ヒステリシス抵抗を含む回路の族のカオスに関連した現象について

(1)

NII-Electronic Library Service

MsMo:Rs or S"AMI

tserrmr- eT TIeemvoLoGT

Vel 22,No. 1,19S8

On

Chaosand

_Related

_Phenomena

_from

a

Circuit

Containing

a

Hysteresis

Resistor

Family

ToshimiehiSAITO"

ABSTRACT: This article discusses chaos and relatecl _phenemena from a circuit familyconsisting of

R, L, C,-R

_(linear

negative resistor), and one hysteresis resistor. This circuit famity isdescribed

by a canonical form of n-dimensional _{piecewise-linear} equation containing a small parameter E. If

e issuMciently small the analysis isreturned intoa hystereticconnecting !inear systems. On this simplified system, follewingresults are shown:

(1)

For three dimensionalcase, a chaos _generating

condition {in_Lasota _& _Yorke's sense) isrigerously derived.

(2)

For a fourdimensionalexample, two

dimensionalPoincar6 map isrigorously formulatedand an interestinghysteretictransitionalphenomenon

between torus and chaos isdemonstrated.

1. Introduction

This

article

discusses

chaos and related

phenomena

frorn

a circuit

family

consisting of

R,L,C,-R

_(linear

negative resistor), and one

hysteresis

resistor.

We

_propose an

n-dimen-sional canonical

form

equation for _the circuit

family.

It

is

piecewise-linear and contains a

small

_parameter

e.

In

the case whereeis

sufficiently _small _the mathematical treatment

is

simplified and

following

results are shown:

For

a three

dimensionai

case,

Poincar6

re-turn map

is

defined

as an one dimensional

mapping,

hence

_a chaos generating conditien

is

rigorously

derived.

The

rneaning of chaos

is

that the mapping

has

an absolutely

conti-nuous

invariant

measure

[1].

For

a

four

dlmensional

example, a rigorous

formulation

of a two

dimensional

Poincar6

map

is

_given and some numerical results are

shown.

Especially,

an

interesting

hysteretic

transitional _phenomenon

between

torus and

chaos is

demonstrated.

In

_addition, some

Iaboratory

experimental results are shown.

"Chaos in electric circuit"

have

been

stu-died

with

great

interests.

There

are some

excellent analytic works such as "Double

Scroll

Farnily

_[2]"

and "Forced

Josephson

Junction

circuit

[3]".

However,

various

pro-blems

should

be

considered _to

clevelop

the

study.

For

example,

1)

Construct

sorne general theories.

Pre-sent works mainly

depend

on some example

clrcults.

2)

Consider

higher

order systems.

We

can certainly

discover

new

interesting

phe-nomena even

for

four

dimensional

auto-nomous circuits.

This

article _gives some impact for these

problems:

Then-dimenslonalcanonical

form

seems to

be

a starting step

for

the

construc-tion of a general theory.

The

"hysteretic

transitional _phenomena"

in

the

four

dimen-sional example suggests the great

interests

of

higher

order systems.

And

rigorous _proof

for

the three

dimensional

case seems to

be

developed

to more _generalized one.

* _ptx=#a _man

vafn62 rp 11 A 8 HMN

2.

Canonical

Form

Fig.

1(a)shows the circuit

family

defined

as below.

A)

NR

is

a current-controlled nonlinear

resistor which

is

characterized by

Fig.

1(c):

.,..f<,,,=l";(,S-2Efr)･(i.E./r,.

,,,

k

r(i,+2E!r),

i,f{-E/r

B) Lo isa small inductor and itis

assum--,13

(2)

re

ecrees

\re

et

Loi ilNR>v,5v Alinearnetwork iconsis.tingef -R,R,LC e HN,-(a) N rlrl . ₊ (b) -']1Vl r= Vl2E''tr N r N ']]-'''-2E ±2E (c) (d)

Fig. 1. (a)The circuit family,

(b)

A realization

of IVR,

(c)

Characteristicof NR,

(d)

acteristic of the hysteresis resistor as Lois

shorted.

ed that

NR

operates as a

hysteresis

resistor

of

Fig.

1(d)

if

L,

is

shorted.

