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International Journal of Bifurcation and Chaos, Vol. 18, No. 11 (2008) 3183-3206 © World Scientific Publishing Company

MEMRISTOR OSCILLATORS

MAKOT O ITOH Departm ent of Information and Communication Engineering,

Fukuo ka Institute of Technology, Fukuoka 811-0295, Japan

LEON O. CHUA Departm ent of Electri cal Engineering and Compu ter S ciences,

University of California, B erkeley, B erkeley, CA 94120, USA

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

The memristor has a ttracted phenomenal worldwide attent ion since its debut on 1 May 2008 issue of Nature in view of its many po te nt ial applications, e.g. super-dense nonv ola tile com­puter memory and neural synapses. T he Hewlett-Packard memristor is a passive nonlin ear two­t erminal circuit element t ha t maintain s a fun cti onal rela t ionsh ip between the time integrals of current and voltage, res pectively, viz. charge an d flux. In thi s paper, we derive seve ra l nonl inear oscillators from Chua's oscillators by rep lac ing Chua's diodes with mem ristors.

K eywords: Memristor ; memrist ive devices; memristive systems; charge; flux; Chua's oscilla tor ; Chua 's diode; learning; ne urons; synapses; Hodgkin-Huxley; nerve membrane model.

1. Memristors T he HP m em rist or shown in Fig. 1 is a pas­sive two-terminal elect ronic device described by aIn a seminal paper [Strukov et al., 2008] which nonlin ear constitutive relation appeared on 1 May 2008 issue of Nature, a team

led by R. Stanley Williams from the Hewlet t ­ v = M( q)i, or i = W (cp )v , (1) Packard Company announced the fab ricat ion of '. between the device terminal voltage v and termin al a nanometer-size solid-state two-termina l dev ice

current i . The two nonlinear functions M (q) and called the m em ristor, a contraction for memory

W (cp), called the mem ristance and memductance, resis tor, which was postulated in [Chua , 1971;

respect ively, ar e defined by Chua & Kang, 1976]. This passive elect ronic device has generated unprecedented worldwide interest .' M ( ) ~ dcp(q) (2)q dq ' because of its potent ial app lications [Tour & He, 2008; Johnson , 2008] in the next gener ation com­ and puters and powerful brain-like "neural" computers.

W ( ) ~ dq(cp)One imm ediate applicat ion offers an enablin g low­ (3)cp dcp ' cost technology for non-volatile memories/ where future computers would turn on instant ly without representing the slope of a scalar function ip = cp(q)

the usual "booting time", currently required in all and q = q(cp), respect ively, called the memristor

personal computers. constitutive relat ion.

IMore than one million Google hits were registered as of June 1, 2008. 2The Hewlett- Packard memristor is a t iny nano, passive, two-terminal device requiring no bat teries. Memristors characterized by a nonmonotonic constitutive relat ion are called active memristors in this paper because t hey require a power supply.

3183

v

3184 M. Itoh & L. O. Chua

+ + v

V=M(q)i i=W('P)V F ig. 1. Charge-controlled memristor (left) . Fl ux-cont rolled memristor (r ight ).

A memristor characterized by a differentiable q - cp (resp. cp - q) characteristic curve is pas­sive if, and only if, its small-signal memristance M(q) (resp. small-signal memductance W( cp)) is non-negative; i.e ,

M(q) = d~~q ) ~ 0 (resp. W( cp ) = d~~) ~ 0) (4)

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

cp (q) = bq + 0.5(a - b)(lq + 11 - Iq - 11), (5)

or

q(cp) = dip + 0.5(c - d)(l cp + 11-Icp - 11), (6)

r

slope =

"­slope = b

q

a

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

Iql < 1, M (q) = d<jJ(q) = {a, (7) dq b, Iql > 1,

and

W( cp) = dq(cp) = {C, Iwl < 1, (8)

dip d, Iwl> 1,

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

p(t) = M(q(t))i(t)2 ~ 0, (9)

or

p(t) = W( cp(t ))V(t )2 ~ 0, (10)

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

it p(r )dr ~ 0, (11) to

for all t ~ to. Thus , the memristor constitutive rela­t ion in Fig. 2 is passive.

Consider next the two-terminal circuit in Fig. 3, which consists of a negative resistan ce' (or a negative conductance) and a passive memristor. If the two-terminal circuit has a flux-cont rolled

....

slope = c

Fig. 2. The const itut 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, powered by batteries.

v

Vl =M(q)i

-R

+ V -G

il =W('P)V

Two-term inal circuit

! + v

+

V=M(q)i i=W('P)V

Active memristor

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

Memristor Oscillat ors 3185

memristor, we obtain the following .p - q curve

q(ep ) = j i(T)dT

= j (i1 (T) + i2(T))dT

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

= j (W(ep) - G)VdT

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

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

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

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

W (ep ) = dq(ep ) = {C- G, Iwl < 1, (13) dep d - G, Iwl > 1.

If c - G < 0 or d - G < 0, then the instantaneous power does not satisfy

p(t ) = W (ep(t ))v(t )2 2: 0, (14)

-1

- d' q

1

f

q slope = d'

slope ­ c'

r

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

3186 M. It oh & L. O. Chua

for all t > 0. In this case, there exists cp(to) = CPo and

it p(T)dT < 0, (15) to

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 emristor" . We illustrate two kinds of characteristic curves in Fig. 4. Similar char act eristic curves can be obtained for charge-controlled mem risto rs. In this pap er , we design several nonlinear oscilla tors using act ive or passive memristors .

2. Circ uit Laws

In thi s section, we review some basic laws for elec­trical circuits. Recall first t he following principles of conservation of charge and flu x [Chua, 1969]:

• Charge and flux can neith er be created nor destroyed. The quantity of cha rge and flux is always conserved .

