Mar 5, 2011 - in parallel with a pass,iue memristor, or an act'iue memristiue deuice, can be described explicitly by a Hami,ltonian eqttation, whose solutions ...
International Journal of Bifurcation and Chaos, Vol. 21, No. 9 (2011) 2395-2425 @ World Scientific Publishing Company DOI: 10. 1 142 I 50218127 4r103012X
MEMRISTOR HAMILTONIAN CIRCUITS MAKOTO ITOH 20-203, Arae, Jonan-ku Fulcuoka 8 1 1- 021 4, Japan
1-1
9-
i,toh- makot
o @j
com. home. ne.
jp
LEON O. CHUA Department of Electrical Engi,neering and Computer Sc'iences, Uni,uersi,ty of Cali,forn'ia, Berkeley, CA 94720, USA chu a@ eec s. berleeley. edu
Received November 25, 2010; Revised March 5, 2011
We prove analytically that 2-element memristiue c'ircuits consisting of a pass'iue I'inearinductor in parallel with a pass,iue memristor, or an act'iue memristiue deuice, can be described explicitly by a Hami,ltonian eqttation, whose solutions can be periodic or damped, and can be represented analytically by the constants of the motion along the circuit Hamiltonian. Generalizations to 3-element and 2-l[-element memristive Hamiltonian circuits are also presented where complex
bifurcation phenomena including choos, abound.
Keywords: Memristor; memristive devices; Hamiltonian; Pfaff's equation; Nos6-Hoover oscillator; H6non and Heiles equation.
1. Introduction Flom both classical and quantum mechanics, it is well known that the equations of motion of. a conseruat'iue (i.e. lossless) physical system can be derived from a scalar function, or operator, called the Hami,ltoniar?, representing the total energy of the system. In particular, each contour of the Hamiltonian of a mechanical system defines a periodi'c orbit of the system whose Hamiltonian is a constant of mot'ion fAndronov et al., 1987]. In electrical circuits, Hamiltonian equations are formulated for lossless, nonlinear circuits fChua & McPherson, 1974; Andronov et al., L987; Bernstein & Liberman, 1989]. As a consequence) it is generally believed that di,ssi,pati,ue systems do not have Hamiltonian equations because the trajectories are damped and hence not periodic.
We will show in this paper that memristors fChua, 1971] and memrist'iue deu'ices [Chua & Kang, 1976] when connected with capacitors, and/or inductors, can be described by an analogous Hami,lton'ian whose contours are constants of motion. This result is counter-intuitive and quite surprising because memristors and memristive devices are basically di,ssi,patiue devices, albeit endowed with memory. In this paper, we will present simple analytical examples of 2-element memristive circuits made of a passive linear inductor and a memristor, or a memristive device, and derive their Hamilton'ian, and orbits, in explicit analytic form. We will show that Z-elemenf memristive circuits have a continuum of periodic orbits, when the memristive device is active, and is hence capable of generating energy.l Such memristive devices can be built
lNote that this memristive circuit consists of fuo elements. If the circuit includes more than two
elements, there are many
nonmemr.r.stiue circuits which have a continuum of periodic orbits and is capable of generating energy.
2396 M. Itoh k L. O. Chua
chaos.
+ Memristive device
:
1. A 2-element Hamiltonian circuit, which consists of an inductor and a memristive device.
Fig.
＞ ｌ ｌ ｌ ヽ ｌ ｌ ｌ ノ
memristor is passive, and hence all trajectories are damped and must tend to the origin. What is truly astonishing here is that this 2-element memristor circuit has a Hamiltonian whose contours are precisely the damped trajectories! In other words, the Hamiltonian contours are not closed, We will also present several 3-element memristive circuits by adding either a passive linear capacitor, or a sinusoidal voltage source, in series with the above cited 2-element memristive circuit. Once again, an explicit Hamiltonian will be derived for the 3-element inductor-capacitor-memristive-device circuit. F\rrthermore, we present a 2N-element memristive circuit, which can be recast into an N-degree of freedom Hamiltonian equation. We will also present numerical simulations of bifurcation diagrams which depict the evolution of rather complex nonlinear dynamics, including
＋ Ｖ 一
with off-the-shelf components, and a power supply, as described in [Chua, 1971]. We will also exhibit a 2-element memristor circuit where the
島 (害 )=等 島 (多 )-1鵠・ ,
This equation can be recast into the form of Pfaff's equati,on describing a conservative system (for more details, see [Andronov et al., 1987; Nemytskii & Stepanov, 19891 and Appendix A):
1
2. 2-Element Memristive Circuits
(の
υ =M(″ )j・
＞ Ｉ Ｉ ヽ １ ︰ プ
(0, 各=∫ ク
1
∂∬
g(″ )づ
∂
"'
where
H(r,i): I Lq\r) J Y!P,d,+ J[90u,
and the symbol / denotes the operation of primitive integral or indefinite integral. The Pfaff's equation (4) has the solution H(r, i): Ho (Ho is any constant), since
dH(r,i,) _ 0H dr 9H di + dt 0r dt Ai dt 1 g(3)を
＞ ｌ ｌ ｌ ヽ ｌ ｌ ｌ ノ
０ ↓
的 ４
∫ 一 一 一 〓
洸 山一 洗 磁 卜
2-element memristive circuit
I
0, Eq. (2) can be written as
(b)
Z
equations:
It g(r)i
∂を
+而 =0.