C>

N

is

a

linear

network consisting of

R,

L,

C,

and -R.

D)

In

the

graph

of the circuit family,

there are no

C-only

_cutset _and no

L-only

tleset.

Applying the

general

theory of state

equa-tions

_[4],

the

state equation of the circuit

farnily

is

given

by

the

form:

,d,

(?i)-B

(i)-J'ti

(,)

d

it=V-f<ii)

,

Lo

dt

where v,=(v,i,･･.,v,.)T

is

a

branch

voltage

vector of capacitor

branches

in

N,

iL=:(i",...,

iL.)

is

a

branch

current vector of

inductor

branches

in

JIV;

B

is

a coeMcient matrix

de･

termined

by

the structure of

N,

and J=(1,

O,

･-･,

_o)r.

ca

22 ig M1e

Next,

via the rescalings:

xiis v/E , xi+i =- v,t/E , xj+.+i =-riJfE ,

z=ri,!E,

<x=(xi,･･･,xpt)',

IV'==1+n+m,

i--1,･

1=1,.･･,m),

r=-tl(rC), e=-L,1(r2C)

the equation

(2)

is

transformed

into

(e2==xi-h(z)

be=Ax-J(z-x,)

,

where

.d

X-d,X, h(2)=2+lz-11-lz+1].

Moreover,

we try to transform the

tion

(4)

into

a cannonical

ferm.

Define

notations:

t,:'E".;fti'.';

£

,xi'f'

l

1sl-Al==sN+aN-isN"i+･･.+cris+aoJ

and define the transforrnation:

M.8L&i&

1,

O, ･-･O al atA : . a,AN-t X=(Xl, xiX2X3: .X4

(X=

T･x, ･･･,

XN)T)

By

using this transformation _with

Hamilton's

theorem, the equation

(4)

formed

into

the

form:

I/:i2

'-Xh

-(.);

e(z-J<),

where

Here,

it

tions

of all simplified

if

T

is

not

O

1 O----O

10 '

o

'･ 1 -cro,-al, ''''',-crN-1 e=(2, e,,･･･,e.) =2･

V

is

remarkable that all

elements

in

the

into

the

form

of the

singular.

Namely,

. ..,nt(3)

(4)

equa-some

(5)

(6)

Cayley-.Is

trans-(7)

state equa-circuit

family

is

equation

(7),

solving

(7)

is

(3)

.-NII-Electronic Library Service

On enaos and Relatedllhenomena

from

a arcuit fumily

Xi X,=h(Z) b1 . N s. J ss-e oNs 2Z N -t s. _-1a

Fig. 2. Vectorfieldfors-O.

suMcient _to _clarify _the _complete

dynamics

of the circuit

family.

The

equation

(7)

is

the

N+1

dimensional

_{piecewise-linear}

equa-tion containing a small _parameter E,

i.e.

a

piecewise-linear version of so-called

"Con-strained equation"

[5].

For

the case where

NR

operates

asahy-steresis resistor, the mathernatical treatment

issimplified

dramatically.

Namely,

_this _case

is

equivalent to the

following:

In

the case of where s issmall enough, z

rnoves

far

faster

than

X

and alrnost all

solu-tions are constrained on stable

_parts

of the

manifold _given

by

Xl==h(z):

(see

Fig. 2)

S.={X]

a1

Xl!)-1,

Xl=z-2}

,

or

Srv={Xz1

Xlsg1,Xl==z+2}

The

dynamics

on

S.

or

S-

are

described

by

X=A(X-P), for XES.

(8-a)

X=A(X+P),

for

l!CES-

(8-b)

wherer

P=

(P,,

2,e,,･･･,e...,),

1 Pi=-

(eN+crN-LeN-i+･･･+a2e2+2ai)

ao

In

addition,

if

a solution on

S.

exceeds the

threshold

Xl=-1,

then it

_jumps

onto

S-,

holding

X==const.

and

if

a solution on

S-exceeds the threshold

X=1,

then

it

jumps

onto

S.,

holding

X=const.

_(In

following

parts, we ornit z).

That

is,transitions

be-tween

S+

and

S-

is

as

follows:

g::gl

[Ig

gl-N,[gib.li

ll

i

il

g::ggg:

ri}

(9)

Thus,

the equation

(7)

is

simplified

into

the

two syrnmetric linear equations

(8-a)

and

(8-b)

with

the

hysteretic

transition

(9).