We can restate thi s principle as follows:

• Cha rge q and voltage vc across a capacito r can­not cha nge inst antan eously.

• Flux sp and current i t. in an inductor cannot cha nge instantaneously.

Ap plying this principle to the circuit, we can obtain a rela tion between the two fun damental circuit vari­ables: the "c harge" and the "flux" . However, we usu ally use th e other fundamental circuit var iabl es, namely the "voltage" an d th e "current" by apply­ing 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 Vn around any closed circui t is zero:

(17) n

They are a pair of laws that resu lt from the con­servation of charge and energy in elect rical circuits. If we apply t he Kirchhoff 's circu it laws to a mem ­ristive circuit, we need th e f our fundam ental circuit variables, namely the voltage, cur rent, charge, and flux to describe th eir dyn amics , because the relation

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 conservation of charge and flux:

(18) m

and

(19) n

where qm and CPn are defined by

qm = .[t i-« dt , (20) oo

and

(21)

resp ectively. The relationsh ip between voltage v and cur ­

rent i for th e four fundamental circuit elements is given by

• Capacitor

dv c = i (22) dt

• Inductor di

L-= v (23) dt

• Resistor

v = Ri (24)

• Memri stor

v = M(q) i (or i = W( cp )) (25)

Using t hese relati ons and the Kirchhoff's circuit laws, we can descri be t he dynamics of electrical circuits.

Integrating Eqs. (22)- (25) with respect to t ime t , we obtain th e following equa tions:

• Capacitor

q = Cv (26)

• Inductor

ip = Li (27)

• Resistor

ip = Rq (28)

Memristor Oscillators 3187

• Memristor

ip = JM(q)dq (or q = JW( <P)d<P) (29)

where q = J~ CX) idt and <P = i': vdt. T hey provide the relationship between Eqs. (16)-(17) and Eqs . (18)-(19).

3. Memristor-Based Canonical Oscillators

Chua's circuit in Fig . 5 is the simp lest electronic circuit exhibit ing chaotic be hav ior [Mad an, 1993]. It is well know n that the canonical Chua 's oscil­lator [Chua & Lin, 1990] in Fig. 6 also has a chaotic attractor. In th is section, we design a non­linear oscillator by replacing the "Chua's diode" in the canonical Chua's oscillator with a memris­tor characterized by a "m onotone-increasing" and "piecewise-linear" non linearity.

3.1. A fourth-order canonical memristor oscillator

Consider the canonical Chua's oscillator in Fig. 6. If we replace the Chua 's diode in Fig. 6 with a

R

+ + Chua'sL V2 V1 diodeC2 C1

A

..

Fig. 5. Chua's circuit.

+ + V2 V1-G Chua 's

C2 C1 diode

Fig. 6. Canon ical Chu a' s oscillator.

+ Flux-controlledV2-G

memristor

Fig. 7. Canonical Chua 's oscillator with a flux-controlled memristor.

L2 Ll + V

C -R Charge-contro lled

memristor

Fig. 8. Dual circui t with a charge-controlled memristor.

flux-controlled memristor, we would ob tain the cir­cuit of Fig. 7. Its dual circuit/' can be easily obtained by using a charge-contro lled memristor (see F ig. 8) .

Applying Ki rchhoff's circuit laws to the nodes A, B and the loop C of the circuit in F ig. 9, we obtain

i l = i3 - i , } V3=V2 - Vl, (30)

i2 = - i3 + i4.

Integrati ng Eq. (30) wit h respect to time t, we get a set of equations which define the relation among two fundamental circuit variables, namely, the charge

AB + V3

+ ... - ......+ +I \

V4

,C

V2 I I,I Flux-controlled V IC2 \

I memristor '.l..-G ­

Fig . 9. Currents ij , voltages Vj, nod es A, B , and loop C are indicated .

4 A pai r of circuits N and Nt are dual if the equations of the two circ uits are identical , after a trivial cha nge of symbols. For more details, see [Chua, 1969].

3188 M. ltoh & L. O. Chua

and the flux:

ql = q3 - q(<.p) , } <.p3 = <.p2 - <P I, (31)

qz = -q3+q4 ,

where

6jt ql = - 00 i l( t) dt, 6jt

<P I = - 00 VI (t) dt ,

6jt<.p2 = - 00 v2(t)dt,

6jt <.p.3 = - 00 v3(t )dt,

6jt <.p = - 00 v( t) dt = <.p l · 6jt

q = - 00 i( t )dt ,

(32)

Here, t he symbo ls ql , q2,q3, q4 and q deno te t he charge of the capac itors Gl , G2 , the inductor L , th e conductance -G, and the memri stor , respectively, and the symbols <P I, <.p2, <.p3 and ip denote th e flux of the capacitors Gl , G2, the inductor L , and th e mem­ristor , respectively. The sp - q characterist ic curve of the memristor is given by

q(<.p ) = lxp + 0.5(a - b)(I<.p + 11 - l<.p - 11) . (33)

Solving Eq . (31) for (q3' qs , <P I , <.p2) , we get

Thus, (ql' q2, ip, <.p3) can be chosen to be the inde­pendent vari ables, namely, the cha rge of the capac­itors Gl , G2 , and the flux of the inductor L and the memristor , respectively.

