International Journal of Bifurcation and Chaos, Vol. 7, No. 6 (1997) Ll95-1223 @ World Scientific Publishing Company

SYNTHESIS OF TOPOLOGICALLY CONJUGATE CHAOTIC NONLINEAR CIRCUITS MAKOTO ITOH Dθ pα rtmθ η ιげE′ cctrJcaι Eη gづ 2ccrづ ng

α ηαθθ mpし tcr

Scづ cη cc,

Nagα sα たO yη づ υersづ ι ν, Nagα sα たづ∂52, Jα pα η

Received」 uly 17,1996;Revised November 14,1996

In this letter, we present a method for synthesizing topologically conjugate chaotic nonlinear circuits. One advantage of our approach is that numerous results from classic circuit synthesis theory can be directly applied to generate a large variety of distinct but equivalent highdimensional chaotic circuits.

1. Introduction It is known that the n-dimensional

First, we define a generic class of nonlinear systems with one scalar nonlinearity. Next, we realize this system by using three kinds of nonlinear circuits, namely, an n-dimensional Chua's oscillator, an n-dimensional canonical Chua's oscillator, and an n-dimensional R-C oscillator. They are all topologically conjugate to the given system. Furthermore, the synthesis of the circuits with multiple nonlinearities is discussed. Finally, we show some examples of the synthesized circuits.

Chua's oscilla-

tor is topologically conjugate to almost every vector field with a scalar nonlinearity. Two systems are said to be topologically conjugate if there is a homeomorphism h which maps the trajectories of one system onto the trajectories of the other system and preserue the parametrization of time. In

●

▼

the case of secure communication systems based on chaotic signals, topologically conjugate systems are indispensable for the recovery of information signals [Feldmann et a1.,1995). The problem for finding an n-dimensional Chua's oscillator which is topologi cally conjugate to an nth-order differential equation with a scalar nonlinearity is completely solved by Wu and Chua [1996]. In this paper, we present a method for synthesizing nonlinear circuits which are topologically conjugate to a given system. We study this probIem from the view point of classical circuit synthesis theory. This approach has the following

2. Circuit Synthesis Approach Consider the nonlinear system defined by

u: f(r), u: L(r),

(1)

/ is a scalar function and .L is a linear operafor whose Laplace transform (transfer function) is given by G(s). In the case of nonlinear circuits, the system (1) is defined by

where

advantages:

(a) It is completely systematic and unified. (b) Many results on classical circuit synthesis are available, which were obtained by R. M. Foster, W. Cauer, O. Brune, R. Bott, R. T. Duffing,

づ=∫ (υ ), づ=L(υ ),

and so on. 1195

(2)

1196

1イ

fι

οん

exist many systems which have the same transfer functiOn C(S)and the scalar nonlinearityノ If G(S)iS a mι づ θηαιfunctiOn

i==f(V)

Chua's

diode

=

的

+

(").

(3)

then the system(1)Can be recast into the following

v

j7η

cjι Pι づ

differential equatiOn

bれ υ(m)+bm_lυ (m_1)十 ろ m_2υ (m-2)

十・… +blυ (1)+bOυ

=∫ (α ηυ(η )+α π_lυ (れ Fig. 1. Nonlinear circuit with a Chua's diode whose u-i characteristic can be realized by numerous methods developed by Chua [1969] and others.

1)+α

2)

η_2υ (π

+・ … +α l υ(1)+α Oυ ),

(4)

where

z=bη υ(m)+bm_lυ (れ -1)+bm_2υ (m-2) where

I

: l@) is the u-i characteristic

+・ … 十 bl υ(1)+bOυ

,

(5)

″ =α πυ(2)+α π_lυ (η -1)+α π_2υ (π +・ … +α l υ(1)+α Oυ

→=り ・ が

2)

,

(6) ″′

of a two terminal nonlinear resistor (Chua's diode)1 [Madan, 1993] and G(s) is equal to the admittance Y(s) of the linear circuit (see Fig. 1). A number of methods are known which realize the network function G(s). A systematic method for synthesizing a 2-terminal u-f driving-port characteristic was pioneered by Chua [tO6S] and many methods are now available for synthesizing Chua's diodes.2 Therefore, there

