A Chaotic Network Based on Intermittently Coupled Capacitors
A CIIAOTIC INTERMITTENTLY
NETWORK COUPLED
BASED ON CAPACITORS
EEE Dept., Hosei T_lrniveGtv. Tolxo,., 1S-L0002, .I~IXUI [email protected]
2. UNIT
ABSTR,ACT
This paper studies a network of’ S chaotic ~11s n-it 11 time-variants h)-st eresis using a concept of Intermittent loCoupled Capacitors ( ab. ICY’ j . The net work exhibits intleresting phenomena: synchronization of chaos, q-nchronizatlion of periodic attractors and their co-existence phenomena. We clarifv ” these phenomena t heoret icallv _ Some of’ the theoret ical reusing a mapping procedure. sults ar? verified in the lahorat ory. An approach t on-ard ion of the ICC technique is also discussed a avneralizat -
1. INTRODUCTION Set works of chaot ic elements have been studied inbenA\-el!- from the view point1 of fundamental problems. Thq- are also ime.g.. modelings of biological qstems. applicatNions, e.g., neural comport ant fo r engineering puting and communication q-stems [I]-[A]. There exist many- possible connection methods [.rj]-[T]. Han-ever the theoretical amlysis of such networks is very diKim11 . 111 order to studv\- the basic dJ-namics of tlhe nctn-ok, we have conklered a feedback connectlion met hod in [6] [;I ( 17-1iic* l i is suit at+ for piecewise (or local j linear q-stems. In this paper we propose a simplified coupling met hod, the Int crmlt t entlJ- Coupled C’apacit’ors (ab. ICC j. and Apple- it to a network of’ A’ (haotic cells. The ICY-’ is a\-ailable not only- for piecewise linear s>-st#ems but also for sinoot 11 qst ems. Roughly- speaking, it consists of home capacitors coupled b\- a periodic switch: the capacit or volt ages are equalized instant aneouslv v at the moment when the sn-itlclr is closed. Basically\-, t lie network d>-namics is described I~!- (A’ + 1 j-L) map with one real and S binaryT- variables. A basic version of the ICC technique i 5 shown in [8].
0-7803-5474-5/99/$10.00 1999 IEEE
First we consider by
GENERATOR
a cell (unitI chaos generatlor)
dx 1, -1, C
described
= p-r + y, y = h (.I+,7-) ,
l
% h(.r. T) =
CHAOS
(11
for .c < D-C+). for -7 3- --J%(r),
T/?(T) = 1 + (a - 1) 2 (C(T - n) - I’(r r-l=0
- 0.5 - n)) (
v-here .I+E R is a st#ate, y f { - 1, l} is an outSput, T is a normalized time and L,-(T) is the unit step function. A periodic square wave Th ( T) controls thresholds of the time-variant hc\-steresis h (s, T) : h is swi khed from 1 to - 1 if .r hits the upper threshold Th (T) and vice wr~a (see Fig. 1 j . For simplicit>-, n-e a.ssume t!hat 0 < p, 0.5 < Tri < 1 and - l/p < --CL < -.r’, , n-here Tp = (X/p)
lni( 1 +p)/jl
-11))~ z: = (G + l/p) =p(y/(2X))
-
l/p and xc = (l-l/p) esp(--p/(ZX))+l/p (see F&l(b)). In t hi.5 case, (1 does not affect tlhe svst em dvnamics and we focus on the parametlers (A. pi E 13. “The sj-stem cl!-namics can be reduced into the discret’e tlime map: .(.(17+ 1) = .f(+),y(n).p). y(rl + 1) = j+(n). y(n);p).
i
’ -y((-y.r-;)tf+;). for .rh < -ys .fi*r,,!/j
= (
y ((-y.r
- :-> tf----+
< ,c,,
!1->.
for .q7 < - y-r < .rb ,
where 17is a positive integer, .rn = - ( 1 + l/1,) esp( -y/ (2X)) + l/1;, and ,~‘h= -( 1 + l/p) espj -1)/X) + l/p (see Fig.1 (13)). Fioq b *7- shows a circuitI
V-414
model
of the cell.