From Eq. (30) (or differentiating Eq . (31) with respect to time t), we obtain a set of four first-ord er differential equations , which define the relation among the four circuit variables (VI, V2, i3, <.p):

du, Gl dt = i3 - W(<.p)V l,

di 3 L&= V2 - Vl ,

(35) dV2

G2 dt = -i3 + GV2 ,

d<.p dt = VI,

where

dql _ i _ G dVl dt - 1 - 1 dt '

dq2 _ i _ G dV2 dt - 2 - 2 dt '

d<.p3 = V = L di 3 dq3 .-;It = 23, dt 3 dt '

W( <.p) = dq(<.p). d<.p

(36)

Note that the two kinds of independ ent vari ables are related by

.. (37)

T hus, Eq. (35) can be recast int o the following set of differential equations using on ly charge and flux as variables:

dq, <.p3 W (<.p )ql

dt L Gl (38)

dq2 <.p3 Gq2 -;It = -y + G '

2

Memristor Oscillators 3189

We next study the behavior of this circuit . the piecewise-linear fun ctions q(w) and W( w) are Equation (35) can be transformed into the form given by

dx dt

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

dy

dt = z - x ,

dz dt = - f3y + , z,

dw di =x,

q(w) = bw + 0.5(a - b)( lw + 11- Iw - I I),} dq(w) { a, Iwl < 1, (40)

W(w) = -- = dw b, Iwl > 1,

respectively, where a, b > O. Note that the unique­(39) ness of solutions for Eq. (39) cannot be guaranteed since W (w) is discontinuous if ai-b. If we set a = 4, f3 = 1, , = 0.65, a = 0.2, an d b = 10, our com­puter simulation'' shows that Eq. (39) has a chaotic attractor as shown in F ig. 10. By ca lculating the

where x = V I , Y = i3, z = V2, W = ip , a = Lyapunov exponents from sampled time ser ies, we 1/C1 , f3 = I/C2 , , = G/C2 , L = 1, an d found t hat this chaotic attracto r has one posit ive

z

1.5

1

0 .5 o

-0.5 -1

-1.5 -2

- 4 2

y

2 1.5

1 0.5

o -0.5

- 1 -1.5

-2 -2.5

- 1.5

Fig. 10. 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.

3

3190 M. Itoh & L. O. Chua

Lyapunov expo nent Al ~ 0.27.6 Furthermore, the which corresponds to the w-axis. The J acobian divergence of the vecto r field matrix D at this equ ilibrium set is given by

div (X ) = - a W (w) + "( o 0] - 1 o 1 0 = - 4W(w) + 0.65 D= (42)o -(3 "( 0 '

- 0.15, Iwl < 1, 1 o o 0r(41) {- -39.35, Iwl > 1, and its characteristic equa t ion is given by

is negative. It follows that the Lebesgue measure of this chaotic attractor is zero, and at leas t one p4 + (aW(w) - "()p3 + ((3 + a - a"(W(w))p2

Lyapunov exponent must be negative," + a((3W(w) - "()p = O. (43) The equilibrium state of Eq. (39) is given by set

The four eigenvalues Pi (i = 1,2, 3, 4) of the equi­A = {(x,y, z,w ) lx = y = z = 0, w = constant.}, I libri um state (0,0 ,'0, w) can be writt en as

P I,2 ~ -0.267093 ± i 2.148, P3 ~ 0.384186, P4 = 0, for Iwl < I,} (44)

PI,2 ~ 0.274905 ± i 0.928318, P3 ~ - 39.8998, P4 = 0, for Iwl > 1.

T hus, they are characterized by an unstable saddle­ I focus except for the zero eigenvalue. Furthermore, writ ten as Eq. (39) can be t ransformed into the form dVI .

CI dt = 1.3 - W(<P)VI , d3 d2zz dz dt 3 + (aW(w) - "() dt 2 + ((3 + a - a"(W (w )) dt di 3

L - = V2 - VI dt '

(49)+ a((3W(w ) - "()z = O. (45) dV2 .

C2 dt = - 1.3 , If we subst itute

d<pdi = VI,

u (t ) = itz(t)dt + uo, (46) Eq uation (35) can be transformed into the form

into Eq. (45), we would obtain a fourth-order dif­ dx dt = a(y - W(w)x),ferent ial equation in t he variable u; namely,

d4u d3u d2u dydt = -~( x + z),

dt 4 + (aW(w ) - "() dt 3 + ((3 + a - a"(W(w )) dt 2 (50) dz

du dt = (3y,+ a((3W(w) - "()di = 0, (47)

dwwhere di =x,

d2udu (3u-"( - + ­ 2

w(t) = ~ dt + woo (48)

Here, Uo and Wo are constants . Thus , its cha racter­istic equat ion also has a zero eigenvalue.

Consider next the fourth-order oscilla tor in Fig. 11 obtained by removing a resis tor from the Fi g. 11. A fourth-order osci llator wit h a flux-con t rolled circuit of F ig. 7. T he circuit equa t ion can be memristor .

6We used the software pac kage MATD S [Govorukh in , 2004] to calcula te the Lyapunov exponents. 7Note t hat if the system is a flow, one Lyapunov exponent is always zero, which corresponds to the dir ection of t he flow.

Flux-controlled memristor

where x = VI, Y = is , Z = - V2, W = ip, a = I/C1 , (3 = I/C2 , ~ = I/L , and the piecewise-linear functions q(w) and W (w) are given by

q(w) = bw + 0.5(a - b)(lw + l l - lw - I I),}

W (w) = dq(w) = {a, Iwl < 1, (51) dw b, Iwl > 1.

From Eq. (50), we obtain

2 2 d {I ( x y z2) } 2- - - + - + - = - W (w)x < 0 (52)dt 2 a ~ {3 - ,

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

Memristor Oscillators 3191

st able, and Eq . (50) does not have a chaotic attrac­tor. 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 memristor is active as shown in F ig. 13 (see [Barboza & Chua, 2008]) .

T he J acobian matrix D at the equilibrium set is given by

- a w (W)

D= -~ (53)o r 1

60 50 40 30 20z 10 o

-10 -20 -30 -40 -50

15

10

-5 y-10

2 -1 5

15

10

5

Y 0

-5

-10

-15

-5 -10

-1 5

o 5

15 10

x

20

w 1.5 -2 0

Fig. 12. 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

L I ~

+

+ V2 Act ive

memristor

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

and its characte rist ic equat ion is given by

p 4 + QW (w )p3 + (Q + (J) ~p2 + Q(J~ W(w) p = O. (54)

The four eigenvalues 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, P4 = 0, for Iwl < 1, } (55)

P I ,2 ~ 0.0546351 ± i 4.46535, P3 ~ -37.9093, P4 = 0, for Iwl > 1.

