LEON O. CHUA. Department of ... 2008; Johnson, 2008] in the next gener ation com and puters and ...... Consider next t he Van der Po l oscillator wit h. Ch ua's ...
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.
+