毕业论文之改进蔡氏电路 MEMRISTOR OSCILLATORS

Tutorials and Reviews International Journal of Bifurcation and Chaos Vol 18 No 11 2008 3183 3206 World Scientific Publishing Company MEMRISTOR OSCILLATORS MAKOTO ITOH Department of Information and Communication Engineering Fukuoka Institute of Technology Fukuoka 811 0295 Japan LEON O CHUA Department of Electrical Engineering and Computer Sciences University of California Berkeley Berkeley CA 94120 USA Received July 15 2008 Revised September 18 2008 The memristor has a ttracted phenomenal worldwide attention since its debut on 1 May 2008 issue of Nature in view of its many potential applications e g super dense nonvolatile com puter memory and neural synapses The Hewlett Packard memristor is a passive nonlinear two terminal circuit element that maintains a functional relationship between the time integrals of current and voltage res pectively viz charge an d flux In this paper we derive several nonlinear oscillators from Chua s oscillators by replacing Chua s diodes with memristors K eywords Memristor memristive devices memristive systems charge flux Chua s oscillator Chua s diode learning neurons synapses Hodgkin Huxley nerve membrane model 1 Memristors T he HP memristor shown in Fig 1 is a pas sive two terminal electronic device described by a In 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 the Hewlett v M q i or i W cp v 1 Packard Company announced the fabrication of between the device terminal voltage v and terminal a nanometer size solid state two terminal device 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 respectively are defined by Chua Johnson 2008 in the next gener ation com and puters and powerful brain like neural computers W dq cp One immediate application offers an enabling low 3 cp dcp cost technology for non volatile memories where future computers would turn on instantly without representing the slope of a scalar function ip cp q the usual booting time currently required in all and q q cp respectively called the memristor personal computers constitutive relation IMore than one million Google hits were registered as of June 1 2008 2The Hewlett Packard memristor is a tiny nano passive two terminal device requiring no batteries Memristors characterized by a nonmonotonic constitutive relation are called active memristors in this paper because they require a power supply 3183 v 3184 M Itoh i e M q d q 0 resp W cp d 0 4 see Chua 1971 In this paper we assume that the memristor is characterized by the monotone increasing 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 memris tance M q and the memductance W cp in Fig 2 are defined by Iql 1 M q d 1 and W cp dq cp C Iwl 1 respectively Since the instantaneous 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 i t p r dr 0 11 to for all t to Thus the memristor constitutive rela tion in Fig 2 is passive Consider next the two terminal circuit in Fig 3 which consists of a negative resistance or a negative conductance and a passive memristor If the two terminal circuit has a flux controlled slope c Fig 2 The constitutive relation of a monotone increasing piecewise linear memristor Charge controlled memristor left Flux controlled memristor right 3The negative resistance or conductance can be realized by a standard op amp circuit powered 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 terminal circuit which consists of a memristor and a negative cond uctance G or a resistance R Memristor Oscillators 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 continuous func tion satisfying q O 0and G O Thus the small signal memductance W ep of this two terminal cir cui t is given by W ep dq ep C G Iwl 1 If c G 0 or d G 0 In this case there exists cp to CPo and i t p T dT 0 15 to for all t E to tl Thus the two terminal circuit in Fig 3 can be designed to become an active device and can be regarded as an active memristor We illustrate two kinds of characteristic curves in Fig 4 Similar characteristic curves can be obtained for charge controlled mem risto rs In this paper we design several nonlinear oscillators using active or passive memristors 2 Circuit Laws In this section we review some basic laws for elec trical circuits Recall first the following principles of conservation of charge and flux Chua 1969 Charge and flux can neither be created nor destroyed The quantity of charge and flux is always conserved We can restate this principle as follows Charge q and voltage vc across a capacitor can not change inst antaneously Flux sp and current it in an inductor cannot change instantaneously Applying this principle to the circuit we can obtain a relation between the two fundamental circuit vari ables the charge and the flux However we usually