TherefOre, if the twO circuits contain the same Chua's diOde and the same adnlittance,then their dynanlics are given by the same implicit difFerential

equation. This implies that their vectOr flelds are 1To avoid the frequent appearance of

a

negotiue sign, the the nonlinear resistor

reference direction of the current of (Chua's diode) is chosen opposite to the standard associated reference in circuit theory [Chua, 1996]. 2Prior to the fundamental work done by Chua, multi-terminal nonlinear devices are identified by various names, such as

vacuum tubes, diodes, triodes, transistors, ferrite coils, sil_ icon controlled rectifiers, zener diodes, varistors, Josephson junctions, etc. Using an axiomatic approach, the basic family of circuit elements was classified and precisely defined by Chua into nonlinear resistors, capacitors, and memrislors [Chua, 1969]. Moreover, the principles for synthesizing arbitrarily prescribed nonlinear characteristics for these

Next we cOnsider the problem of realizing the

admittance y(s)(=G(s)).The admittances y(s) of the capacitor(3,inductOr ι,and resistor」 R are

given by sC,金 ,andお ,respectively(See Fig.2).

町1露「 品晨撃弔 晏3鷲 翼F品 ∬sね

▼

1∬

of the partial fractiOns,that is, Ⅳ

y(S)=Σ

乙 (s),

(8)

づ =1

then y(s)is realized by the circuit h F増

.3(a).For

『な 屁鵜ま I:蹴 l霧 猟樹i勇 F爛濫蹴

世猟;]flhiWIn‖ 性 鳳ifr跡 7殊 ∬

the partial fractiOns

〓

Z(S)

Ⅳ〒んＨ

basic circuits elements were developed for the first time by Chua, even though the technology in the 1960's were not quite ready to implement Chua,s principles. In this context, 2-terminal resistors with o-i curves synthesized using circuit_ theoretic techniques, as opposed to device physics, i, gu.r.r_ ally called Chua's d,iod,es. With the phenomenal advance in uery large scale integrated, circuit (VLSI) technology over the last 3 decades, Chua's diodes with virtually any prescribed r.,-i characteristic can be easily built using discrete compo_ nents, or IC chips when demand calls for a large number of them in applications.

topologically cottugate.

る (S),

(9)

.

Synthesis oJ Topologically Conjugate Chaotic Nonlinear

Capacitor

Name

Inductor

Ｌ

Resistor

Z(s) =Zr1t1

.-

.

+ Zp(s)

L

C

１ 一Ｒ

SC

*rrr.5)+Zr(5)+

R

Ｌ １ 一Ｓ

Adlnittance

︱︱︱ｌａつ︱︱︱ｌｌ

Svmbol

Circuits lL97

ａ ｄ

︲ 一Ｃ ｓ

呼Ｚ

Y(s)

Z(s)=sL+ sc l

＞ ｓ

sL

R (b)

Fig.

2.

Basic linear circuit elements.

Fig.4 Realization of the impedance Z(s),whiCh is given by the sum of the partial fractions(b)Series connection of ろ

(S)(b)Example of senes connection.

then Z(s) is realized by the circuit in Fig. 4(a). For example, if. Z(s) : Z{s) + Z2Q) : sL + $, the.t Z(s) is realized by connecting the inductor.L series with the capacitor C [see Fig. (b)]. Furthermore, if Y(s) is expanded into the continued fraction y(S)=ン 7(S)+

る(S)+ Y(s)=Yl(s)+Y2(S)+Y3(S)+…

■(S)+

・ +I《 S)

40+計

(a)

(10)

then Y(s) is realized by the circuit shown in Fig. 5(a) by using the above two methods alternately. For example, consider the admittance Y(s) :Y1(s) + z,ra-',1:rg",:*+r+ First,

Y(s)=

sC

+|

(b)

Fig. 3. Realization of the admittance Y(s), which is given by the sum of the partial fractions. (a) Parallel connection of Y; (s). (b) Example of parallel connection.

4i, :

UY sL * ZLG) + connecting the capacitor C series with the inductor tr. Then, we can realize Y(s) by connecting it parallel with .R [see Fig. 5(b)]. Without loss of generality, we can assume that Y(s) has the form: 2+… .+bls+b0 _2Sπ y(S)= bmsm+bm_lSm 1+bπ 2+… 1+α .+α _2Sπ _lSη α

we realize Z'r(s) +

nsれ +α れ

ZL:

π

#

lS+α 0

(3)