Thus , they are characterized by an unstabl e sad dle­Integr ating Eq. (56) with resp ect to time t , wefocus excep t for the zero eigenvalue .

obtain a set of equat ions which define the relation between the charge and the flux:

3.2. A third-order canonical ql = m emristor os cillator CP3 =

Removing a capacitor (resp. an inductor) from the circuit of Fig. 7 (resp. F ig. 8), we obtain the third ­ where order oscillator in Fig. 14 (resp. Fig. 15). Applying Kirchh off 's circu it laws to node A and loop G of the circuit in' Fig. 16, we obtain 6jt

ql = - 00 i l(t)dt,

(56) 6jtq3 = -00 i3(t) dt,

6jtq = -00 i( t)dt,

-R Flux-controlled memristor

q3 - q(cp ),} (57)

CP4 - CP I,

6jtCP I = -00 VI (t )dt,

6jtCP3 = -00 v3(t)dt,

6jt CP4 = -00 v4(t) dt,

6jtcP = -00 v(t)dt = CP l ·

(58)

Here, the symbols ql , q3 , and q denote the charge of capacitor GI , indu ctor L , and th e memris­

Fig . 14. A th ird -ord er oscillato r with a flux-contr olled tor, resp ectively, and the symbols CP I, CP3, CP4 and memris to r. cP denote the fl ux of capacitor GI , indu ct or L ,

--- -

Ll +

-G V

C

Charge -contro lled memristor

Fig. 15. Du al circu it with a charge-cont rolled memri stor.

A

,-- ....., , -R I \

I I \

' -' I

::... C

Flux-cont rolled mem risto r

+ V3­

+ V4

F ig. 16. Currents i j , volt ages Vj, node A, and loop Care indicated .

resist ance - R, and the memri stor, respectively.f The sp - q characteristic curve of the memristor is given by

q(cp ) = bcp + 0.5(a - b)(lcp + 11- Icp - 11). (59)

Solving 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 indepen­dent vari ables , namely, the charge of capaci tor C I ,

th e flux of inductor L , and the flux of th e memris­tor , respectively.

From Eq. (56) (or differentiating Eq. (57) with respect to time t ), we obtain a set of three first­ord er differential equations, which defines the rela­tion among the three variables (V I , i3 , cp):

dVI CI dt = i3 - W (cp )VI,

di3 . (61) L~ = R Z3 - VI

dt '

dcpdi = VI,

Memristor Oscillators 3193

where

dq, _ C dVI dCP3 = V = L di3 zl - I dt ' dt 3 dt ' dt

dq3 . dCP4 .ill = V4 = R Z3, dt = z3,

dCPI W(cp ) = d~;) .----;[t= VI'

(62)

Note that the two kinds of independent vari ables are related by

(ql,cp,CP3) <II (vI, cp , i3) (63) ql = CIVI , CP3 = Li3

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

dq, CP3 W(cp )ql -dt L CI

dCP3 R CP3 ql (64) dt L CI'

dcp ql - CI ' dt

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

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

dy(65)dt = -~x + f3y,

dz - = x dt '

where x = VI , Y = i3, z = ip, ex = l / CI , ~ = 1/L, f3 = R / L , and the piecewise-linear functions q(z ) and W( z ) ar e given by

q(z) = bz + 0.5(a - b)(lz + 11- [z - 11), }

W( z) = = {a, Izl < 1, dq(z ) (66) dz b, Izl > 1,

respectively, where a, b > O. The equilibrium st ate ofEq. (65) is given by the

set A ~ {(x ,y, z )lx = y = 0, z = constant}, which corresponds to th e z-axis. The Jacobian matrix D

8T he term "charge" and "flux" are just names given to th e definition in Eq . (58) , and should not be interpret ed as a phys­ical cha rge or flux in the classical sense . The imp ortant concept here is that they ar e m easurable qu ant it ies, obtained via int egr ation .

3194 M. Itoh & L. O. Chua

at this equ ilibrium set is given by

- aw (Z) a 0]

r D = -~ (3 0 , (67)

1 0 0

and its characterist ic equation is given by

p3 + (aW( z) _ (3) p2 + a( ~ - (3W(z))p = O. (68)

If we set a = 1, (3 = 0.1 , ~ = 1, a = 0.02, and b = 2, then it has three eigenvalues Ai (i = 1, 2, 3):

Al,2 ~ 0.04 ± i 0.998198, A3 ~ 0, for Izi < I , }

Al ~ - 1.27016, A2 ~ -0.629844, A3 = 0, for [z] > 1. (69)

f'::, • f'::,Thus, the set B = {(x ,y, z )lx = y = O,lzl < I} IS unstable, and the set C = {(x , y , z )lx = y = 0,

Izl > I} is stable. Our computer simulation shows that Eq. (65) has two distinct stable periodic attractors as shown in Fig. 17. Observe that they are odd symmetr ic images of each other , as expected in view of the odd -symmetric characterist ic q = q(<p) of the memristor in Eq. (66).

