PATTERNCONTROLPOSSIBLE

IN A COUPLEDAPPLICATIONCHAOTICNETWORKANDITS

IN COMMUNICATIONS
Akio USHIDA

Yoshifumi NISHIO

Tokushima University, 2-1, Minami-Josanjima, Tokushima 770, JAPANnishio@ee.tokushima-u. ac.jp

ABSTRACT

article. The 11 spatial patterns are described as follows.

In this study a controlling method of generated patterns ina simple coupled chaotic network is discussed. By adding
an connection between two subcircuits, the number of generating patterns is restricted. Further, synchronization phenomenon of simple transmitter-receiver systems consistingof two chaotic networks is also examined.

1.

INTRODUCTION

In our previous study [1][2], we have.proposed a simple coupled chaotic network whose structural element is chaotic
autonomous circuit. By computer simulations, we invest igated the generation of interesting irregular self-switching
phenomenon of a large number of spatial patterns characterized by both of attractors’ shapes and phsse diferenceof quasi-synchronization of chaos [3]. We found that 5632
dMerent spatial patterns appear for the size of 3 by 3 andthat switching speed could be controlled by tuning circuit
parameters.
In order to apply our chaotic network to engineering systems, it will be first step to control generating spatial patterns. If we can restrict the number of generating patterns,
it will correspond to embed information in the network.
Further, if the network can produce irregular self-switching
phenomenon of the restricted number of patterns, it willcorrespond to dynamical search of memorized information.
In this study, we investigate irregular self-switching phenomenon of spatial pat tems generated iu a coupled chaotic
network with an additional connection between two chaotic
subcircuits.
By computer simulations we found that wecan successfully restrict the number of generating spatialpatterns. It is also found that the number of restrained
patterns are changed by the location of the additional connection. Further, we consider a transmitter-receiver system consisting of two chaotic networks. We will discuss
about synchronization of irregular self-switching of spatial
patterns.
2.

“’’’(%)(%)

CIRCUIT

where different characters represent independent phase and~ means the opposite phase of A. These patterns are generated by reason of the underlying physical mechanism; thenetwork tends to minimize the currents through the coupling resistors &.
In this study we try to add a connection between two
chaotic subcircuits for the purpose of the restriction of thenumber of generating patterns. An example of coupledchaotic networks with an additional connection is shownin Fig. 1. In the case, a connection is added from sub circuit 1 to 6. In the followings we use simple drawings as,
IPI
L

3.

to represent such additional connections.
JRESTRICTIONOF GENERATINGPATTERNS

There exist many combinations of additional connections
even for the network with the size of 3 by 3. Then, weconsider the case that the number of connections is one.
Further, we ignore some connections which become equivalent after rotation of the network.
After the above restrictions, we can count up all connec-

MODELFl2

In the origjnal chaotic network without additional connection [1][2] can generate 11 different spatial patterns with respect to the phase states. (In the Refs. four patterns could
not be found. ) Actually, there coexist 29=512 patterns withrespect to the attractors’ shapes. Hence, total number ofthe different spatial patterns is 512x 11=5632. However, inorder to concentrate on the control of the phase states, weignore the switching of attractors’ shapes throughout thistions. First connection is~

spatial pattern, modified TypeE :In this case, only one

AAB
AAB() AAB———, is observed.

Fig. 2(a) shows the time series of the Poincar6 map obtained
from the network with the above connection. From the figure we can see that no self-switchiigs occur.

Copyright 1997 IEEE1037

P!I:(W%)E3x,-. II

$L,I

I

X3+775++.

JhP—————==
I

X9Iv’--J.v~vtiviA-’A..jo4(a)I&iteration

7JFig.

1 Coupled chaotic network with a connection fromsubcircuit 1 to 6.

Other connections
are listed as

and the corresponding

spatial patterns.W

I

X9..-o–(b)xl——X2+.-—;Ob1.
———

iteration——1X3p-) (%).

Fig.

2 Time series of Poincar6 map. (a) Additional
connection from 1 to 2. (b) Additional connection
from 1 to 3. (c) Additional connection from 1 to 5.