働働 紛閉
The nonlinear function M(r) is called the memristance [Chua, 1971; Chua k Kang, 1976]. Applying Kirchhoff's voltage and current laws to this circuit, we obtain the following differential
＞
g(・ )を
Consider the circuit in Fig. 1, which consists of an inductor and a memristive device. The currentcontrolled memristive device is described by
∂∬
＞ ｌ ｌ ｌ ヽ ｌ ｌ ｌ ノ
＞
協 暢
Pfaff's equation
current-controlled memristive one port
(3)
1
)
After time scaling by dr : (g(r)i,)dt (for more “ details, see [Andronov et al., lgBT; Nemytskii & Stepanov, 1989], and Appendix A), Eq. (a) assumes the equivalent form
Memristor Hamiltonian C'ircutts
2397
ｔ ｒ ｌ ｌ ノ ヽ ︱ ｌ
酔 流 寧 山一
ｎ ａ
ｎ Ｏ
ａ
ｍ
Ｈ
Note that H* is a function of r alone, while f16 is a function of z alone. The Hamiltonian H(r,i,) is the sum of the pseudo-potential energy fI, and the pseudo-kinetic energy I{, that is, ″ (",t)=夏 ″(・ )+島 (`)・
dτ
(12)
Example l. Choose 0・
M(")="+β
α ″ ・ し=/懺考 "+/響 流
Note thtt the Hamiltonian I(2,づ )iS COnserК d
β
＞ Ｉ ｌ ｌ ノ ヽ ｌ ｌ ｌ
=0・
(10)
: i + 0, Eq. (14) can be transformed
into
Pfaff's equation A
=― (″ +β ).
Hence, they are conseruat'iue
(g(r)1.) dt : i,dt, we obtain the associated Hamiltonian system with respect to the scaled time r:
After time scaling by dr :
Let us introduce the symbols fI, and Il, called pseudo-potenti,al energy and pseudo-k'inet'ic energy, respectively,3 which are defined by
﹂
・
ｚ
ヽ ｌ ｌ ｉ ｔ ｒ ｉ ｌ ｌ ノ
χ ｄ
一卵
)
／
(づ
咆
／
島
一 一 〓
夏π(χ )
2This
(10
the form of Pfaff's equation
2-element Mernristor Circuit Hamiltonian Property 1 Equat'ions (2), (/r) and (7) haue the same ti,me-i,nuari,ant 'integral H(r,i) : Ho for all
:0.
一 ＋
(χ )司
ー
各=0× レ
ば し
(χ ,づ
It g(r)i
一 〓 一
0
現)lξ ]sttnsRITattctl鴛 ll席 l L設 ぶ α ∬ )=屏
two-dimensional memristive system A α 一
=等 等一 竿等=0
(13)
where a and B denote some constants.4 Then Bq. (2) is recast explicitly as 洸 洗 疏 一 山一
on each trtteCtOry,χ (t)and t(ι )With initial condi¨ tiOn(χ (0),づ (0)),fOr all scaled― time τ≧ 0;namely,
=等 多十 竿#
,
L=1,
,の
where g(r)l systems.