Equ ation (65) can be transformed into the form

d2y dydt2 + (aW (z ) - (3) dt + a( ~ - (3W (z ))y = 0, (70)

or equiva lent ly

d2y dy- + (co - (3)- + a(~ - a(3)y = 0 for Izi < I,}dt2 dt ' (71)

d2y dy- + (ba - (3 )- + a(~ - b(3 )y = 0 for [z] > 1.&2 & '

Thus, Eq. (65) can be interpreted as a secon d-orde r linear different ial equation over the domain of the state variable z whose dynamics evolves according to dz/dt = x in Eq. (65). Furthermore, if we substitute

t

u(t) = in y(t)dt + c,f'::, r )

(72) z (t) ~ £.:1 { flU(!) - d~;t) } + d,

1.5 Z 1

0 .5 o

-0 .5 -1

-1.5

-0.0 8

-0.06

-0 .0 4

-0 .0 2

o x 0 .02

0 .04

0 .06 -0 .06 0.08 ­ 0 .00

Fig. 17. Two pe riodic a t t rac tors of the thi rd-order canonical memristor oscillator.

Memristor Oscillators 3195

into Eq. (70) (c and d are constants), we would obtain the following third-order differential equation

~:~ + [aw(C'{ I3U(t)- d:;t) }+ d)-13] ~~ +a[, -I3W (C'{ I3U(t)- d:;t) }+ d)] ~~ ~ 0

(73)

in terms of u, or equivalently

d3u d2u du dt 3 + (aa - (3) dt2 + a(~ - a(3) dt = 0, for I ~- l { (3u (t ) - d~~t) } + d I < 1,

(74) d3u d2u du dt 3 + (ba - (3) dt 2 + a( ~ - b(3 ) dt = 0, for I C 1 { (3u(t ) - d~~t) } + d I > 1.

They can be written as

~ [~~ + [aW(C' {I3U(t) - d:;t) }+ d) - 13] ~~ + a[, - I3W(C' {I3U(t) - d:;t) }+ d) ]U] = 0,

(75)

and 2

d { d U du }- - + (aa- (3 )- + a (C- a(3 )u = 0 for I e:' { (3u (t ) - d~~t) } + d I < 1, ) dt dt2 dt <, ,

(76) d {~U ~ }- - + (ba - (3)- + a(C - b(3 )u = 0 for IC 1 { (3u (t ) - d~~t) } + d I> 1, dt dt 2 dt <, ,

respectively. Note that dW( z)jdz = O. Since the characteristic equation of Eqs. (73) and (74) have a zero-eigenvalue everywhere, and Eq. (76) can +be interpreted as a second-order linear different ial

L V Flux-controlledequation, Eq. (65) does not have a chaotic attractor, (1even if the circuit elements are active. memristor

Consider next the th ree-element circuit in Fig. 18, obtained by shor t circuit ing the resistor from Fig . 14 (its du al circuit is shown in Fig. 19).

F ig. 18. A th ird-order circuit wit h a flux-cont rolled The dynamics of this circuit can be written as memristor .

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

dy (77)dt = -~x ,

dz dt = x. L1

From this equat ion, we obtain Charge-controlled

memristor2 +-d { -1 (x-2 + -y2) }= - W (z )x < O. (78)

dt 2 a ~ ­VC

Hence, the z-axis is globally asymptotically stable. From Eq. (77), we obtain

dz dy c = O. (79) dt + <, dt Fig . 19. Dual circu it with a charge-cont rolled memristor.

3196 M. ltoh & L. O. Chua

Thus, y(t) and z(t ) satisfy

(80)

where Yo is a constant. Since W(z) W(Yo - y), Eq. (77) can be transformed into the form

2

--.J!... + aW Yo - Y -.J!... + ay = O.d ( ) d (81) dt2 ~ dt

Thus, Eq. (77) is equivalent to a one-parameter fam­ily of second-order different ial equat ions. Since the mini ma l dimension for a continuous chaotic system is 3, Eq. (77) cannot have a chaotic attractor, even if the circu it elements are active. We will d iscuss thi s observat ion in Sec. 4.2.

3.3 . A second-order canonical memristor circuit

If we remove an inductor (resp. a capacitor ) from Fig. 14 (resp. Fig . 15), we would obtain the second­order circuit in Fig . 20 (resp. Fig . 21). Appl ying the Kirchhoff's circuit laws to the circuit in F ig. 22, we obtain

(82)

Integrating Eq . (82) with resp ect to time t , we obtain a set of equat ions which define a relation between th e charge and the flux:

ql = q3 - q(cp), (83)

where

6jtql = - 00 i l (t)dt ,

(84)

q "= ].- 00 i(t)dt ,

6jt cp = - 00 v(t )dt .

Here, the symb ols qi , qz , and q den ote the charge of capacitor Gl , conductance - G, and the memr istor, and the symbol ip denotes the flux of memristor,

+ Flux-controlled -G V

memristor

. Fig . 20. A second -order circuit with a flux-controlled memristor .

Ll

-R

Charge-controlled memristor

Fig. 21. Dual circui t with a charge-cont rolled memristo r.

respectively. The cp - q characteristic curve of th e memristor is given by

q(cp ) = bcp + 0.5(a - b)( lcp + 11- Icp - 11). (85)

Solving Eq . (83) for q3, we obtain

q3 = ql + q(cp) . (86)

Thus, ql and cp can be chosen as independent variables.