1198

』 イ ftο ん

expanded into the form ylSl― dlS・ d升

(t2) then the admittance Y(s) and the system (1) are

一 ︲ ，

Y(s) =

Yi(s) + Z:(S)+

可(S)+

Zl(S)+

realized by one of the linear ladder circuits shown in Fig. 7 and the n-dimensional canonical Chua's oscillator in Fig. 8, respectively. Recently [Wu & Chua, 1996] expanded Y(s) into the form: Y (s)

- ets*ez-l e3 ト

e$l

1 e5S‐ 卜

(a)

C6‐ 1

1

C7+θ

L

1

83+… ・ (13)

and realized Y(s) by using one of the linear ladder circuits shown in Fig. 9. For this case, the system (1) can be realized by the n-dimensional Chua's oscillator [Madan, 1993] in Fig. 10. Furthermore, if Y(s) is expanded into the form ︲ 一

Y(s) :6rs-u

＋

--1R

上

=

﹂ ｓ

Yts)

ｃ ｓ

ん2+

h3s* ん4+

(b)

Fig.

ん5S十

5.

Realization of the admittance Y(s), which is given by the continued fraction. (a) Ladder circuit realization (series-parallel connection). (b) Example of series-parallel

connection.

If Y(s) can

ん6+計

(14)

then the admittance Y(s) and the system (1) can be realized by the linear ladder circuits shown in Fig. 11 and the n-dimensional R-C oscillator circuit in Fig. 12, respectively. Next, we consider the following problem.

be expanded into the form

Y(s):c1s*c2*!

[Problem] Synthesize a nonlinear circuit which is topologically conjugate to the given

j %S+%+1

Σた

system (1).

1 ,(11)

c6s

*

c641

*

c1rq2s

*

cpa3

then the admittance Y(s) and the system (1) can be realized by the linear "ladder circuit,' shown in Fig. 6(a) and the nonlinear circuit in Fig. 6(b), respectively. Note that the realized circuit may contain negative linear circuit elements. If f (s) can be

In order to solve this problem, we first

calculate the Laplace transform of the linear operator tr, that is, the transfer function G(s). If the transfer function G(s) is realized by the admittance Y(s) of the circuit in Fig. 7 or Fig. 9, then we can solve the problem. That is, we can generate the topologically conjugate vector field by connecting a Chua,s diode characterized by i : /(u) with these realized

Synthesis oJ Topologicallg Conjugote Chaotic NonLineat

Circuits

1L99

i=f(v)

+ Chua's v diod€

(b)

Fig.

6.

Circuit realization oI the system. (a) Realization of the admittance Y(s). (b) Realization of the nonlinear system.

Ｎ 上 ｄ

６ ︲・ ｄ

d*t

ｌ

６ ︲・ ｄ

       ﹁

”         ２ ・ Ｎ

ｌ

１

２     上ｄ ．

dN-3 1

dN

(b)

Realization of the admittance Y(s); Circuit in (a) is used wheu the cortinued flaction (12) is terminated at the term dri+rs + drr+z (t is an integer). Circuit (b) is used when the expansion (12) is terminated at d'$+zs * dtr+s'

Fig.7.

1200

1イ

fι

οん

d8

dN_3

dN_2

i=f(V)

+ ︲ 一ｄ

I oe dro

６

Chua's diode

o

du-l

dN

i=К V) +

Chua's diode

1

:l

d5

t

d9

d10

V

(b)

Fig. 8. n-dimensional canonical Chua,s oscillator.

eN

V

eN_1

(b)

Fig.9.RealizatiOn Of the admittance y(s);Circuit in(a)iS uSed when the cOntinued fraction(13)is terminated at the term e2た S(ん iS an integer).Circuit(b)is used when(13)is terrninated at e2た +1

Synthesis of Topologically Conjugate Chaotic Nonlinear C'ircu'its 7207

句 ゴ  ︲ ︲ カゴ ・ ・ Ｎ Ｎ ｅ ｃ

ｌ

VV―

i=f(V)

l ell

寺

Ｎ １ 一Ｃ

?:::S V

寺%

上 ％

ｒ

diode v

Ｌ ＦＩＩＩＩ

+ Chua's

e4

V

(b)

‐

10. n-dimensional Chua's oscillator.

句ゴ ︲ ・ Ｎ ｈ

Fig

▼

(b)

Fig. 11. Realization of the admittance Y(s); Circuit in (a) is used when the continued fraction (14) is terminated at the term hzr+rs (k is an integer). Circuit (b) is used when (14) is terminated at hz*-

1イ

句ゴ ︲ ・ Ｎ ｈ

1202

ftο ん

Chua's diode

hN

1=flV)

+

Chua's diode

(b)

Fig.