(づ
g(")=1,
Hamiltonian
屏
2_1),
∫(j)=α
deflned by:
＞ Ｉ Ｉ Ｉ ′ ヽ ︱ ︱ ︱
where g(″ )を
2 The∬ α ηof Eq.(7)is ηづ α mt;ι θ ≠
Hamiltonian system A
(11)
ti*e scaling maps orbits between systems (2) and (7) in a one-tG-one manner except at the singularity (2, i) where S@)i:0, although it may not preserve the time orientation of orbits. 3In physical systems, kinetic and potential energy are traditionally denoted by 7 and V, respectively. The Hamiltonian 11 represents the total energy of the system, which is the sum of 7 and V, that is, II : T + V. aln thir example, the memristive device \s actiue [Chua, 1971; Chua & Kang, 1976] because M(") 0 and on i ( 0, respectively. The r-axis is the horizontal axis and the i-axis is the vertical axis. Labels on contour lines denote the elevation (height) representing the
10
H
10 0
total
5
Fig 2.Hamiltonian(17) with
o - l,p : 0, namely,
″(",。 )=(1/2)"2+(1/2)を 2 - ln lil, where i I
0.
energy.
The Hamiltonian (17) and its contour plot for L1 0 : 0 are shown in Figs. 2 and 3, respectively. Two trajectories of Eq. (14) with the same parameters are shown in Fig. 4, which move on ct :
Memristor Hamilton'ian C'ircuits 2399 l
4
Ho%2.31 ／ ／
//
く
く
2
、 、
リ
0
X 2.0 1.5 1.0
…2
0.5
10
-4
(b)
…4
…2
Ttajectories of Eq. (14) with o : l,P :0. Initial conditions: r(0) : 0.1, i(0) : O.t (top) and r(0) : O.f, r(0) : -0.1 (bottom). They are located in the neighborhood of the origin. Here, Hg x 2.31.
Fig.
4.
…1 …2
-3 the contour ０
hQl =写 +平 ―
ヽ、′ノ
+(平
ｎ
一
∬ 0=ギ
π23L
of the Hanliltonian, where the initial conditions
are given by
χ(0)=0・
1,t(0)=±
state"(t),Current t(t),V01tage υ(ι )J(ι ),and energy E(ι
0・ 1・
υ(ι ),pOWer
)=lP(ノ
)α ノ Ofthe
The
‐2 ‐4 ‐6
P(t)= mem―
(d)
ristive device are shown in Fig. 5 as a function b also show づ ?η c t. ヽ ヽ of the original unscaled ι ″(ι ),t(ι ),夏π(ι ),島 (ι ),〃 (ι )and E(ι )as a fllnction of tilne t in Fig.6.Observe that the IIanliltonian∬
1
is conserved.A quiよ glimpse of Figs.5(d)and 5(e) seems to suggest the lnemristor in this example θcη ― cratcs energy for all tilnes t≧ O in the sense that E(ι )0 0Ver a
К ry
small neighborhood centered around t=0,ι
=5.9, t = 11.7 and t = 17.5. ]During these extremely brief time intervals where p(ι )>0,the memristor is absθ rbjη θ the energy initially stored in the inductor
whereづ (0)>0・ Hence,du五 ng each cycle,the mem― ristor acts brie■ y like a stη たand does not generate energy.
(")
5. Numerical solutions of c(t), i(t),u(t),p(t) : a(t)i(t) E(t) : Iip!')at'of Eq. (14) as a function of time, with initial conditions r(0) : 0.1, r(0) : 0.1, and parameters o : 1, 0 : 0. Here, r(t) and i(i) are calculated numeriFig. and
cally from Eq. (14) using the Mathematica function NDSolve (general numerical differential equation solver). Since Hamiltonian equations are known to be very sensitive to numerical round off errors, the numerical integration is shown only for a short time interval.
2400 "[■ οん &五 .0.aん しα
2.0 1.5 1.0
0.5
10 (b)
2.0
2.0
1.5
1.5
1.0
1.0
0.5
0.5
0.0
0.0
10
10
(C)
(d)
2.0
0
1.5
-1
1.0
‐2
0.5 0.0
…3
10
0)
(f)
Fig.6.Numerical solutions of π(ι ),0(ι ),Ic(ι ),Ife(t),I(ι )and E(ι )Of Eq(14)as a function of time ι,with initial conditions
ω(0)=0.1,づ (0)=0 1 and parameters α=1,β =0.
Remark. Since i(l) ) 0 for all t > 0 in Fig. 5(b), the loci of i,(t) versus u(t) will be a symmetrical nonintersecting "hysteres,is loop" , and not a
two-dimensional memristive svstem B
弊=―
"p'inched hysteres'is loop" fChta,2077). Observe also
that since the memristance M(0) : 0 in Example 1, the memristor in this example is current controlled, but not voltage controlled because its memductance W(r) is infinite, and hence undefined, at
r:0.
Example
2.
解
(づ
+β )",
=_α ("2_1)づ
.