From Eq . (82) (or differentiating Eq . (83) with resp ect to t ime t) , we obtain a set of two first -order differential equations:

(87) dcp7ft --v1 ,

Mem ristor Oscillators 3197

i3

-G + VI

il

(1

i + V

Flux-controlled memristor

q

r slope = a

Fig. 22. The circuit from Fig. 20 and th e <p - q charac te rist ic of t he flux-c ontrolled memristor defined by Eq. (85). Currents ii, i 3 and voltage Vi a re indicated .

where

dqI _ i _ C dVI dt - 1 - 1 dt '

dq3 . dt = '/,3 = e VI ,

W( ) = dq(e.p) e.p de.p ·

given by

q(y) = by + 0.5(a - b)(ly + 11 - Iy - 11), }

W (y ) = dq(y) = {a, Iyl < 1, (92) dy b, Iyl > 1, (88)

resp ect ively, where a, b > O. The first equat ion of Eq. (91) can be wri t ten as

dx { a(f3 - a)x, Iyl < 1, dt = a(f3 - b)x , jyj > 1. (93)

Note that the two kinds of independent variables are related by Thus, the solution of Eq . (93) for Iyl < 1 and Iyl > 1

can be expressed as x( t) = xoeo{B-a)t, (94)

(q I, e.p ) .. ~ (VI, e.p) (89) and qI = C I VI x (t ) = xoea(.B- b) t , (95)

respecti vely, where x(O) = Xo is the ini tial condi­Thus , Eq. (87) can be recast in terms of the charge

t ion for t = O. If we set a = 0.01, b = 0.05. a = 1, and the flux as state variab les: f3 = 0.03 and e = 10, our computer simulation

shows that x (t) -> 0 for t -> 00 as shown in Fig. 23. I ddqt = (e - W( e.p )) ~II' 1 Thus, t his second-order circuit does not oscillate.

(90) de.p q: - - 4. Memristor-Based Chua Oscillators dt C I '

In t his sect ion, we design a nonlinear oscilla tor by replacing "Chua's diode" with an act ive two­We next study the behavior of t his circuit . terminal circuit consist ing of a nega t ive conduc­

1 Equation (87) can be t ransformed into th e form

tance and a memristor (or an act ive memristor ). We derive a set of differential equat ions from t he

dt = a(f3 - W(y )) x , nonlinear circuit directly. dx (91) dy 4.1. A fourth-order memristor-baseddt = x,

Chua oscillator

where x = VI , Y = ip, a = 11C, f3 = e, and Consider Chua 's oscillator in F ig. 24. If we replace the piecewise-linear fun ctions q(y ) and W(y) are Chua' s d iode with an active two-t erm inal circuit

3198 M. Itoh & L. O. Chua

0.45

0.4

0.35

0.3

0.25 X

0.2

0.15

0.1

0.05

a

F ig. 23.

a 250 300 350 400

Solution of th e second-order circuit in Fig. 22.

consisting of a conductance and a flux-controlled memristor , we would obtain the circuit as in Fig. 25. The dynamics of the circuit in Fig. 25 is given by the following set of four first- order differential equations

dVI V2 - VI CI dt = R + GVI - W (<P)VI ,

dV2 _ VI - V2C2 dt - R - i,

di .L- = V2 - rz

dt '

d<pdi = VI ,

where

q(<p) = b<p + O.5(a ­

W (<p) = d~~) .

R

+ V2

C2

r

(96)

b)(I<p + 11- l<p - 11), }

(97)

+ Chua 's VI diode

CI

Fig. 24. Chu a 's oscillator .

Equation (96) can be transformed into the form dxdi = O'(y - X + ~x - W( w)x),

dy dt = x - y + z ,

dz dt = - f3y - ,,(Z ,

dw&= x,

where we set

(98)

x = VI, Y = V2, Z = - i, w = sp ;

1 1 r (99) 0' - , "( = ~ = G, f3 = L' CI L '

C2 = 1, R= 1,

and the piecewise-linear functions q(w) and W(w) are given by

q(w) = bw + O.5(a - b)(lw + 11- Iw - 11), }

W (w) = { a, Iwl < 1, b, Iwl > 1,

(100)

respectively, where a, b > O. If we set 0' = 10, f3 = 13, "( = 0.35, ~ = 1.5, a = 0.3 and b = 0.8, our computer simulat ion shows that Eq. (98) has a chaotic attractor as shown in Fig. 26. By calcu ­lating the Lyapunov exp onents from sampled time series, we found that this chaotic attractor has one positive Lyapunov exponent Al = 0.0779.

L

Memristor Oscillator's 3199

R --------------- -----1

L

r

+ V2

C2

+ VI -G

CI

Flux-controlled memristor

~-------------------

R

L + + V2 VI Active

CI memristor

Fig. 25. Chua's oscillator with a flux-controlled memristor and a negative conductance.

The equilibrium state of Eq . (98) is given by A = {(x ,y,z,w) lx = y = z = 0, W = constant} , which corresponds to t he w-ax is. The Jacobian mat rix D at this equilibr ium set is given by

a(- 1+ ~ - W(w)) a 0 0

1 - 1 1 0 D=

0 -{3 -, 0 , (101)

1 0 0 0

and its four eigenvalues Pi (i = 1, 2,3, 4) can be written as

Pl ,2 ~ - 1.31104 ± i 2.74058, P3 ~ 3.27207, Pl ,2 ~ 0.0786554 ± i 2.84655, P3 ~ -4.50731,

A4= 0, P4 = 0,

for Iwl < I,} for Iwl > 1.

(102)

Thus, t hey are characterized by an unstabl e saddle-focus except for t he zero eigenvalue.

4.2. A third-order memristor-based Chua oscillator

I where

Consider next t he Van der Po l oscillator wit h Ch ua's diode as illust rated in F ig. 27. If we replace Chua's dio de with a two-terminal circuit consist ing of a cond uctance and a flux-controlled memristor , we would obtain the circuit shown in F ig. 28. The

W( ) = dq(cp )cp d'

q(cp) = bCP : 0.5(a ­ b)(lcp + 11­ Icp -

}

11). (104)

dynamics of t his circuit is given by

dv C dt = -i - W( cp)v + Gv,

di L dt = v,

dcpdi = v,

Equation (103) can be transformed into the form

dx dt = a( - y - W(z)x + , x ),

dy (105) (103) dt = {3x ,

dz dt = x,

3200 M. ltoh & L. O. Chua

6

4

Z 2

0

- 2

-4

-6

1.5

Y 0.5

0

-0.5

- 1

-1.5

-3 5

F ig. 26. Chaotic attractor of Chua 's oscillator with a flux-controlled memristor and a negat ive cond uctance.

where x = v , y = i , z = sp, a = l/C, /3 = 1/L, respect ively, where a, b > O. From Eq. (105), we ,=G, and the piecewise-linear funct ions q(z) and obtain W (z) are given by

(107) q(z) = bz + 0.5(a - b)(lz + 11- [z - 11),

Thus, y(t ) and z(t ) satisfy (106) a, Izl < 1,

_ y(t ) + cW( z) = { ( ) (108) z t - /3 'b, Izi > 1,

L

L

L

+ V Chua 's

C diode

Fi g. 27. Van de r Pol oscillator.