12.

n-dimensional R-C oscillator.

circuits. Therefore, all we have to do is realize the admittance Y(s). Assume that Y(s) is defined by the rational

yい )

function ..

*

a1s

*

a9

(3)

> n. If. m I n, simply realize Z(s) : Next, we define deg(Y(s)) : rn - n. If Y(s) does not satisfy deg(f(s)) : 1, then replace Y(s) with the perturbed Yr(s), which is obtained from the following algorithm: where rn

Y(r)-t.

=Ⅲ

網

(stt

*

(Realization Algorithm) Case 1. Canonical Chua's oscillator Extract the term d1s * d2 from y(s), that

=Ⅲ

…

(15)

We are now in a position to state the realization algorithm.

(Perturbation Algorithm) First, expand Y(s)-l into a form: 珀

-ai

brn-ts*-L

t/t ^\ _b^s* I b^-ts^-r * b*-2s*-z + ... + 61s * 6s

+ an-tsn-l * &n_2sn-2.

ゴ ≒ F+平

s+θ た β02+γ :'

Bた

and - 0x * 1pi are the roots of b-s* * b7n-2sn-2 + ... * brs * b6 : 0. Next replace .47 with Aj - r, or Bp with .B7. e for sufficiently small e. Then the perturbed admittance Y"(s) satisfies the relation deg(fe(s)): t. where

ansn

1=平

is,

的

…

WithOut 10ss of generalitL we can assume that deg(K(s))=た

一ブ=1.If this cOndition is not satisied, replace n(S)with the perturbedィ (s)。 Next,extract the term α 3S+α 4缶 Om yl(s) 1,that is yl(S) 1= α3S+α 4+y2(S)・ Repeating this process,we wi11 lnally get the expansion:

ylSl=α r十 ご2+ ぬS+α 4+「

戸

ァ

丁TT

ご5S+α 6+

二

T

α7S+ご 8+ご

一―

9s+α 10+…

.

(12)

of Topologically Conjugate Chootic Nonlinear Circuits

Synthesis

1203

and therefore the admittance Y(s) can be realized by one of the circuits shown in Fig. 11. We note that the set of the circuits which does not satisfy deg(f3(s)) :1has measure zero. Case

2. Chua's oscillator

Step

1.

Extract the term els from Y(s), that is, Y(s)

Step

2.

Extract e2 from Yr(r), that Y(s)

: ef +

e2+ yz(s)

:

is, Y2(s)-1

e3

*

elsr +'

Y3(s) and deg(r3(s))

e4s

Ql-tk-!,qL-fk-'L'* qx-ztk-2 -+ "' + qrt + qo . pjsi * pi-1si-r *pi-2si-2...*prs *po

-L* ",

- -1.

- k- j :0.

(1つ

Then es can be extracted from Y2(s)-1. That

Thus, we obtain

Y(s): ef Next, extra ct

els + yr(t).

is,

Here, e2 is some constant satisfying deg(Y2(s))

:

:

from Y3(s)-r, that is, Y3(s)-l

+e2.#, :

*

e4s

(18)

Ya(s), and so we have (19)

""' -

Here, we choose

l_

"nr+Ya(s) the value of e2 so that ess can be extracted from Ya(s)-l, that is, Ya(s)-l

:

e5s

Step S. Repeat the Step 2. If it is difficult to extract eft or ers, then replace Ye(s) with Y6(s) sufficiently small) or the perturbed fl(t). Finally, we get the following expansion:

Y(s): efi+e2+

+ Ys(t).

*

e (e is

(13)

C3+ e48

* e5s

* _1_ eo

‐

V

*

e7 + --TeB, =

and therefore the admittance Y(s) can be realized by one of the circuits shown in Fig. 9.