It -g(r)i : -(-ri) : ir *
0, Eq. (21) can be transformed into the form of Pfaff's equation
Choose
Pfaff's equation B
l"l < 1.
、 ィ ー ー ー ヽ ︱ ︱ ︱ ノ
0 for
＋ ″
(
、 ︲︲′′′ ， ヽ ． ︲︲ ′／ １一 一 ″ ヽ、 ρν 〓 ０ ・
(携 )
一 一
5In this example, the memristive device is also actiae since M(o)
)
１ ／ １ ＼ 、 ′ ︲ ヽ α
where a and p denote some constants.s Then Eq. (Z) can be recast as
／
分(各 分
υO)
o2)
-1 …2 ‐3
０卜 Ｏ Ｈ
Fis.7. Nuhericslslutionofr(r)oversbdefinter\alinthercighborhoodoft!0,t*5.9,1-11.7edr*17.5.Oben€thatE(t)>0inth€€eneishborhoods.
2402
μl■ οん &L.0.
Chじ α
γ ご ια t,祀 =― → 隼 う =ね
After time scaling byご
obtain the Halniltonian systel
Hamiltonian system B ヽ
、 ︲︲′′′ ， ヽ ． ︲′ ノ ー‘ 一″ ヽ、 ρν 〓 ０ ・
／
ｌ ｌ ＞ ︰ ︱ ︱ ノ
／ １ ＼ ＼ α
十 ″
１
′︲ヽ
一 一
一 一 〓
ｌ
山一 ″ ＆ 一 ″
In this case,the∬ αmづ ι ι οηづ αη of
Eq。
(23)is giVen by
∬ し=― /懺3山 /撃 硫
5-5 Fig. 8. Hamiltonian (24) with a : 1,0 : 0.6, namely, H(r,i): (tl2)r2 -lnlrl -i-0.6lnlil, where r l0 and. i+0.
,の
=/α
― (π
=α whereづ
(考 i-ln l″
″―
￨)一
/(1+:)α
j―
β
ln lづ
￨.
づ
(24)
0.Thus,Eq.(21)has the integral ５ ２
助
〓
ｎ
β
一
一
″
ヽ ︱ ノ
ｎ
一
α
／ １ １ ヽ
′丁
"≠
:)α
t x
Example
x
two-dimensional memristive svstem C 3.71,
＋
6I.t
Ｚ
of the Hamiltonian, where the initial conditions are given by r(0) : +0.8, ,(0) : -4.0. The state r(t), current z(t), voltage u(t), power p(t) : u(t)z(t), and energ"y E(t) : Iip!)at'of the memristive device
＞ １ １ １ ヽ ｌ ｌ ｌ ノ
0.6In 4.0
＞ Ｉ Ｉ Ｉ ヽ ｌ ｌ ｉ プ
-
＋
4.0
ｌ ″ １
-
we set n : e, the memristance M(q1 : q2 + 1 defines a pass'iue memristor with g : q + (qslZ) because M(r) > 0 for all r and i. Here, q and rp denote a charge and a flux, respectively. In this example, Eq. (Z) assumes the form
一 一 〓
In 0.8
(2つ
If
Ъ 一 ・
;
-
一 一 一 一 一 一 〓
0.92
l"(0)l _ i(0) _ 0.6In li(0)l
Choose
洸 山一 洸 流一
:
,_
3.
Ъ ・
(26)
The Hami,ltonian (24) and its contour plot for (t : 1,0 : 0.6 are shown in Figs. 8 and 9, respectively. Two trajectories of Eq. (21) with the same parameters are shown in Fig. 10, which move on the contour
_ r(0\2 -\-/
10 and t = 20 (see Fig. 12), where the memristive device generates energ"y. We also show r(t),i(t),H"(t),Hilt),H(t) and E(t) as a function of time t in Fig. 13. Observe that the Hamiltonian f/ is conserved.
胴嗣咆ｚ
ｔ ｒ ｌ ヽ ｌ ノ
χ ・
ヽ、 ′ ノ
ヽ Ｚ
ｎ 勝Ｐ
′︰︲ヽ
α 一 一 一 〓
勢慟
／
聯 聯
where I/s is a constant and i,r I 0. The pseudopotenti,al energy and the pseudo-ki.netic energy are given respectively by
rrItO---tlt
are shown as a function of time t in Fig. 11, in which E(t) is positive except in a small neighborhood of I = 10 and I ry 20. Thus, the memristive device behaves like a passive nonlinear capacitor most of the time. However, it becomes active, that
thi. case, we did not use the time scaling a, : k@)i.)d,t: -i.t:dt as applied in Eq. (4), but its reversed time, since it does not preserve the orientation of orbit in the region i,r > 0.
