iwl> 1,

(8)

and

V=M(q)i

W( cp) = dq(cp) =

i=W('P)V

dip

F ig. 1. Charge-controlled memristor (left) . Fl ux-cont rolled memristor (r ight ).

respectively. Since the instantaneo us power dissi ­ pated by t he above memristor is given by

A memristor characterized by a d ifferentiable

q - cp (resp. cp - q) characteristic curve is pas­ sive if, and on ly if, its small-signal memristance M(q) (resp. small-signal memductance W( cp)) is non-n egative; i.e ,

M(q) =

d~~q) ~ 0

(resp. W( cp ) =

d~~) ~ 0) (4)

(see [Chua , 1971]). In this paper, we ass ume t hat the memristor is characterized by the "monotone­ increasing" and "piecewise-linear" non linearity shown in Fig. 2, namely,

cp (q) = bq + 0.5(a - b)(lq + 11 - Iq - 11),

(5)

q(cp) = dip + 0.5(c - d)(l cp + 11-Icp

(6)

or

-

11),

d,

p(t) = M(q (t))i(t)2 ~ 0 ,

(9)

p(t) = W( cp(t ))V(t )2 ~ 0,

(10)

or

the energy flow into the memristor from time to to t satisfies

i

t

p(r )dr

~ 0,

(11)

to

for all t ~ to. Thus , the memristor constitutive rela­ t ion in Fig. 2 is passive. Consider next the two-terminal circ uit in Fig. 3, which consists of a negative resist an ce' (or a negative conductance) and a passive memristor. If t he two-terminal circuit has a flux-cont rolled ...

r "­ slope = b

q slope = a

slope = c

Fig. 2. The const it ut ive relation of a monotone-in creasing piecewise-linear memristor: Ch arge-controlled memrist or (left). Flux-controlled memristor (r ight) . 3 T he

negative resistance or conductance ca n be realiz ed by a standard op amp circuit, power ed by batteries.

Memristor Oscillat ors

3185

memristor, we obtain the following .p - q curve

-R

+

q(ep ) = j i(T)dT

V

-G

= j (i 1 (T) + i 2(T))dT

= j (W( ep)v - GV)dT

Vl =M(q)i

il =W('P)V

= j (W(ep) - G)VdT

=

j (W (ep) - G) dsp (~~

=

dsp + 0.5(c - d)(lep + 11- lep - 11) - Gep

Two-term inal circuit

!

=

v)

= (d - G)ep + 0.5(c - d)(lep + 11- lep - 11), (12)

+

+

v

v

V=M(q)i

where we assumed that q(ep) is a cont inuous func ­ tio n satisfying q(O ) = 0 and G > O. Thus , th e sm all signal memductan ce W (ep ) of this two-termin al cir­ cui t is given by

W (ep ) = dq(ep ) = dep

i=W('P)V

If c - G < 0 or d - G power do es not satisfy

Active memristor

Fig . 3. Two-t erminal circui t which consists of a memr ist or and a nega ti ve cond uctance - G (or a resistance -R).

{C- G , d - G,

Iwl < 1, Iwl > 1.

< 0, then the instantaneous

p(t ) = W (ep(t ))v (t )2 2: 0,

(14)

- d'

q

q

slope

=

d'

slope ­ c'

1 -1

f

Fig . 4. 'P - q charac te rist ic of the two-t erminal circu it .

(13)

r

3186

M. It oh & L. O. Chua

for all t and

>

0. In t his case, there exists cp(to)

i

t p(T)dT to

< 0,

=

CPo

(15)

for all t E (to, t l )' Thus, the two-terminal circuit in Fig. 3 can be designed to become an active device, and can be regarded as an "ac tive m emri stor " . We illustrate two kinds of characteristic curves in Fig. 4. Similar char act eristic cur ves can be obtained for charge-controlled mem risto rs. In this pap er , we design several nonlinear oscilla tors using active or pas sive memristors .

2.

between current i and voltage v of the memri st or is defined by Eq. (1). If we integrate th e Kir chhoff's circuit laws with respect to time t , we would obtain th e relati on on the conse rva tion of charge and flux:

(18) m

and (19) n

where qm and CPn are defined by

.[t

qm =

Circ uit Law s

i-« dt ,

(20)

oo

In thi s section, we review some basic laws for elec­ trical circuits. Recall first t he following principl es of conservation of charge an d flu x [Chua, 1969]:

and

• Cha rge an d flux can neith er be created nor destroyed. The qu a ntity of cha rge an d flux is always conserved .

resp ectively. The relationsh ip between voltage v and cur­ rent i for th e four fundamental circuit elements is given by

We can restate thi s principle as follows: • Cha rge q and volt age vc across a capacito r can­ not cha nge inst antan eously. • Flux sp and current it. in an inductor cannot cha nge instantaneously.

