ON SYNCHRONIZATIONPHENOMENA

IN COUPLEDCHAOTIC CIRCUITS NETWORKS
Yoshifumi NISHIO
Tokushima

University,

Akio USHIDA

2-1, Minami-Josanjima,
nishio@ee.t okushima-u.ac.

Tokushimajp

2.

ABSTRACTIn this study spatial patterns of synchronization
states observed from two types of chaotic circuits networks are investigated.
For the parametervalue at which chaotic subcircuits generate two asymmetricattractors,
number of synchronization
states increases and more complicatedirregular self-switching
phenomenon
is observed.

1.

INTRODUCTION

In our previous study [1], we have proposed
two types ofcoupled oscillators networks and have investigated
the generation of various phase pat t ems. Because the coupled oscillators networks have only local interconnection
and generat e various synchronization
states, it has possibility y toprocessing system suchbe utilized as a parallel informationas cellular neural networks.
However, the stability of thephase patterns observed from the Lattice type network is
weak and some generating patterns sometimes vanishes bysmall error of real circuit elements.
Further, the Honeycomb type network can generate only two phase states inspite of network size.

770, JAPANCIRCUIT

MODELThe two types of chaotic circuits networks [2] [3] are shownin Fig. 1. These networks have local connection
structuresuch that four or three sub circuits around one coupling resistor R are coupled.
Though we treat the networks withthe minimum
size in this article, it is easy to extend totwo-dimensional
arrays.
Each chaotic sub circuit can produce chaotic attractors asshown in Fig. 2, In our previous study we treated the casethat each sub circuit generates symmetricchaotic attract orin Fig. 2(a), In this study we consider the case that each
sub circuit generates asymmetricchaotic attractors
in Fig.
2(b).
‘At first, we approximate e the v - i characteristicsof thediodes by the function ‘Ud =9 @.
The circuit equation
for the Lattice type network is described by 42nd order ordinary differential equation including 9 nonlinear functions,while for the Honeycomb
type network by 30th order differential equation including 6 nonlinear functions.
In thefollowings, The differential equations are calculated
by using Runge-Kuttamethod.
Because of the limitation of the page space, we show only
the normalizationof variables and parameters.

——

drlxk’ ak‘a / z’yk’ ‘k ‘azk’

On the other hand, we also proposed two types of chaotic
circuits networks whose structuralelement is chaotic autonomous
circuit [2] [3]. These networks were obtained byreplacing oscillatorsin the above-mentioned
coupled oscillators networks
by chaotic circuits.
By computer
simulations, we investigated
the generationof interesting
selfswitching phenomenonof spatial patterns characterizedby
phase-differenceof quasi-synchronizationof chaos [4]. However, we considered
only the case that each sub circuit generates symmetricchaotic attractor.
Namely, we have notinvestigatedwhat types of synchronization
phenomena
areof each sub circuit changes.
observed as control parameter
Further, because only phase difference was available freedom, total number of spatial pat t ems were limited; especially, Honeycomb
type network could produce
only twopatterns in spite of network size.

mationIn this study we pay our attentions to the case that eachchaotic sub circuit in two types of chaotic circuits networks
generates chaotic attractors
with asymmetricshape,
Because of the symmetryof the sub circuit, two asymmetricattractors
located symmetricallywith respect to the origin
coexist in each sub circuit. Hence, each sub circuit can taketwo different states. This means that the network have twodifferent freedoms;
attractors’
shape and phase difference,and can produce more complicatedirregular self-switching
phenomena.

(:::)
whereU and L means chaotic attractors in Figs.
the notations
2(a) and 2(b), respectively.
Simply said, there coexist
29 = 512 different synchronizedpatterns with respect tothe attractors’
shape.
AlthoughFig.
3 cannot show the informationon thephase difference,
the two pairs of opposite
phases quasisynchronizations[2] [4] generates around each coupling resistor. The informationof the phase difference is represented
by the notation in [2] and the number of different spatial

Ik‘a“i‘=’;‘=’efi‘f=%(where3.

‘1)a=s{~),SYNCHRONIZATIONPHENOMENA

Figure 3 shows an example of synchronizedpatterns observed from Lattice type network. We represent the inforof the attractors’

shapeas

Copyright (C) 1996 IEEE. All Rights Reserved.

.—.,

$L,

t1?

Lxe2L

(a)

II-1

II-1

I(b)

Fig. 2 Chaotic
attractors
observed feom each subcir- cuit.
(a) Symmetricattractor(a = 24.0 and ~ =0.300). (b) Coexistenceof two asymmetricattractor(a=24.0 and /?= 0.283).

3L,

At% X1-Z1$.ttiX2-L?

Att: X3-73

1Re1?.

oIk‘k(.

~kc!

~2–r \——
Fig.

