ANALYSIS

OF CHAOTICTWO-DIMENSIONAL

WANDERING
COUPLEDCHAOTIC

Yoshifumi NISHIO
Dept.

OF PHASE

PATTERNSCIRCUITSIN A

NETWORKand Akio USHIDA

of Electrical and Electronic Engineering, Tokushima University
2-l Minami- Josanjima, Tokushima, 770-8506 Japan
Email: nishiobe. tokushima-u.
ac . ix>

1. INTRODUCTION
Studies on spatio-temporalphenomena
observed fromcoupled chaotic networks,
namely coupled systems ofmany chaotic cells, are classified into two categories;
namely discrete time systems and continuous time sysFor discrete time systems Kaneko’s
coupledtems.
map lattice is the most interesting
and well-studied
He discovered
various nonlinear
spatiosystem [l].
temporal chaotic phenomena such as clustering, Brownian motion of defect and so on. -41~0 -4ihara’s chaos
neural network is the most importantchaotic networkfrom an engineering
point of view [2]. His studv indicated new possibilityof engineering
applicationsofchaotic networks, namely dynamicalsearch of patterns
embedded in neural networks utilizing
chaotic wandering. Further;applicationof chaos neural network tooptimization
problems is widely studied. However, forcontinuous
time systems there have been few studies
on spatial patterns observed after vanishing the effectof initial patterns.
Therefore,
in order to fill the gap
between the studies of discrete-time
mathematicalabstract and studies of continuous-timereal physical systems, it is important
to investigate
simple continuoustime coupled chaotic circuits networks generating clustering or pattern switching phenomenon.
We have proposed a continuous-time
coupled chaotic circuits
network[3]-[6] and have reported generating spatial patterns,
irregular
self-switching
phenomenon of spatial patterns,
patterncontrol and soon. The importantfeature of our chaotic network is its
Namely adjacent four chaotic circouplmgstructure.
cuits are coupled by one resistor. Because such a coupling exhibited
quasi-synchronization
with phase difference [7], various spatial patterns could be generated
in the chaotic network.
Therefore,
the network would
be a good model to make clear physicalmechanismof spatio-temporalchaotic phenomena
in continuoustime systems. However, chaotic wandering phenomenaof phase patterns observed from a relatively
larger sizeof the network has not been investigated.
In this study, we investigate
chaotic wandering phenomena of spatial patterns characterized
bv phase differences in the coupled chaotic circuits network with
10x10 size. Snapshots of the Poincare map tell how

0-7803-5474-5/99/$10.00 1999 IEEE

switchings of phase states start and spread over the network. Further, we can show that an independent
angular variables derived from higher-dimensional
Poincaremap can be effectively
utilized in order to check theswitching of phase states around coupling resistors.
2.

CHAOTICCIRCUITSNETWORK

Fig. 1 shows coupled chaotic circuits network. We approximate
the si - ,Wcharacteristicsof the diodes by atwo-segment
piecewise-linear
function as~d(i~,~~) = 0.5 (jrd &By changing+ E - 1‘rd im,n - E 1) .

the variables

and parameters,n!=R&

6=rd/$

the normalized
circuit equations
Nr x N2 size are described asn=wm,ndYm 1n-=a{P(xm,ndzm,n--

d7
d7

dwit2

3ndwm

bdT

Xm,7x8(dr+X n2,nU)k:n+(2)of the network

Wh9n+ Ym,n)-Ym,71)Zm,n+qLl.f-2z,1n9-nf (Ym,n)}Ynz,n+--^I(UJ&n+‘w&, lin-1 + W&-l :n + u’$,-l 9n,- i)>n--41 x m,n. + Ym,n>+wkz

V-422+xm,n7n+(1)IwE2-1

t n+lZm,n-r(W&+ wZ2-l3n) ’n+lwith

dwk,n-d7-

ddw m,ndT

--

dwt2
1--

9ndr

PC Xm,n+ Ym,n)+wR+l9n-l-zm,n+ wiia,nPCXm,n + Ym,n)-+Wk+l,n+ wh9n+l

PC xm,n+ Ym,n)+wfn+l,nAll of the results in this study are obtained by calculating Eq. (3) using the Runge-Kutta
method with step
size h = 0.005. In the following computer calculations
we fix the parameters
a! = 7.0, ,0 = 0.13 and 6 = 50.0
and vary the coupling parameter y as a control parameter.

Y(Wh+,,n+ wiz ?n-1)
Zm,n--- Y(w&+l,n+l.

+ wkzm,n+ wkt 9n+l-, n)Y(w&+l,n+l+ w& 3n>(3)( m=l,2, 3l* Nl andn=l,2, 3, --N2)

wheref (Ym,n)=dw6,n=wO,n6Ym,n+l-Ifiym,n-112b=wm,OI

0d=wm,O=w&l+l,n=w>l+l,n== 0 and the value of y must be= wh,N2+1
neglected for the calculationof the inductors
on theedge.

wk,N2+lhc

nnnFig.

2An example
the chaotic(7=0.30).

3.

of phase patterns
observed fromcircuits
networkwith 10~ 10 size

SPATIALPATTERNS

Fig. 2 shows an example of frozen spatial patterns observed from the network with 10x 10 size. In Fig. 2
all horizontal
axes are x1,1 and vertical axis of the attractor located (mp) is xm n. Hence, if the attractor
located (mp) is almost on 4i” line, xm,n is almost synchronized to x1,1. 7Vhile, if the attractor located (m,n)
is almost on -45’line, xm,n is almost synchronizedto x1)1 with 7r phase difference.
If the attractor
is almost on a circle, the phase sift is k90°. In Fig. 2 wecan see that four phase synchronizations
around coupling resistors can generate spatial patterns. Under theassumptionthat four-phase synchronizations
are generated around all coupling resistors, we can prove thatthe number of phase patterns appearing in Nl xNZ network is 6 (21v1-' + 2Nz-2 - 1). This means that 10x10
network can generate 3; 066 different frozen spatial patterns and that 20x20 network generates 3,143,722 patterns.
4.