In the figure,
3. THE
ICC
NETWORK
V-415
- x,
Xh
0 r(n)
[X(17
,]
Figure 1: A esarrqde of’ the t-d w-t-t map f. clott ccl line shows the t-et urn imp F for cl ( n ) = (p = (0.275.0.125) c $I().)
The 0 5.
O.’ initial
a (0j
LO
Figure 6: Plots of .T( h ) for mrious t-at io ck(0 ). (p = ppo.) The dots indicated by (a), (43) and (c) shows ~(0) of Fig.i( a ). Fig.S(\.‘) and Fig.S(c). respectiwlv. . Sot e again that the uppi~ ai~4 the lower ligut-e, 5 are ohtaitwd ft-m-n differmt
(a)
state ratio
.F( 0 ) .
0 0.1 0 p 0.08 .it2n+2
)
I 5 ic1
chaos 1
t
turo
t
0.04: l;perifkys 0.5
Figure
5: Esa~rqh3 of the return map G’ ant4 at tractors f 01‘ p = p,3 . (a) I-petiociic attractors (0 (0) = 0.6. ) ( t3) L-pc-tiodic atti-acttotx (n (0) = 0.8. ) (c ) CIaotic attmctot.. ( c\ ( 0 ) = 1 .o. ) T11e upp Cl’ and the lo\\-er at t r-act 01‘5 are ohhind fi-cm cMfei~ei~t hitid states of .1.(0).
t
chaosI
:_ 1
initial sbte ratio a ( 0) l *O
diagram from the JC’C’ tletn-wk. Fb3 we 7: Bifurcation 0.275.) The dots correspond to the c4ots in Fig.6.
4. GENER,ALIZATION
V-416
(x =
r u” -
-5!
t,’ -
II
II
Z
. . --u -
*
,
-. --
c
V-417
NETWORK COUPLED
BASED ON CAPACITORS
EEE Dept., Hosei T_lrniveGtv. Tolxo,., 1S-L0002, .I~IXUI [email protected]
2. UNIT
ABSTR,ACT
This paper studies a network of’ S chaotic ~11s n-it 11 time-variants h)-st eresis using a concept of Intermittent loCoupled Capacitors ( ab. ICY’ j . The net work exhibits intleresting phenomena: synchronization of chaos, q-nchronizatlion of periodic attractors and their co-existence phenomena. We clarifv ” these phenomena t heoret icallv _ Some of’ the theoret ical reusing a mapping procedure. sults ar? verified in the lahorat ory. An approach t on-ard ion of the ICC technique is also discussed a avneralizat -
1. INTRODUCTION Set works of chaot ic elements have been studied inbenA\-el!- from the view point1 of fundamental problems. Thq- are also ime.g.. modelings of biological qstems. applicatNions, e.g., neural comport ant fo r engineering puting and communication q-stems [I]-[A]. There exist many- possible connection methods [.rj]-[T]. Han-ever the theoretical amlysis of such networks is very diKim11 . 111 order to studv\- the basic dJ-namics of tlhe nctn-ok, we have conklered a feedback connectlion met hod in [6] [;I ( 17-1iic* l i is suit at+ for piecewise (or local j linear q-stems. In this paper we propose a simplified coupling met hod, the Int crmlt t entlJ- Coupled C’apacit’ors (ab. ICC j. and Apple- it to a network of’ A’ (haotic cells. The ICY-’ is a\-ailable not only- for piecewise linear s>-st#ems but also for sinoot 11 qst ems. Roughly- speaking, it consists of home capacitors coupled b\- a periodic switch: the capacit or volt ages are equalized instant aneouslv v at the moment when the sn-itlclr is closed. Basically\-, t lie network d>-namics is described I~!- (A’ + 1 j-L) map with one real and S binaryT- variables. A basic version of the ICC technique i 5 shown in [8].