Memristor Hamiltonian Circuits 2403 (b)
(a)
-5 -4 -3 -2 -1
0 (d)
︱︱ん句／／
0
-1 -2 -3 -4 -5
-4 -3 -2 -1
4
3
5
Fig.9.Contours of the Hamiltonian(24)of the Hamiltonian circuit in Fig.l with α =1,β =0.6.Several curves along axis is the horizolltal axis and the夕 axis which the Hamiltonian has a constant value are illustrated on the(π ,じ )― plane.The π― < 0,じ > 0, is the vertical axis Labels on contour lines denote the elevation (height)representing the total energy.(a)“ 0,こ (b)">0,を >0,(c)π 0, ゲづ (0)>0, (40)
αηご
づ (t) 0, Eq. (A.14) can be transformed into the Hamiltonian system. That is, by transforming the independent variable I by means of the formula d"r : dtlQ(r,y), we obtain
︱
″■ ﹃
は ぃ 枷 動
︱ ︱
(A.16)
＝ り
In this transformation, we multiply the velocity at the point (r,a) by the value of the function Q@,y) at this point, however, it does not alter the trajectories fNemytskii & Stepanov, 1989]. If Q@,U) < 0, the time orientation of orbits is reversed. Note that Eqs. (A.14) and (,{.16) are recast into the same form
１ ・
ｎ静 寺 妊ｂ ａ ｃ
イ
αν 山
働一働
ｏ ヽ︱︱＞︱︱り ｌ
ｄ 一 一 一 一 一 一 一 一 一 一 一 一 〓 ｎ
一 缶 職 缶Ｉ Ｃ イ イ イ
2424
,
(A.17)
２ ３
Ａ Ａ
刹イ
ｑ
＜
ち ０ α 〓
δ
Ｅ
if Q@,y) + 0 and 0F l1y 10.
Appendix B Nos6-Hoover Oscillator Equation Consider the following Hamiltonian from et a1.,1986):
fPosch
H(P,Q, P", s)
:G(e).t (:)
*Ins
++,
(B.1)
where F(.) and G(.) are scalar functions. Here, Q denotes the position, P denotes the momentum, s denotes the time scale variable (o, effective mass variable), and P, denotes the conjugate momentum. The equations of motion follow from Eq. (B.1):
απ σづ rcπ づ ι s ■イεη2れ sι οr lαm`Jι οれを
α0
dt=:吾 =F′ (1):, αP O),
#=F′ (p),
就―絡―α αs ごι
α鳥 αι
=α 島
畿
2425
多=― 手=α
(B.2)
,
G′
(B.6)
(g)― (P,
(PF′
(p)-1)・
F\rrthermore, from the third equation in Eq. (B.3), =F′
― 等
we obtain the following relation
―
(1)多
:・
dtns 1/ar\ _:;:aPs:e' ;\a"rl
α τ=dt/s,We get
After time scaling by
Integrating the equation
写 (1), 事=― SdO), =ノ
:(#)=C,
#=α
生L=F′
7ξ
ln S(7)=ズ
,
(B.9)
(t)dt・
Hence, the Hamiltonian (B.1) can be written as
(1)I_1
H:G(e)."(:) +hs.+
Changing the variables
'o:? q:
(B.8)
we get (B.3)
S島
(B'7)
n(q) + f -r a Lttd'1F(p)+ + -2 \p) -T tr, - u\q/
-
(B.4) \-''/
]
:G(q) + F(p) +
oP",)
rr Jo
12
e({)dt,
*\.
(B.10)
Assume that we obtain
y: #: .' (:) : F'(p), tdP
dp d /P\
(B.11)
Pds
E:E\;i:;E-?E
Then, from Eqs. (B.a) and (B.10), we obtain the
=卜 鉄の一 得)鵡 ′
=一 θ (9)一
Nos6-Hoover oscillator d9 ごτ
(B.5) α鳥
(1)
多
一
１ ′
ノ
ヽ、 ′ ノ
ヽ
Ｐ 一ｓ
／ｒ ｔ ＼
的
Ｆ ｌ 一
Ｐ 一ｓ
＜
卜 α α 〓 一 一
∝所
=― G′ (9)一 ξ P,
=-9-(P,
(B.12)
#=α (p2_1), and the Hanliltonian
70乙 ∬ =誓 +誓 +手 +ズ
Hence, we have a set of differential equations for the
variables (q,p, e)
=P'
respectively.
Gη