Wecall

thissimplifiedsystemas"Idealsystem". This

system

is

not correct

in

the rigorous

mathe-rnatical sence but well explains the actual

dynamics

of the circuit.

In

addition, the

justifiability

of "Ideal _system"

is

_analytically

explained in an engineering sense for N=2

[6],

Next,

we show some results

for

this

system.

3. The Three Dimensional Case

F.or

N=2,

"Ideal

System"

is

(ilf')=(-e

21)(XiLPi), for

_(x;y)es.

.

(i')==(-9

2:)(Xy++P2i),

for

(10)

with the transitional condition

(9),

where

1

v==cro, -2a=-ai,

Pi-

(4a-02).

v

For

a simplicity, _pararneters are assumed to

be

Pi=1,

a>O, v>a2

(di=-Vv-o2).

(11)

In

this case, eigenvalues are a±j'toNand

Pi=1

implies

that equilibrium _points _are _on lines

X==-1 or X=1.

Next,

define

some objects

on

S,,

for

the definition of the Poincar6

re-turn map

(see

Fig.

3).

Li{X

Y[X=1,

Y}l-2}

,

-1Y, lr.s.s. _Y MsN 21!r1xL,X ₂ --x}o-1N!1-/-xLl[L72----DO-p1 s'-DL!1.-X

/Y,

-r2--1'o Lt-)rJ2-ie -M

Fig. 3. Trajectories on S+ and S-.

(4)

-reecr* Jk\re pt

za

22 if M 1 e LF (Yo)MT(D)L--r i _l-T--'-: 1 -LD-2 _L, Lz

:

---e-.V-: 1. .,.;i..4 -11 1 , 2-D lx LLi

I/- -1---LZ--d x F--;----, (-2) xe,- 1' ,l Lt11t 1 v-- 1-d -M.' --1 dL;L' ,L,L, -M -D2-2D _rvI

Fig. 4. An outline of Poincar6

Yomap.

M:

The

point

on

L

such that the

tory started

frorn

it

_passes

(1,

-2).

D:

The

point

on

L

such that the

tory started from ithits

(-1,

2),where

D>-2

is

assumed.

L,i{X

YIX=1,

DE{

Y<M}

,

L,!{X;

Y'[X=1,

-2"<

Y<D}

.

And

define

syrnmetric objects on

S-.

L,={X,

YIX=-1,

Y:f{l2}

,

L,,={X; YIX=-1, -M<

Ys-D}

,

L,,={X]

YlX=-1, -D<

Yf{2}

.

Consider

trajectories started

trom

L,ULt

at

r=O. A trajectory started

from

a _point

Yh

on

Li

intersects

L

at r:=n/w

(Let

Yl

be

this

intersected _point).

Also,

a trajectory started

from

a _point

Yh

on

L2

hits

the threshold

X=-1

_(Let

}1 be this

hit

_point),and at this

mornent,

it

jumps

onto a point

Yl

on

L. Hence

it

follows

that we can

define

the next

one

dimensional

mapping:

T:

(L,UL,)-(LUL,),

YhHY,.

(12)

Since

the vector fieldon

S-

is

symmetric to

that on

S.,

we can also

define

the mapping:

T:

(L!uL,,)

-

(LuL,)

_, Y6b Yl .

(13)

Hereby

behaviors

of trajectories are

simpli-fied

into

the

following

one

dimensienal

Poincar6

map whose outline

is

shown

in

Fig.

4:

IZ

q

z

8

i

Uilii

11 1liiiii

'

tli･t

l

' '

I'

6 .es.1.15.2

Fig. 5. Chaos generating region.

T:

(L,uL,uLIuL,r)->(LuLt),

YheM.

(14)

This

rnapping can be formulated rigorously

hence

the

fellewing

result

is

obtained.

[THEOREM]

Let

2 a

q='

v-iJ-

, 6=

v-iJ'

,

(g*Etan-i-2-

, to=

tw)

.