use the other fundamental circuit variables namely the voltage and the current by apply ing th e following Kirchhoff s circuit laws Chua 1969 The algebraic sum of all th e currents im flowing into the node is zero 16 m The algebraic sum of branch voltages Vn around any closed circuit is zero 17 n They are a pair of laws that result from the con servation of charge and energy in electrical circuits If we apply the Kirchhoff s circuit laws to a mem ristive circuit we need the four fundamental circuit variables namely the voltage current charge and flux to describe their dynamics because the relation between current i and voltage v of the memristor is defined by Eq 1 Ifwe integrate the Kirchhoff 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 ti dt 20 oo and 21 resp ectively The relationship between voltage v and cur rent i for the four fundamental circuit elements is given by Capacitor dv c i 22 dt Inductor di L v 23 dt Resistor v Ri 24 Memristor v M q i or i W cp 25 Using these relations and the Kirchhoff s circuit laws we can descri be the dynamics of electrical circuits Integrating Eqs 22 25 with respect to time t we obtain the following equations 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 They 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 exhibiting chaotic behavior Madan 1993 It is well known that the canonical Chua s oscil lator Chua It 23 dt 3 dt W p dq p d p 36 Note that the two kinds of independ ent variables are related by 37 Thus Eq 35 can be recast into the following set of differential equations using only 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 functions 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 II dq w a 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 com puter simulation shows that Eq 39 has a chaotic attractor as shown in Fig 10 By calculating the where x V I Y i3 z V2 W ip a Lyapunov exponents from sampled time series we 1 C1 f3 I C2 G C2 L 1 and found that this chaotic attractor has one positive 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 attractor of the canonical Chua s oscillator with a flux controlled memristor 5We used the fourth order Runge Kutta method for integrating the differential equations 3 3190 M Itoh namely d4u d3u d2u dy dt x z dt4 aW w dt3 3 a a W w dt2 50 dz du dt 3y a 3W w di 0 47 dw where di x d2udu 3u 2 w t dt woo 48 Here Uo and Wo are constants Thus its character istic equation also has a zero eigenvalue Consider next the fourth order oscillator in Fig 11 obtained by removing a resistor from the Fig 11 A fourth order oscillator with a flux controlled circuit of Fig 7 The circuit equation can be memristor 6We used the software package MATD S Govorukhin 2004 to calculate the Lyapunov exponents 7Note t hat if the system is a flow one Lyapunov exponent is always zero which corresponds to the direction of the 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 II W w dq w a Iwl 1 From Eq 50 we obtain 22 d I x yz2 2 W w x 0 and b O In this case the equi librium state A x y z w lx Y z 0 w constant i e the w axis is globally asymptotically Memristor Oscillators 3191 stable and Eq 50 does not have a chaotic attrac tor However if we set a 4 2 3 20 1 a 2 and b 9 our computer simulation of Eq 50 gives a chaotic attractor in Fig 12 By calculating the Lyapunov exponents from sampled time series we found that this chaotic attractor has a positive Lyapunov exponent Al 0 050 In this case the capacitance C2 and the inductance L are both negative active and the memristor is active as shown in Fig 13 see Barboza CP3 can be chosen to be the indepen dent variables namely the charge of capacitor CI the flux of inductor L and the flux of the memris tor respectively From Eq 56 or differentiating Eq 57 with respect to time t we obtain a set of three first order differential equations which defines the rela tion among the three variables V I i3 cp dVI CI dt i3 W cp VI di3 61 L R Z 3 VI dt dcp di VI Memristor Oscillators 3193 where dq C dVI dCP3 V L di3 zl I dt dt 3 dt dt dq3 dCP4 ill V4 R Z 3 dt z3 dCPI W cp d t VI 62 Note that the two kinds of independent variables are related by ql cp CP3 II vI cp i3 63 ql CIVI CP3 Li3 Thus Eq 61 can be recast into the 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 transformed 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 C I 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 respectively where a b O The equilibrium st ateofEq 65 is given by the set A x y z lx y 0 z constant which corresponds to the z axis The Jacobian matrix D 8The term charge and flux are just names given to the definition in Eq 58 and should not be interpreted as a phys ical charge or flux in the classical sense The important concept here is that