0-7803-5474-5/99/$10.00 1999 IEEE
First we consider by
GENERATOR
a cell (unitI chaos generatlor)
dx 1, -1, C
described
= p-r + y, y = h (.I+,7-) ,
l
% h(.r. T) =
CHAOS
(11
for .c < D-C+). for -7 3- --J%(r),
T/?(T) = 1 + (a - 1) 2 (C(T - n) - I’(r r-l=0
- 0.5 - n)) (
v-here .I+E R is a st#ate, y f { - 1, l} is an outSput, T is a normalized time and L,-(T) is the unit step function. A periodic square wave Th ( T) controls thresholds of the time-variant hc\-steresis h (s, T) : h is swi khed from 1 to - 1 if .r hits the upper threshold Th (T) and vice wr~a (see Fig. 1 j . For simplicit>-, n-e a.ssume t!hat 0 < p, 0.5 < Tri < 1 and - l/p < --CL < -.r’, , n-here Tp = (X/p)
lni( 1 +p)/jl
-11))~ z: = (G + l/p) =p(y/(2X))
-
l/p and xc = (l-l/p) esp(--p/(ZX))+l/p (see F&l(b)). In t hi.5 case, (1 does not affect tlhe svst em dvnamics and we focus on the parametlers (A. pi E 13. “The sj-stem cl!-namics can be reduced into the discret’e tlime map: .(.(17+ 1) = .f(+),y(n).p). y(rl + 1) = j+(n). y(n);p).
i
’ -y((-y.r-;)tf+;). for .rh < -ys .fi*r,,!/j
= (
y ((-y.r
- :-> tf----+
< ,c,,
!1->.
for .q7 < - y-r < .rb ,
where 17is a positive integer, .rn = - ( 1 + l/1,) esp( -y/ (2X)) + l/1;, and ,~‘h= -( 1 + l/p) espj -1)/X) + l/p (see Fig.1 (13)). Fioq b *7- shows a circuitI
V-414
model
of the cell.
In the figure,
3. THE
ICC
NETWORK
V-415
- x,
Xh
0 r(n)
[X(17
,]
Figure 1: A esarrqde of’ the t-d w-t-t map f. clott ccl line shows the t-et urn imp F for cl ( n ) = (p = (0.275.0.125) c $I().)
The 0 5.
O.’ initial
a (0j
LO
Figure 6: Plots of .T( h ) for mrious t-at io ck(0 ). (p = ppo.) The dots indicated by (a), (43) and (c) shows ~(0) of Fig.i( a ). Fig.S(\.‘) and Fig.S(c). respectiwlv. . Sot e again that the uppi~ ai~4 the lower ligut-e, 5 are ohtaitwd ft-m-n differmt
(a)
state ratio
.F( 0 ) .
0 0.1 0 p 0.08 .it2n+2
)
I 5 ic1
chaos 1
t
turo
t
0.04: l;perifkys 0.5
Figure
5: Esa~rqh3 of the return map G’ ant4 at tractors f 01‘ p = p,3 . (a) I-petiociic attractors (0 (0) = 0.6. ) ( t3) L-pc-tiodic atti-acttotx (n (0) = 0.8. ) (c ) CIaotic attmctot.. ( c\ ( 0 ) = 1 .o. ) T11e upp Cl’ and the lo\\-er at t r-act 01‘5 are ohhind fi-cm cMfei~ei~t hitid states of .1.(0).
t
chaosI
:_ 1
initial sbte ratio a ( 0) l *O
diagram from the JC’C’ tletn-wk. Fb3 we 7: Bifurcation 0.275.) The dots correspond to the c4ots in Fig.6.
4. GENER,ALIZATION
V-416
(x =
r u” -
-5!
t,’ -
II
II
Z
. . --u -
*
,
-. --
c
V-417