4

e:(g--PD<q< 21a ,-Zcot-(ii,2.e2)

(15)

e2-.Vg'He-:=

<1

is

satisfied,

then

there

exists a set

J

and

its

subset

i

(lcJ)

in

the

domain

of

T such -that

T(I)cL

Tes(Yh)e4for

all

Yh

in

L

fbr

aay

integer

n, and then

IDT(Yb)[>1

for

almost

all

Y6

in

L

where

DT

is

the

dt:ffbrential

coelf-ficient

of

T,

This

theorem means

that

T:

IL>l

is

stable,

is

expanding

(DTI(Yb)1>1),

and

has

an

ab-solutely continuous

invariant

measure

[1].

Hence

almost all orbits of T are chaotic.

Proof

of

this

theorem

is

shown

in

[6].

The

condltion

(15)

is

satisfied

in

the shaded

re-gion

in

Fig.

5. We

have

actually observed

(5)

-16-NII-Electronic Library Service

this

Le-OiL

On Chaos and Related Rhenomena

from

a

Nclv,v,rcx

Fig. 6. An

chaotic

-g

example circuit forIV=3.

attractor

from

some

circuit

_[6].

4. A Four

In

this section,

four

dimensiona1

circuit

dynamics

(i):-C

i･ eth=x-h(w) where

h(w)==w+lw-11-1w+11.

Moreover,

by using the transformation:

(ilil)=-(,,),l

0

21

-iXi),

the equation

(17)

is

transformed

into

the

canonical

form:

(

X

9=(,[2agi),

-2(2

'

-6), _2s2i)(ll)

L

where _.f<･)

describes

and

is

given x-v,IE, y=v,IE r!t/(rC) ,

The

equation Dimensional

Example

we consider _an _example of

circuit

_(see

Fig. 6).

The

is

described

by

dv,

･･

c

=-t-11

dt

dv!

. C ==gv2+1

dt

(16)

diL

=Vt-V2

dt

di,

=Vi-jC<ii) ,

Lo

dt

the characteristic of

NR

by

_(1).

Next,

via a rescalings:

, z=ri/E, w!riilE, 26=- rg _, _pii!r2c!L _, E=-

L,f(r2C)

_,

(16)

is

transformed

into

2i

ij,

-g･xi)-(g)(w-.,

Circuitthmily

-( ITL I

Ip

)

(w-x)

eto==X-h(w)

(18)

Ifsis

small _enough, "Ideal

system" of this

equation

is

_glven

by

X=A(X-P),

for

XES.

(19-a)

X=A(X+P),

for

XeS-

(19-b>

where

Ar=(X,

Y;Z)', _and

.--..(g

,i

?)

Np(2E-1),

-2(p-6),

26-11,

2

P=-ip,,

2, -2),

Pi=26-1

and the transitionalcondition

is

(9).

For

a simplicity, eigenvalues of

A

is

as-sumed to

be

s=2,a ±]'w

(ca=

Vp-ff2),

where

a>O, v>o2, 2>2a.

In

this case, a, _p, and v are _given explicitly

by

2

and a:

26==A+2a+1, _{p=(262+-22a.)A} , ,=2+22ap

Fig. 7.

-z

Vector fieldson S+ with some objects.

(6)

-17-相模工業大学紀要　第 22 巻　第 1 号コ」 r 、「こ（a ）　　　　ユ臼_一 1 ● ワ」 5　　z　　日

2v

ロ　　エ

2v

つ」！尸二（

b

）　　　　1 　ド−・！「　　丶｛　　 ’・t ＼　　　 s．　　　　　　　　　＼ _」図

L

−＿ −₁ _記 ₅_z_日 3Y2 （c ）　　　

1

払 − 臼一，，辱曾・ ● ，「孕 1 2 5　 Z　 8 ヨ Y2

（

d

）ユ　　　　　　　2 　　　　　　　−

1

　　　 2　　　 5　 Z　 8

Fig ．8．　 Some 　attractors 　by　numerical 　and 　laboratery　experiment 　for　p＝12．（a）： Periodic_attractor