they are m easurable quantities obtained via int egration 3194 M Itoh 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 periodic attractors of the third order canonical memristor oscillator Memristor Oscillators 3195 into Eq 70 c and d are constants we would obtain the following third order differential equation a w C I3U t d t d 13 a I3W C I3 U 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 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 1 dt dt2 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 differential L V Flux controlled equation Eq 65 does not have a chaotic attractor 1 even if the circuit elements are active memristor Consider next the three element circuit in Fig 18 obtained by short circuiting the resistor from Fig 14 its dual circuit is shown in Fig 19 Fig 18 A third order circuit with a flux controlled The dynamics of this circuit can be written as memristor dx ill a y W z x dy 77 dt x dz dt x L1 From this equation we obtain Charge controlled memristor 2 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 circuit with a charge controlled memristor 3196 M ltoh 11 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 1 100 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 simulation shows that Eq 98 has a chaotic attractor as shown in Fig 26 By calcu lating the Lyapunov exponents 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 the w axis The Jacobian matrix D at this equilibrium 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 A 4 0 P4 0 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 this circuit is given by dv C dt i W cp v Gv di L dt v dcp di 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 Fig 26 Chaotic attractor of Chua s oscillator with a flux controlled memristor and a negative conductance where x v y i z sp a l C 3 1 L respectively where a b O From Eq 105 we G and the piecewise linear functions 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 L L L V Chua s C diode Fig 27 Van der 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 mem ristor and a negative 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 equivalently d2y dy dt2 a a dt a 3y 0 z 1 Thus Eq 105 is equivalent to a one parameter family of second order differential equations parametrized by the constant e via Eq 108 Since the minimal dimension for a continuous chaotic system is 3 Eq 105 cannot have a chaotic attractor even if the circuit 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 attractors as shown in Fig 29 Observe that these two limit cycles are Memristor Oscillators 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 constant which corresponds to the z axis The Jacobian matrix D at this equilibrium set is given by a h W z a D 3 o 111 1o Its characteristic equation and eigenvalues Ai i 1 2 3 can be written as p p2 a a p a 3 0 Izl 1 and A1 2 0 2 i 1 4 A3 0 for Izl 1 respectively Thus the equilibrium set is unstable if jzl 1 4 3 A second order memristor based circuit Consider again the Van der Pol oscillator with the voltage controlled Chua s diode from Barboza Chua 2008 as illustrated in Fig 27 If we let the capacitance C 0 we would obtain the relaxation oscillator shown in Fig 30 which exhibits a jump behavior Chua 1969 Furthermore by replacing Chua s diode in Fig 30 with a two terminal circuit consisting of a conductance and a flux controlled memristor we obtain the circuit of Fig 31 The dynamics of the circuit in Fig 31 is given by where i G W p v di L dt v 114 d p di 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 2 z 1 o 1 2 3 4 5 1 o x 2 3 L Chua s VD diode Fig 29 Two periodic attractors of the third order memristor oscillator ID Fig 30 A relaxation oscillator where the VD iD curve of Chua s diode is given by Fig 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 3dz 0 dt dt Thus y t and z t satisfy y t 3z t c 1 I I I I I I I I G Flux controlled 1 L I I memristor 116 I I I I I l L G From Active memristor V 117 L 118 Fig 31 A second order oscillator 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 parameter fam ily of first order differential equations namely dz y dt t W z 3z c W z c r z l t b 120 The solution of Eq 120 for Iz 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 respect 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 thecorrect 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 computer simulation shows that Eq 124 exhibits the in correct irregular oscillation shown in Fig 33 This erroneous computer generated solu tion is caused by the numerical integration error at z l 4 4 First order m emristor based circuit Consider the circuit in Fig 34 which consists of a current source J and a two terminal circuit consist ing of a conductance and a flux con