—3 An example of synchronizedpatterns observed
from Lattice type network (a = 24.0, ~ = 0.280 andy = 0.01).

patterns

is seven as follows.

i+Fig.

1

Coupledoscillators

networks.

Copyright (C) 1996 IEEE. All Rights Reserved.

-.—.
/_“..

. . ----

where different(mcharacters

represent‘2)
independent

phase and~ means the opposite phase of A.
Because attractors’
shape and phase difference seem to be
independent,
it is consideredthat there coexist 512 x 7 =3584 spatial patterns.
On the other hand, Fig.
4 shows an example of synchronized patterns observed from Honeycomb
type network.
The informationof the attractors’
shape is represented as(ULL]Uu.
U(“)

Simplysaid, there coexist

the values of ~i (i=l,2, . . .. 9) on ~i - n (n denotes thenumber of iteration of the Poincar6 map) plane when thesolution hits the Poincar& section.
Figure 5 shows the time series of the Poincar6 map obtained from computer simulations where the position of subcircuits correspondingto z] –zg is as follows.

(i::)

2C = 64 dif-ferent synchronizedpatterns with respect to the attractors’
shape.
The Honeycomb
type network produce only the following
two different phase states in spite of the network size.

‘ithphasestatesof(!!

~Oadattrac’Ors’shape~)

0$:07[0;:07(3)where O, 27r/3 or 47r/3 means the phase difference with respect to the’ sub circuit located at ~he upper left corner.
Hence, it is consideredthat there coexist 64 x 2 = 128
spatial patterns.

iappear in the shaded area. Also spatial‘f(~patterns[‘4)From the figure we can see the self-switching
phenomenaof various spatial patterns.
For example, a spatial pattern

withphasestatesof(!
fi)appemsin
the left side of the figure. Further, switching of attractors’
shape can be observed at many points.
Figure 6 shows the result obtained for different parameter sets. Switching speed of phase states becomes slower
and switching
speed of attractors’
shape becomes
faster.
Namely, it is possible to change the switching speed of phasestat es and attractors’
shape by varying parametersof coupling and negative resistance.
For Honeycomb
type network, we investigated time seriesof Poincar& map. However, we cannot observe se~-switching
between two different phase states.
The phase difference
seems to be decided by given initial conditions.
Hence, wecan observe self-switchingof 64 spatial patterns with different attract ors’ shape.
5.

CONCLUSIONSIn this study we have investigatedspatial patterns of synchronization
states observed from two types of chaotic circuits networks.
For the parameter value at which chaotic
sub circuits generate two asymmetricattractors,
number ofsynchronization
states increases and more complicatedirregular self-switching
phenomenon
are observed.
For the purpose of engineering applicationsof this network, it will be first step to control generating spatial patterns. Namely, the situation that only some of generating
patterns remains stable and others are unstable is desirable.
We have tried to add one connections
between two subcircuits, it was partly achieved.
The detail will be shown in
the presentation.

——~-”-REFERENCES[1]Y. Nishio,Fig.

4 An example of synchronizedpatterns observed
from Honeycomb
type network (a = 24.0, ~ = 0.280,~ = 0.05 and 6 = 0.01).

4.

SELF-SWITCHINGPATTERNSOFSPATIAL

In order to investigate self-switching
phenomenonof spatial
patt ems observed from the Lattice type network, we define
the Poincar& section as Z1 = O where dzl /dt >0 and plot

S. Mori and A. Ushida, “On Coupled Oscillators Networks; for the Cellular Neural Network,”
Proc. ISCAS ‘9$, pp. 2327-2330,
May 1993.
[2] Y. Nishio and A. Ushida, “Spatio-Temporal
Chaos inIEEE Trans. onSimple Coupled
Chaotic
Circuits,”
Circuits Syst. I, vol. 42, no. 10, pp. 678–686, Oct. 1995.
of Spatial[3] Y. Nishio and A. Ushida, “Self-Stitchings
Proc.
Patterns in a Simple Chaotic Circuits Network,”
NOLTA ’95, pp. 961-964, Dec. 1995.
Phe[4] Y. Nishio and A. Ushida, “Quasi-Synchronization
nomena in Chaotic Circuits Coupled by One Resistor,”
IEEE !lYans. on Circuits Sy9t. 1, (will appem).

Copyright (C) 1996 IEEE. All Rights Reserved.

,/”,>,”-(. 1,’”1‘i/1

Ao+Fig.

50000
n: iteration

5 Time series of Poincar6 map:
Self-switchingofvarious spat ial patterns observed from Lattice typenetwork (a = 24.0, @ = 0.280 and T = 0.30).

1500000+Fig.

n: it erat ion

6 Time series of Poincar6 map:
Self-switchingofvarious spatial patterns observed from Lattice typenetwork (a = 24.0, ~ = 0.2805 and -y = 0.10).

Copyright (C) 1996 IEEE. All Rights Reserved.

-.
,,~