# memristor oscillators

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Tutorials and Reviews Internat ional Journal of Bifurcation and Chaos, Vol. 18, No. 11 (2008) 3183-3206 World Scientific Publishing Company MEMRISTOR OSCILLATORS MAKOTO ITOH Department of Information and Communicati on Engineering, Fukuo ka Institute of Technology, Fukuoka 811-0295, Japan LEON O. CHUA Department of El ectri cal Engineering and Compu ter Sciences, Uni versity of Californi a, Berkeley, Berkeley, CA 94120, USA Received July 15, 2008; Revised Sept ember 18, 2008 The memrist or has a ttracted phenomenal worldwide attent ion since it s debut on 1 May 2008 issue of Nature in view of its many potent ial applications, e.g. super- dense nonvolatile computer memory and neural synapses. The Hewlett-Packard memristor is a passive nonlin ear twot erminal circuit element t hat maintains a functi onal relat ionship between t he time int egrals of current and voltage, res pectively, viz. charge an d fl ux. In thi s paper, we derive seve ra l nonl inear oscillators from Chua's oscillat ors by rep lacing Chua's diodes with memrist ors. K eywords: Memristor ; memrist ive devices; memristi ve systems; charge; flux; Chua's oscillat or ; Chua's diode; learning; neur ons; synapses; Hodgkin-Huxley; nerve membrane model. 1. Memristors T he HP memrist or shown in Fig. 1 is a passive two-terminal elect ronic device described by aIn a seminal paper [Strukov et al., 2008] which nonlinear constitutive relation appeared on 1 May 2008 issue of Nature, a team led by R. Stanley Williams from t he Hewlet t v = M( q)i, or i = W (cp )v , (1) Packard Company announced t he fabricat ion of '. between the device terminal volt age v and ter minal a nanometer-size solid-state two-t erminal device current i . The two nonlinear functi ons M (q) and called the m em ri stor, a contraction for memory W (cp), called the mem rist ance and memductance, resis t or, whi ch was postulat ed 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 applications [Tour & He, 2008; Johnson, 2008] in the next gener ation comand puters and powerful brain-like "neur al" computers. W( ) ~ dq(cp)One immediat e applicat ion offers an enabling low (3)cp dcp ' cost technology for non-volatile memories/ where future comput ers would t urn on instant ly wit hout represent ing t he slope of a scalar function ip = cp( q) t he usual "booting time", currently required in all and q = q(cp), respect ively, called t he memristor personal computers. constit utive relat ion. IMore than one million Google hits were regist ered as of J une 1, 2008. 2The Hewlett- Packard memristor is a t iny nano, passive, two-terminal device requiring no bat teries. Memristors charact erized by a nonmonotonic const it ut ive 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 Fig. 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 t hat the memristor is characterized by the "monotoneincreasing" and "piecewise-linear" nonlinearity 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 Fi g. 2 are defined by Iql < 1, M (q) = d 1, and W( cp) = dq(cp) = {C, Iwl < 1, (8)dip d, Iwl> 1, respectively. Since the instantaneous power dissi pated by t he 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) t o 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 resist an ce' (or a negative conductance) and a passive memristor. If t he two-terminal circuit has a flux-cont rolled .... slope = c Fig. 2. The const it ut ive relation of a monotone-increasing piecewise-linear memristor: Charge-controlled memrist or (left ). Flux-controlled memristor (r ight) . 3 The negative resistance or conductance ca n be realized by a standard op amp circuit, power ed by batteries. v Vl =M(q)i -R + V -G il =W('P)V Two-terminal circuit ! + v + V=M(q)i i=W('P)V Active memristor Fig . 3. Two-t erminal circui t which consist s of a memr ist or and a negati ve 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) + i 2(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 function satisfying q(O) = 0 and G > O. Thus, th e small signal memduct ance W (ep ) of this two-t ermin 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 t he 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 terist ic of the two-t erminal circuit . 3186 M. It oh & L. O. Chua for all t > 0. In t his case, there exists cp(to) = CPo and it p(T)dT < 0, (15) to for all t E (t o, 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 memri stor" . We illustrate two kinds of characteristi c curves in Fig. 4. Similar char act eristic cur ves can be obt ained for charge-controlled mem risto rs. In this pap er , we design several nonlinear oscillators using act ive or passive memristors. 2. Circ uit Laws In thi s section, we review some basic laws for electrical circuit s. Recall first t he following principl es of conservation of charge and flu x [Chua, 1969]: Charge and flux can neither be created nor destroyed. The quantity of charge and flux is always conserved. We can restate thi s principle as follows: Charge q and volt age vc across a capacitor cannot change inst antaneously. Flux sp and current i t. in an inductor cannot change instantaneously. Applying t his pri ncipl e to the circuit, we can obtain a relation between the two fundamental circuit variabl es: the "charge" and the "flux" . However, we usually use the other fundamental circuit var iabl es, namely the "volt age" and the "current" by applying th e following Kirchhoff 's circuit laws [Chua , 1969]: The algebraic sum of all th e currents im flowing int o 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 t hat result from the conservation of charge and energy in elect rical circuits. If we apply t he Ki rchhoff's circu it laws to a memristi ve circuit, we need the f our fundamental circuit variables, namely the voltage, cur rent, charge, and flux to describe their dynami cs, because the relation between current i and voltage v of the memri st or is defined by Eq. (1). If we integrate the 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 ecti vely. The relationship between volt age v and current i for the four fundamental circuit elements is given by Capacitor dv c = i (22) dt Induct or diL-= v (23) dt Resistor v = Ri (24) Memri stor v = M(q) i (or i = W( cp )) (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 respect to t ime t , we obtain the following equat ions: Capacitor q = Cv (26) Inductor ip = Li (27) Resistor ip = Rq (28) Memristor Oscillators 3187 Memristor ip = JM(q)dq (or q = JW( 0 and b > O. In t his 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 Lyapunov 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 act

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