(21)

• Capacitor

c dv dt

= i

(22)

L-= v

(23)

v = Ri

(24)

• Induct or Ap plying t his pri nciple to the circuit, we can obtain a rela tion between the two fun damental circuit vari­ abl es: the "c harge" and the "flux " . However, we usu ally use th e other fundamental circuit var iabl es, na mely the "volt age" an d th e " curre nt" by apply­ ing th e following K irchhoff 's circuit laws [Chua , 1969]: • The algebraic sum of all th e currents i m flowing into the node is zero: (16) m

• The algebraic sum of branch volt ages any closed circui t is zero:

Vn

around

(17)

di dt

• Resistor

• Memri stor v = M(q) i

(25)

Using t hese relati ons and t he Kirchhoff's circuit laws, we can descri be t he dynamics of electrical circuits. Integrating Eqs. (22)- (25) with resp ect to time t , we obtain th e following equa tions: • Capacitor

n

They are a pair of laws t hat resu lt from the con­ servation of charge and energy in elect rical circuits. If we apply t he Ki rchh off's circu it laws to a mem­ ristive circ uit, we need th e f our fundam ental circuit variables, nam ely the voltage, cur rent, cha rge, and flux to describe th eir dyn ami cs , becau se th e relation

(or i = W( cp ))

q

= C v

(26)

ip

= Li

(27)

ip

= Rq

(28)

• Inductor

• Resistor

Memristor Oscillator s

• Memristor ip

=

J

M(q)dq

(or

q=

J

W( 1,

a(y - W(w)x) ,

Iw - I I),} (40)

respectively, where a, b > O. Note that the unique­ ness of solutions for Eq. (39) cannot be guaranteed since W (w) is d isconti nuous if ai-b. If we set a = 4, f3 = 1, , = 0.65, a = 0.2, an d b = 10, our com ­ puter simulation'' shows that Eq. (39) has a chaotic attractor as shown in F ig. 10. By ca lculating the Lyapunov exponents from sampled time ser ies, we found t hat this chaotic attracto r has on e posit ive

1.5 1

0 .5

z

o -0.5 -1

-1.5 -2

2

-4

2 1.5 1

y

0 .5

o

-0.5 -1

-1.5 -2 -2.5

- 1.5

Fig. 10.

3189

3

Chaotic att ractor of t he canonical Chua's osci llator with a flux-controlled memristor.

5We used t he fourth-order Rung e-Kutta method for integrating t he diffe rential equations.

M. Itoh & L. O. Chu a

3190

Lyapunov expo nent Al ~ 0.27.6 Furthermore, the divergence of t he vecto r field

which corresponds to the w-axis. The J acobian matrix D at this equ ilibrium set is given by

+ "( 4W(w) + 0.65

div (X ) = - a W (w) = -

Iwl < 1, Iwl > 1,

- 0.15, { -39.35,

D= (41)

p4

-0.267093 ± i 2.148, 0.274905

± i 0.9 28318,

+

d 2z

(aW(w) - "() dt 2 + ((3

+ a((3W (w ) -

~

0.384186,

P3

~

- 39.8998,

dz

+ a - a"(W (w ) ) dt

= O.

"()z

-(3

1

o

= 0, P4 = 0,

P4

2

(43)

I,}

(44)

dVI

.

C I dt =

1.3 -

W( 1.

(51)

Fr om Eq. (50), we obtain

{I

2

2

-dtd -2 ( -xa + -y~ + -z{32)

} = - W (w )x 2 < 0 - ,

(52)

3191

st able, and Eq . (50) does not hav e a chaotic attrac­ tor. However , if we set a = 4.2, (3 = - 20, ~ = -1, a = -2 and b = 9, our computer simu lation of Eq . (50) gives a chaotic attractor in Fig. 12. By calc ulating the Lyapu nov exponents from sampled time series, we found that t his chaotic attractor has a positive Lyapunov exponent Al ~ 0.050 . In this case, the capacitance C2 and the ind uctance L are both negative (ac tive) and the mem ristor is active as shown in F ig. 13 (see [Barb oza & Chua, 2008]) . T he J acobian matrix D at the equilibrium set is given by

- a w (W)

assuming a > 0 an d b > O. In t his case, the equ i­ librium state A = {(x ,y , z ,w)lx = Y = z = 0, w = constant} (i.e. the w-axis) is globally asymptotically

-~

D=

r

(53)

o 1

60 50

40

z

30

20

10

o

-1 0

-20

-30

-40

-50

15

10

y

-5

-1 0 2

-1 5

15 10

Y

5 0 -5 -10 -15 20 15

10

-5 -10

w F ig. 12.

1 .5

o

5

x

-1 5 -2 0

Chaotic attractor of the four t h-order oscillator with active elements (a = - 1, b = 5).

3192

M. It oh & L. O. Chua

+

I

------------------- ,I

I I

I I

: ,

Flux-controlled : memristor ,

I

I

I

I I

L

~

+ V2

Act ive memristor

Fig . 13. A four-element fourth-orde r oscillator with three active eleme nts, one linea r capac ito r, one linear ind uctor , an d a memristor.

and its characte rist ic equat ion is given by 4

p

+ QW (w )p3 + (Q + (J) ~p2 + Q(J~ W(w) p =

O.