3. n-dimensional R-C oscillator Extract the term d1s from Y(s), that is, Case

y(s)

:

d1s

*

gL-rs&-1-+ qt-zse- 2j "'+ qrs + qo yr(s) :- uru d,rs *qtse--+ I prsi +pi-1si-r *pi-2si-z... *prs *po

(20)

Without loss of generality, we can assume that

the expansion: If this condition is not y(s)=ん ls+一 一 deg(yl(S))=た ブ=0・ : Yr(s) + e for ん2+ satisfied, replace Y1(s) with Y/(s) h3s * sufficiently small e. Next, extract the term d2 yz(s). : If dz * from Yi(s)-l, that is Y1(s)-1 deg(Y2(s)-t) * t, replace Yz(r)*l with the perturbed YiG)-'. Then, we can extract d3s from yz(r)-1. Repeating this process, we will finally get

λ4+

ん5S+摩 (14)

l2O4 M. Itoh and therefore the admittance Y(s) can be realized by one of the circuits shown in Fig. 11. Thus, we conclude the following:

(a) If the transfer function G(s) has the expansion (12) [respectively (13) and (14)] without perturbation, we can realize the system by using the n-dimensional canonical Chua's oscillator (respectively the n-dimensional Chua,s oscillator, the n-dimensional R-C oscillator). Both systems are topologically conjugate to each other. (b) If the transfer function G(s) is not expanded into (12) [respectively (13) and (14)], then we can approximate the transfer function G(s) by G'(r), which satisfies lG'(r) - G(r)l < O(r) and obtain the expansion (12) [respectively (13) and (14)]. Then we can realize the perturbed G(r) by using the n-dimensional canonical Chua's oscillator (respectively the n-dimensional Chua's oscillator and the n-dimensional R-C oscillator).

Fig.

3. Multiple Scalar Nonlinearities Consider the following nonlinear system with multiple scalar nonlinearities

: f j(r j) , (j :1,2,. u: Lr,

uj

.. , n)

(21)

Nonlinear circuit with multiple nonlinear resistors.

Consider the nonlinear circuit with three nonlinear resistors in Fig. 1 (a). This circuit is topologically equivalent to the circuit in Fig. 14(b), if the linear circuit is reciprocal, that is, if the relation y;,i : yj,i holds. This is due to the following reason: The admittance matrix ya(") in Fig. 10(b) is given by ν4

νO

νA ν2

νσ

νス ν B

   

ν3 ν σ

一

νB

νσ

νB

４

Solving the equation Y6(s)

:

Y(s), we get

: lz(uz),..., in : l.(u.1, QJ) where the matrix II(s) : Y(s) : [Ut,i@)|,17, and V1, arc the Laplace transform of L, ij, and u6, re_ spectively, and ii : fi@i) are the u-i characterisi2

tics of Chua's diode. Thus, the problem of finding a topologically conjugate circuit is reduced to the classic realization problem of the transfer matrix Y(s) (see Fig. 13).

UA:

UT,2,

: uz'T ' ac:a2Jt yr: yL,r * At,z * Uz,t, Az: gz,z * y\z * Azt, Us: Us,z * yz,t * Az,s. aB

Q5)

Therefore, if the symmetric admittance matrix y(s) is given, then we can realize it by y6(s). This shows that we can realize the circuit in Fig. 14(a) by that in Fig. 14(b).

︶

ν4

．  

=[「

１１１ｊ ０

知

２ ２

И ち ・ ︰Ｌ

ir : fr(ut),

”   ．︰  ％

νη,2

η   η                η

ν2,2

仇

〓

︰・ 島

︰ ％ 仇 ∽ ．

ム ら

・ ・・・

13.

Y(s)= [ rr,, (s)J

３ ν

where u : (ut,.u2t..., un), x : (rt, r2,..., rn), fi is a scalar function, .L is a linear operator and its Laplace transform is given by the matrix fl(s). In the case ofthe nonlinear circuit, the system (1) can be given by νl,2

Chua's diodes

Synthesis of Topologically Conjugote Chaotic Nonlinear

Circuits

1205

z

diodes

＋  め ・

Chua's

Chua's diodes

Y(s)= [r,,r(s)J

(b)

(a)

Fig.

14.

Example of a nonlinear circuit. (a) Nonlinear circuit with three nonlinear resistors. (b) Equivalent circuit.

4. Chua's Diodes

nonlinearity:

The a-i characteristic of Chua's diode can be realized by numerous methods developed by Chua [1969] and others. The most practical implementation was designed by [Kennedy, 1992]. He realized a three-segment piecewise-linear Chua's diode

with slopes ms and m1 and breakpoints Fig. 15(a)l: l,

:

f (u)

:

Ttut'u

* 0.5(ms - *r)(1, a ael υ  υ  υ

ｍ  ｍ  ｍ

〓

ヽ ｒ ｉ り ヽ １

― (π l―

iBpl

la

-

Bpl)

mO)BP fOrυ