　

forδ＝．7

，（b）：Torus　attractor 　for δ

＝．8，（c）：Periodic　attractor 　forδ＝．905，　and （d）： Chaotic

　　　　　attractor 　for　δ＝．92．

Consider

　the　vector 　

field

in

S

_＋．

Let

　E γ be

the　eigenspace 　corresponding 　to　the 　real　eigen

value 　_and 　

let

Ee

be

　the　_eigenspace 　correspond −

ing

　to　the　_cQmplex 　eigenvalues ．

They

　are

given　

by

E

『 ≡ ｛

X

，】

r

，

Zl

（

Z

十2）＝ 1_（

y

−

2

）：

R2

（

X

−

P1

）｝

Er

≡ ｛

X

，　

Y

，　

Zl

（

Z

＋

2

）− 2σ（

Y

− 2）＋レ（

X

−

_A

_）＝

0

｝　　　　　　（

20

）．

(7)

On Chaos and RelatedPilenomena

from

a crrcuitFlamily

hl" L! tL･/.g :-/.ti.. 1 ; ]) 2 e .99 .9

Fig. 9.

(a):

Bifurcqtion.'diagram for increasing

6

(p=12).

In

addition,

define

some objects

in

S. (see

Fig.

7):

B+={X,

Y;

ZIX=

1},

B-={X;

Y,

ZIX=-1},

B.i{X,

_Y

ZIX=1,

(Z+

2)

-2a(Y- 2)+vT(1 -Pi) <O} ,

B.i{X

Y,

Z

1X=1Y>

-2}

,

B'n={X;

Y;Z

[X=ml,

Y<2}.

(21).

Consider

a trajectory started

from

a point

(1,

Yli,Zli)eB'.

Such

a trajectory must

hits

Bk

at some

tpositive

time T,, since the

ex-ponential component of

X

is

negative and

since

X<O

on

BS.

Let

(-1,

Yl,Z,)GBS

be

this

hit

point.

At

this moment, it

jumps

onto the

_point

_(-1,

Yl,Z,)

in

S-.

Here

itis

remarkable that the trajectory started

from

(-1,

Yl,

Zi)

in

S-

is

symmetric to the

trajec-tory started from

(1,-Y,,-Zi)

in

S..

Hence

.9t fi

(p==12),

{b):

I l/ _.".(a) (b) D _e 'S2

Bifurcationdiagram fer decreasing

it

follows

that

behaviors

of trajectories are

simplified into orbits of two dimensional

Poincar6

map:

T:

B.-B.,

(Yh,ZD)->(-Yl,-Zl)

via

(Z,Z,)

(22).

Since

the equation

(19-a)

is

linear,

an implicit

forrnulation

of this rnapping

is

easily

obtain-ed as

below:

Yl=

oa.

F(T,

Y6,

Zo)ir-tt+2

,

(23)

02

Zi=

e.2

F(T,

Yli,

Zh)rt-ri-2

,

whereF(rY6,

Z6)=exp(oT)(A

cos toT+B sin tor)

+Cexp(aT)

c=(Zh+2)-2a(Yh-2)+v(1-P,)

22-2o2+v '

(8)

-19-Npt=*

lt\resiva 22

8 ca

1e 3Y2

Ol(

"-L

s9Y2

2

T5

ze

@i

a.-tA

J

s

z

T

x･

:sp.

_Y:2?sa 1 ":t

t

.--.

fig･

T

, , 'Observable phenomena are O-@-@-@-O, for inereasing

oj

and @-,@.(Fig.8,{c))--@-@.O, fordecreasinge. 5Z8

@

s

-,

se2

2

Ts

zs -t 3Yz 1

B-tz

i . . . .

T5

z8

l

2

T

s

2,

::t

E･@

Y2 "x8 zs

o

: fi==.895,

@

&

@:

sx,

i

sg･

T

Fig.IQ. Hysteretictransitional phenomena,

(!)

ti=.913,

@:

6=.915, and _p= 12.

A=1-P,-C

and 1 ,

B=

to(Y6-2-aA-2C) ,

(24)

T,

is

the solution of the equation:

F(T,,

Yb,

4)+P,+1=O

(2s)

. -1

S

Z Y2 SZ8

T

-t Z 5 6=,8975, (D: Z8 fi=.899,

@

&

_@:

In

actual calculations, we solve thisequation

by

using

Newton-Raphson

method.

Fig.

8

shows some example attractors of

T

with

corresponding

laboratory

experimental results.