I I I I I -G Flux-controlled I I memristor I I I

~ - - - - - - ------ - - - - - - - - - --

Active memristor

Fig. 28. A third-order oscillator with a flux-controlled mem­risto r and a negati ve cond uctance.

where e is constant . Since W (z ) = W ((y + e)/{3), Eq. (105) can be transformed into the form

-d 2y

+ a { W (y+--e) - , } d- Y + a{3y = 0 (109) dt2 {3 dt '

or equivalent ly

d2y dydt2 + a (a - , ) dt + a{3y = 0, [z] < I,}

(110) d2y dydt2 +a(b -,) dt + a{3y =O , [z] > 1.

Thus, Eq . (105) is equ ivalent to a one-p arameter fami ly of second-order differential equations, parametrized by the constant "e" , via Eq . (108). Since th e minimal dimension for a cont inuous chaot ic system is 3, Eq . (105) cannot have a chaot ic attractor, even if the circui t elements are active. If we set a = 2" = 0.3, {3 = 1, a = 0.1 and b = 0.5, our computer simulation shows that Eq. (105) has two periodic at tractors as shown in Fig. 29. Observe th at these two limit cycles are

Memri stor Oscillat ors 3201

odd-symmetric images of each other, as expected . The equilibrium state of Eq . (105) is given by A = {(x,y, z )lx = y = O, z = const ant }, which corresponds to t he z-axis. The Jacobian matrix D at this equilibrium set is given by

a h - W( z)) -a

D = {3 o (111) [ 1 o

Its characteri sti c equation and eigenvalues Ai (i =

1, 2, 3) can be written as

p(p2 + a(a -,)p + a(3 ) = 0, Izl < I,} (112) p(p2 + a (b -,)p + a (3 ) = 0, Izl > 1,

and

A1,2 = 0.2 ± i 1.4, A3 = 0, for Izl < I, } (113) A1,2 = - 0.2 ± i 1.4, A3 = 0, for Izi > 1,

respectively. Thus , th e equilibrium set is unstable if jzl < 1, and stable if Izl > 1.

4. 3 . A s econd-order memristor-based circuit

Consider again the Van der Pol oscillator with the voltage-cont rolled Chua's diode from [Barboza & Chua, 2008], as illustrated in Fig . 27. If we let t he capacitance C -'> 0, we would obtain th e relaxation oscillator shown in Fig . 30, which exhibits a jump behavior [Chua, 1969]. Furthermore, by replacing Chu a's diode in Fig . 30 with a two-terminal circuit consisting of a cond uct ance and a flux-controlled memri stor , we obtain the circuit of Fig. 31. The dynamics of t he circuit in Fig. 31 is given by

where

i = (G - W( <p))v ,

di L dt = v , (114)

d<pdi = v ,

W( <p) = dq(<p ) , d<p

(115) q(<p) = bip + 0.5(a - b)

x (I<p + 11- I<p - 11)·

3202 M. Itoh & L. O. Chua

3 2z 1 o

-1 -2 -3

-4

.5 - 1

o x

2

3

+ L Chua 's VD

diode

F ig. 29. T wo p er iod ic at tractors of the t hi rd-order memri stor oscilla tor.

ID

Fig. 30 . A relax at ion osc illator where t he VD - iD curve of Chua's diode is given by F ig. 15 of [Barboza & Chua, 2008].

Equation (114) can be written as ,-------------------1

y = b - W (z ))x ,

dy dt = /3x ,

dz dt = x,

where x = v, y = i, z = 'P,/3 = Eq . (116), we obtain

dy _ /3 dz =0. dt dt

T hus , y(t) and z(t ) satisfy

y(t ) - /3z( t) = c,

..--------1---+---..., I I I I I I I I -G Flux-controlled 1L I I memristor :

(116) I I I I

I

l / L ,G = , . From

+ Active

memristor V(117) L

(118) F ig. 31. A second -order osc illa tor.

where c is a constant. From Eq . (116) , we obtain

dz y = b - W (z))x = b - W (z)) dt . (119)

We can interpret Eq. (119) as a one-p arameter fam­ily of first-order differential equa tions, namely,

dz y -

dt 't ': W (z)

(3z + c

, - W( z)

cr+[z] < 1, I -a

' {3z + c

[z] > l. "t ': b '

(120)

The solution of Eq . (120) for Iz[ < 1 and [z] > 1 can be expressed as

(J t c z(t ) = de -y- a - /3 ' (121)

and

-.iL t C z(t ) = de ..,-b - /3 ' (122)

resp ect ively, where c and d are constants. If we replace the piecewise-linear function of the

memristor by a smooth cubic function, namely

Z3 }q(z) = 3 ' (123)

W(z) = z2,

Mem ris ior Oscillators 3203

we would obtain

dz dt

{3 z + c , - z2'

(124)

If we set , = 1, (3 = 1 and c = 1, the corr ect solution of Eq. (124) is given by

z(t ) = 1 ± } 2 (e - t ), (125)

where e is a constant , and shown in Fig. 32. Our comp uter simulation shows that Eq . (124) exh ibits the in correct irregular oscillation shown in Fig. 33. T his erroneous comp uter-generated solu ­t ion is caused by the numerical integration error at z = ± l.