(54)

The four eigenvalu es Pi (i = 1,2 ,3 ,4) a t each equilibrium state (0, 0, 0, w) can be written as P I ,2 ~

- 0.189912 ± i 4.37021 ,

P3 ~ 8.77982,

P I ,2 ~

0.0546351 ± i 4.46 535,

P3 ~ -37.9093 ,

Thus , they are characterized by an unst abl e sad dle­ focus exce p t for the zero eigenvalue .

P4 = 0, for Iwl < 1, } P4 = 0, for Iwl > 1.

Integr ating Eq. (56) with resp ect to time t , we obtain a set of equat ions wh ich define the relation b etween the charge and the flux:

3.2. A third-order canonical m emristor os cillator Remo ving a capacitor (resp. an inductor) from t he circuit of Fi g. 7 (res p. F ig. 8), we obtain t he t hird­ orde r oscillator in Fig. 14 (resp. Fi g. 15). Applying Kirchh off's circu it laws to node A and loop G of t he circuit in' Fig. 16, we obtain

ql = q3 - q(cp ),} CP3 = CP4 - CP I,

6jt ql 6jt q3 6jt q =

=

- 00

-00

-00

i l(t)dt, i3(t) dt, i( t)dt,

6jt (t )dt, 6jt v3(t) dt, CP3 6jt v4(t) dt, CP4 6jt v(t)dt = = CP I =

-00

=

-00

=

-00

cP

-R

Flux-controlled memristor

Fig . 14. A th ird -ord er oscillato r wit h a flux-contr olled memris to r.

(57)

where

=

(56)

(55)

-00

VI

CP l ·

(58)

Here, the symbols ql , q3 , and q den ote t he charge of capacitor G I , indu ct or L , and th e memris­ tor, resp ectively, and the symbols CP I, CP3, CP4 and cP denote the fl ux of capacitor GI , indu ct or L ,

Mem ristor Oscill ators

3193

where

Ll

+

-G

dq , dt

dq3 dt

_ C dVI I dt '

=

zl -

dCP3 = V = L di3 dt 3 dt '

. z3,

ill = V4 = R Z3,

dCP4

.

V C

Charge -co ntro lled mem ristor

dCPI ----;[t= VI'

W(cp ) =

d~;) . (62)

Not e that the two kind s of ind ependent vari ables are related by Fi g. 15.

Du al circu it with a charge-cont rolled memri stor.

(ql,cp,CP3)

+ + V4

-R

A

V3­

ql = C I VI , CP3 = Li3

, ,-- ..... , I

\

\

I

I

Flux -cont rolled mem risto r

I

::...

-' ' C

F ig. 16. Currents indi ca ted .

ij ,

volt ages

+ 0.5(a -

b)(lcp

+ 11- Icp - 11).

(59)

di 3 L~ dt dcp

.

= R Z3 -

di =

VI,

VI

'

(61)

W(cp )ql CI

-

dq, dt

CP3 L

dCP3 dt

--- -

R CP3 L

dcp dt

ql

CI'

(64)

ql

-

CI '

We next study t he behavior of this circuit. Equation (61) can be t ransfor med into the form

dx dt = ex (y - W( z )x) ,

Solvin g Eq . (57) for (q3 ' CP4), we get

q3 = ql + q(cp), } (60) CP4 = sp + CP3 · Thus, (ql , sp ; CP3) can be chosen to be the ind epen­ dent vari ables , namely, th e charge of capaci tor C I , th e flux of inductor L , and the flux of th e memris­ tor, resp ectively. From Eq. (56) (or differentiating Eq. (57) with resp ect to time t ), we obtain a set of three first­ ord er d ifferential equations, which defines t he rela­ tion among t he t h ree variables (V I , i3 , cp): dVI C I dt = i3 - W (cp )VI,

(63)

Thus, Eq. (61) can be recast into th e following set of differential equations using only charge and flux as var iables :

node A , and loo p Care

Vj,

resist an ce - R , and t he memri stor, respectively.f The sp - q characteristic curve of th e memristor is given by

q(cp ) = bcp

(vI, cp , i3)

1.

A3 = 0,

•

(69)

f'::,

Thus, the set B = {(x ,y, z )lx = y = O ,lzl < I} IS un stable, and the set C = {(x , y , z )lx = y = 0, Izl > I} is stable. Our com puter sim ulation shows that Eq. (65) has two distinct stable periodic attractors as shown in Fig. 17. Observe that they are od d symmetr ic images of each other , as expected in view of the odd -symmetric cha racterist ic q = q( 1,

(76)

+ L

From t his equation, we obtain dt

d~~t) } + d I < 1, )

= a(y - W( z) x) ,

dy

dt = -~x , dz

dt

,

for

a

~

2

< O.

­

+

(78)

Charge-controlled memristor

C

V

Fig . 19.

Dual circu it with a charge-cont rolled memristor.

Hence, t he z-axis is globally asymptotically stable. From Eq. (77), we obtain dy dt

c dz = O.

+