Copyright 1997 IEEE1038

this case each subnetwork can produce 5 different spatial
patterns. Time series of Poincar6 map is shown in Fig. 6.
From the figure we can see that Z1, 23, Z; and Zj ares}mchronized. However, self-switching of spatial patterns
rue completely independent.
In order to develop a new
type of communication systems using chaos, we wish to exploit irregulzu self-switching phenomenon generated in coupled chaotic networks. Thus, we consider to trsmsmit twoFigs. 2(b) and (c) show the time series of the Poincar6
map obtained from the network with the connections

1&chaotic signals to the receiver side as @\@Pl=dFlrespectively‘romthefisrewe
IILI

can see that only some spatial pat tems descrived above generate. By conbining some connections it would be possible
to constmct networks in which only some desired spatial
patterns cars be appear.
4.

TRANSMITTER-RECEIVER‘s(%))
Wor(%))theSYSTEM

In this section we consider simple transmitter-receiver systems consisting of two chaotic networks. The simplest example is shown in Fig. 3. This system can be represented as

F1w

the receiver side must be also one of these.

While when

the phase state of the transmitter side isABAABA() ABAABA_——.4 B A() BAB

From this system we can observe chaos

synchronizations shown in Fig. 4. Time series of Poincar6
map is also shown in Fig. 5. Because each subnet work withthis connection can produce only one spatial pattern, thisresult is not so interesting. Howe\-er, this results indicate
the possibility of synchronization of coupled chaotic networks.
t&@InkA
7L,/++,L,+’4Lor, the phese state of the receiver side must be alsov/———

one of the two. Namely, spatial patterns of two subnetworks
are not the very same. However, some of self-stitchings at
the transmitter side will transferred to the receiver side andonly some chaotic subcircuits are synchronized by trammitted signals. Detailed results will be introduced in the presentation. Now, we do not know whether we can develop
novel communication systems using our coupled chaotic network whkh can overcome conventional chaos communica.
tion systems. However, we hope that our system would contribute to develop novel types of communication systems.

~~Next, we consider the system of~~In this case, when the phase state of the transmitter side

*7+“,.

x,~ J-

4L,

@

4L,c

2L,

2L,m

* ~,,
4,,
Rm&

4L ,4LImc?)*A-+***RECEIVER

TRANSMITTER
Fig. 3 Transmitter-receiverchaotic networks.

systemusing

Copyright 1997 IEEE

coupled1039

h-HE!El
E!!liilF!HFEil
EEB31zl
TRANSMITTERFig.

RECEIVER

4 Chaos synchronization
system.
I-r.
I,-’in transmitter-receiver

r-r.

1.
.
.

I

x9t.
.

I,.66Fig.

+itemtion

5 Time series of Poincar6 map.
Z1+Z2,
Z: +X5 andxl -+z~.
5.

Connections

arex;oFig.

6Time series of Poincar6 map.
andxl +x; .-~.-+ z3, z\+zj

CONCLUSIONS

Connections

areREFERENCES

In this study a controlling method of generated patterns in
a simple coupled chaotic network has been discussed. By
adding an connection between two subcircuits, the numberof generating patterns was been successfully restricted. Further, synchronization phenomenon of simple transmitterreceiver systems consisting of two chaotic networks was also
examined.
Our future research is to find attractive applications USing synchronization of two coupled chaotic networks and toinvestigate larger size of coupled chaotic networks.

[1] Y. Nishio and A. Ushida, ‘Spatio-Temporrd Chaos inIEEE inns.
On
Simple Coupled Chaotic Circuits:
Cimds Syd. 1, vol. 42, no. 10, pp. 678-686, Oct. 1995.
[2] Y. Nishio and A. Ushida, “On Synchronization Phenomena in Coupled Chaotic Circuits Networks,” Pruc.
ISCAS’96, vol. 3, pp. 92-95, May 1996.
[3] Y. Nishio and A, Ushida, “Quasi-Synchronization Phenomena in Chaotic Circuits Coupled by One Resistor,”
IEEE Tbans. on Circuits Syst. I, vol. 43, no. 6, pp. 491496, Jun. 1996.

Copyright 1997 IEEE1040