4.4 . Fi rst-order m emristor-based circuit

Consider the circuit in Fig. 34, which consist s of a cur rent source J and a two-te rminal circuit consist ­ing of a conductance and a flux-con troll ed memris­tor. The circuit equat ion of Fig. 34 can be written as

J(J + Gv)dt = q(<p), (126)

where q(<p) denotes th e characteristic of th e mem­ristor. Differentiating Eq . (126) with resp ect to time t , we obtain

J + Gv = W(<P)V,} (127)d<p

&=v,

1.2 .------,-----~---,----_.__--~---

0.8

N O.6

0.4

0 .2

0 1<::::...__-'- "-__---"-- "-__----'-- _

o 0 1 0.2 0.3 0 .4 0 .5 0.6

t

F ig. 32. Correct solut ion z(t ) = 1 - V1=2t, which satisfies the initial condition z(O) = O.

3204 M. Itoh & L. O. Chua

10 ,---.-­- ,..-­ .---­ ,...-­ --.----,-----.------.-----.-----,

·5

N ·10

·15

·20

·25

-30 L-_.l...-_ -'-_ -'-_ --'--_ --'-_-'-_---'-_---'-_---'-_---l o 100 zoo 300 400 500 600 700 800 900 1000

t

Fig. 33. Erroneous oscillation of the second-orde r oscillator from Fig. 31 obtained by compute r-simulat ion via a fourth-order Runge-Kutta formu la with st ep size h = 0.002 and the initial condit ion z(O) = O.

where

W (ep) = d~~). (128)

From th e first equation of Eq. (127), we obtain

J depv - -­ (129) - W (ep) - G - di '

Thus , Eq. (127) can be written as

dx e (130)

dt W( x) - {3 '

r-------------------I

V

I I I I

J t Flux-controlled I

memristor : I I

- - - - - - - - - - - - - - - - - - - _I

ActiveJ t memristor

Fig. 34. A first-order oscillator with a flux-controlled memristor.

where x = ep, e = J, (3 = G, and the functions q(x ) and W (x) ar e defined by

q(x ) = {0.05X 2

, x 2 0, 0.05x , x < 0,

W(x) = dq(x ) = { O.l X, x 2 0, (131)

dx 0.05, x < 0.

In this case , q(x) is not a piecewise-linear function. If we set (3 = 0.3 and e = 1, our computer sim­ula tion shows that Eq. (130) exhibits an irregular oscillation as shown in Fig. 35. This computer gen­erated solution is erroneous , and is caused by t he numerical integra t ion error at x = 3, since Eq . (130) can be recast into th e form

dx - 10 for x > 0, (132)

dt x - 3 '

it follows that Idx/dtl tend s to infini ty when x ---+ 3. T he exact solut ion of Eq. (132) is given analyti­cally by

x = 3 ± )2(C - lOt), (133)

where C is some constant , and does not exhib it any oscillations, as shown in Fig. 36. Note t hat solut ion of Eq. (133) does not exist for t > C / 10, implying that a more realist ic circuit model of the physical circuit is needed [Chua et ol., 1987].

Memristor Oscillators 3205

100

50

a

·50

X

·100

·1 50

·200

·25 0 a 100 200 300 400 500 600

t

Fig. 35. Erroneous oscillation of the first-order oscillator from Fig. 34 obta ined by computer-simulation via a fourth-order Runge-K utta formula with step size h = 0.003 and the initial condition x( O) = 0.3.

3.5 ,...---­ ,...------,,-------,------,-­ ----.------.-­ -----r-----,

0 .35 0. 3 0 .25 0 .2 0 .15 0 .1 0 .05

oL-_ _ '-----_ -----.J__-----.J__-----'_ _ -----'__-"_ _ ---'-__---'

o OA

3

0 .5

1.5

X

2

2 .5

t

Fig. 36. Correct solution x (t ) = 3 - )2(C - lOt) (C = 3.645) .

5 . Conclusion

We have derived severa l memristor-based non lin­ear oscillators from Chua's oscillators. These oscil­la tors have many interest ing oscillation proper ties and rich non linear dynamics. We conclude there­fore that the memristors are useful for des igning non linear oscillators.

References

Barboza, R. & Chua, L. O. [2008] "T he four-el ement Chua 's circuit," Int. J . Bifurcation and Chaos 18 , 943-955.

Ch ua , 1. O. [1969] Introduction to Non linear Network Theory (McGraw-Hill, NY).

Chua , L. O. [1971] "Memristor - the missing circuit element," IEEE Trans. Circuit Th . CT-18 , 507-519.

Chua, L. O. & Kang, S. M. [1976] "Memristive devices and systems," Proc. IEEE 64 , 209-223 .

Chua, L. 0., Desoer , C . A. & Kuh, E. S. [1987] Lin ear and Non linear Circuits (McGraw-Hill, NY).

Chua , 1. O. & Lin , G. N. [1990] "Canonical realization of Chua's circ uit fami ly," IEEE Trans. Circuits Syst. 37, 885-902.

Govorukhin , V. N. [2004] "MAT DS -- MAT LAB bas ed program for dy na mical sys te ms investigation ," http)/kvm.math.rsu .ru /matdsj.

3206 M. Itoh & 1. O. Chua

Johnson, R. C. [2008] "W ill memristors prove irr e­ Strukov, D. B. , Snider, G. S., St ewart, G. R. & Willi am s, sistible?" EE Tim es issue 1538, August 18, R. S. [2008] "T he missing memristor found," Nature pp . 30-34. 453 , 80-83.

Madan, R. N. [1993] Chua 's Circuit: A Paradigm for Tour, J. M. & He, T. [2008] "T he fourth element," Nature Chaos (World Scientific, Singapore) . 453 ,42-43.