In

thisexperiment, we

have

verified that

NR

(9)

On Chaes and RelatedPhenemena

Here

we would like

to

demonstrate

an

interesting

hysteretic

transition

between

torus

and chaos.

Fig.

9 (a)

shows a

bifurcation

diagram

which

is

obtained by plotting

Y

co-ordinates of attractors

for

increasing

6

and

for

_p=12. At each step of a,the

last

_point

of the old step

is

used as the

initial

point of

the new _step _and transient states are omitted.

On

the other

hand,

Fig. 9

(b)

shows the

dia-gram

for

decreasing

6

which

is

obtained

by

contrary rnanner.

As

shown

inthese

figures,

different

attractors are _observed

in

intervels

A

and

B. It

irnplies

that at least two

at-tractors exist

in

A and

B,

and that a

hy-steretic _transitionoccur.

Fig.

10shows some

typical attractors observed at points

O

to

@

in

Fig.

9. Consider

the case of increasing a

(Fig.

9(a)).

From

6:=.89 to fi=:a,there exists one torus

attractor as

Fig.

10 (iD.

At

j=a,

the new

periodic attractor as

Fig.

10

@

is

born,

but

observable attractor

is

Torus

of

@

in

inter-val

A. In

this

interval,

this

hidden

new

at-tractor _grows to "Periodic

Torus

like

at-tractor" as

Fig.

10 @.

At6=:b,

the old torus

attractor

dies,

thus "Periodic

_Torus

_like

at-tractor"

is

observed.

After

it,

this attractor

fades

to 11-periodicattractor as

Fig.

8 (c)

or

Fig.

10 @.

These

trans'i`tionsnear

A

are

schematically indicated

by

down-ward

arrow

in

Fig.

10. The

transitions near

interval

B

are

some-what similar to

it.

Namely,

at O=c, the

new

8-periodic

attractor as

Fig.

10

@

is

born,

but

observable attractor

is

the old 11-periodic

attractor as

Fig.

10 @.

And

at 6=d, the old

attractor

dies

thus the 8-periodicattractor is

observecl.

Finally,

this periodic attractor

grows to chaotic attractor

(Fig.

10

@->Fig.8

(d)).

On

the other

hand,

the case of decreasing

6

can

be

explained

by

contrary manner.

It

is

indicated by up-ward arrow in Fig. 10,

frem

a CVrcuit Fizmdy

where "birth _and "growth" _of _up-ward _arrow

correspond to "death" _and "fading" _of

down-ward arrow, and so on.

These

explanation

is

a conjecture.

Now

we are trying to _give some theoretical

con-firmation

for

it.

However,

these results

sug-gest the great interests

in

higher

order

systems.

5. Conclusion

We

have defined a circuit

family

contain-ing one

hysteresis

resistor and

have

_proposed

a canonical

form

equation which

dscribes

it.

The

equation

is

simplified

into

"Idea1

System",

a chaos

_generating

condition

is

rigorously

derived

for

three

dimensional

case, and an

interestinghysteretic transitional phenomena

is

demonstrated

for

a four

dimensional

ex-ample.

The

major _present _problem

is

to_give

some theoretical confirmation

for

the four

dimensional

exarnple.

Acknowledgement

The author deeply appreciates the

sugges-tions frorn

Prof.

S. Mori

of

Keio

university

and

Prof.

M.

Hasler of

Swiss

Federal

Insti-tute of

Technology.

References

[1]

A. Lasota and

J.A.

Yorke, Trans. Amer.

Math. Soc,, 186 (1973),481.

[2]

L.O. Chua, M. Komuro and T. Matsumoto,

IEEE Trans. Circuits Systs.,CAS-33, 11

(1986),

1073.

[3]

F.M.A. Salam and S.S. Sastry, IEEE Trans.

CircuitsSysts.,CAS-32, 8

(1985),

784.

[4]

Y. Kajitaniand S.Shinoda, "Network

lysis",

Japan

Science& Tech. Press, 1979.

[5]

H. Oka and H. Kokubu, Tech. Rep. IECE

Japan,

CAS84-102 (1984),1,

[6]

T. Saito,Trans.IEICE

Japan

(in

press).