Fig.

1 (a) Chaotic
tionaroundcircuits
network.
a cell at position(b) Magnifica(m, n).

CHAOTICWANDERING

Further, by tuning coupling parameter, we can observe
chaotic wandering
phenomenonof phase patterns
inthe network.
In order to investigate
how switchingsof phase states start and spread over the network, wedefine the Poincare section as zl,l = 0 and xl 91 < 0.

V-423

Fig. 3 shows the projectionsof the Poincare map onto
to the result( Xm,n - zrn,n ) planes, which corresponds
in Fig. 2. Because the chaotic attractor
observed fromeach subcircuit is strongly constrained
onto the (x - z)plane when the diode is off, this figure almost corresponds to the phase differences between the subcircuit
located (&I) and the others.

eluded in frozen patterns.
Fig. 4 shows an example
of such frozen patterns.
Four-phase synchronizations
are observed around coupling resistors indicated as l ,while in- and opposite-phases
synchronizations
are observed around coupling resistors indicatedas CL ForIJiII(a >Fig.

3 Poincarb

map of the phase patternin Fig. 2(y=o.30).
Ii

I I
Fig.

II I

II I II Y II I

I

ii 1 III

I_ I

il I fl 4 I

4 Phase Pattern
includingin- and oppositephases synchronizations
and four-phasesynchronizations(y = 0.10).

By computer simulations
we found two critical coupling parametervalues 7~1 and 7~2 at which observed phenomenachange qualitatively.
Namely, forsynchronizaY < Xl E 0.15, in- and opposite-phases
tions [3] as well as four-phase synchronizations
are in-0CFig.

V-424

5 Chaotic wanderingof phase patterns.
6:50:.
( >=(c)6,300
n = 6,500- 6?400.
- 6y600.
(b) n = 6,400 -this parameter
range spatial patterns
includingonly
four-phase synchronizations
as Fig. 3 are also stable.
Hence, if we give random initial conditions for the network, phase states settle down to one of a number ofstable frozen patterns.
For yc1 < y < 7~2 2210.38, inand opposite-phases
synchronizations
become unstable. If we give initial conditions corresponding
to Fig. 4,
the initial spatial pattern changes and will settles down
to one of 6(2Nl-2+2Nz2 - 1) spatial patterns after unpredicted chaotic wandering.
For 7~2 < y, four-phasesynchronizations
also become unstable and no stablespatial patterns exist. Different spatial patterns appear
but they disappear after a while. For relativelylarger
value of y, well-organizedfrozen patternsare rarely
observed. In this case we can always find some partlydisturbed
areas in the network.
Though the area may
be well-organized
after a while, another area will starts
to stir. Snapshots of the Poincare map can show how
switchingsof phase states start and spread over theoriginatedat (1,B)
network.
Fig. 5 shows a switching
generates turbulencealong the edge of the network.
Finally, in order to get informationof switchings oflocal phase pat terns, we define the following independent variables from the discrete data of xm 7n and zm >non the Poincare map.
tan-’zm,n(n >Xm,n, (-n _>. . . Xm,n(n)2m,n(n >Xm,n(n >. . .

Xm,n(n)>0<072-(5>By using these dependent
angular vari ables: we cancount the number of switching
generat ed around all
coupling resistors.
In Fig. 6 the depth of gray at a
position denotes switchingnumber during 1OjOOO iterations of the Poincare map around the coupling resistor
at the position.
These data are utilized for statisticalanalysis of the chaotic wandering
phenomenon.

(b)
Fig.

5.

ACKNOWLEDGMENT
This work was partly supportedby the ResearchGrants for Incentive Research from the Yazaki LMemorial Foundationfor Science and Technology.
REFERENCES
K. Kaneko(ed.), 5% eolr y and Applicationsof Coupled
Map Lattices, John Wiley & Sons, Chichester,
1993.

6 Switchingwork.

CONCLUSIONSIn t his study, we have investigatedchaotic wanderof spatial patterns
characterized
by1% phenomena
phase differences in a-coupled chaotic circuits network.
We showed that independent
angular variables derived
from higher-dimensional
Poincare map could be effectively utilized to check the switchings of phase states.

[I]79.

numberin coupledchaotic(a) y = 0.39. (b) y = 0.40.

[2] K. Aiharaet al., “Chaotic
Neural Networks,”
Lett. A, vol. 144, no. 6&7, pp. 333-340, 1990.
[3] Y. Nishio
et al., “Spatio-Temporal
CoupledChaoticCircuits,”
IEEE42, no. 10, pp. 678-686, 1995.

net-Phys.

Chaos in Simple
Trans. CAS-1: vol.

[4] Y. Nishio et al., “On Synchronization
Phenomena
inCoupled Chaotic Circuits Networks,” , Proc. ISCASW,
vol. 3, pp. 92-95, 1996.
[5] Y. Nishio et al., “PatternControl in a Coupled Chaotic
yetwork
and its Possible Applicationin Communications ,” Proc. LSCAS’97, vol. 2? pp. 1037-1040,
1997.
[6] Y.
Nishioet al..
“OnPhase
Two-Dimensional
CoupledChaotic
NOLTA’98,
vol. 1, pp. 31-34, 1998.

PatternNetwork,”

inaPTOC.

[7] Y. Nishio
et al., “ Quasi-Synchronization
Phenomena
in Chaotic
Circuits
Coupled by One Resistor,”
IEEETrans. CAS-I, vol. 43, no. 6, pp. 491-496, 1996.

V-425