patiotemporal Chaos in Four Chaotic Circuits
Coupled by One Resistor
Yoshifurni NISHIO and Akio USHIDA
Dept. of Electrical and Electronic Engineering, Tokushima University
2- 1 Minami- Josanjima, Tokushima, 770 Japan
Tel. $81-886-56-7470
Fax. $81-886-54-9632
Email nishioQee.t okushima-u.ac.jp
ABSTRACTIn this study, four simple autonomous chaotic circuits coupled by one resistor are investigated. B~ carrying out
computer calculations and circuit experiments, it is shownthat our very simple coupled circuit can exhibit spatiotemporal chaos as well as quasi-.synchronizations of chaos inspite of that the number of chaotic subcircuits is only four.

I. INTRODUCTION
Many nonlinear dynamical systems in the various fields
have been clarified to exhibits chaotic oscillations and recently applications of chaos to engineering systems attract
many researchers' attentions, for example: chaos noise
generator, control of chaos, synchronization of chaos, andso on. Among the studies on such applications, synchronization of chaotic systems or signals is extremely interesting [1~N[31,because the chaotic solution is unstable andsmall error of initid values must be expanded w timegoes. As far as we know, such phenomena have been
firstly reported t o be genareted in simple real circuits bya group of Saito [l]. Since Pecora e t al. have investigated such phenomena theoretically [3]. many papers havebeen published until now. Further, secure communication systems using chaos synchronizations [4][5] and coupled chaoptic circuits generating various types of quasisynchronizations [6]-[8] are also proposed.
On the other hand, a network of chaotic onedimensional maps have been investigated earnestly byKaneko [9]-[12]. He has discovered varios kinds of phenomena called as spatiotemporal chaos such as diffusion
and Brownian motion of defect, clustering, spatiotemporal intermittency and so on. Recently, Chua and his colleagues published their papers on spatiotemporal chaos
observed in a chain of coupled Chua's circuits [13][14].
However, they treated only the parameter sets for which
each Chua's circuits generate simple one-periodic attractor. The study of such systems are very important not
only as models for nonlinear systems with many degrees
of freedom, but also for the clarification of biological information processing and for engineering applications.
In this study, we investigate four simple autonomous

0-7803-2570-2/95 $4.00 01995 IEEE

chaotic circuits coupled by one resistor. By carrying out
computer calculations and circuit experiments, we foundthat our very simple coupled circuit can exhibit spatiotemporal chaos as well as quasi-synchronizations of chaos inspite of t h a t the number of chaotic subcircuit.s is onlyfour. We consider that our model would be good modelto clarify mechanism and characteristics of spatiotemporal chaos because it is the simplest real physical system
exhibiting spatiotemporal chaos. Further, we would liket o emphasize that all phenomena introduced in this paper
have been observed from real physical circuit model madeUP easily in the laboratory.

11. CIRCUITMODEL
Circuit model is shown in Fig. 1. In our system four samechaotic circuits are coupled by one resistor. Each chaotic
subcircuit is a symmetric version of the circuit model proInaba et ". [151. It consists Of threePosed
L1i-r
1777-

*m-*/%-+weFig. 1 Circuit model.

1472

Authorized licensed use limited to: Tokushima Univ. Downloaded on March 14,2010 at 06:51:18 EDT from IEEE Xplore. Restrictions apply.

elwnents, one linear negative resistor and one nonlinear resistor, which is realized by connecting some diodes, and is
one of the simplest chaotic circuits. Fig. 2 shows a typicalexaiiiple of chaotic attractors obtained from the uncoupledchaotic subcircuit. In the following circuit experiments,
the values of the inductars and the capacitor in eachchaotic subcircuit are fixed arid those values are measured
as L1 = 204.15mH f 0.073%, Lz = 9.9331nH f 0.030%
and C = 0.03425pF f 0.29%.

For comput,er calculations, in order to consider the difference of real circuit elements, (4) is rewritteu as follows.
=$kP(.z'k+nY k ) - Zk - yZk]=1i k=21;= (1(5)a { P ( X k -k y k ) - z k - f ( 2 / k ) }+ (IC - l ) A W } ( X k + Y k )( k 1 , 2, 3, 4).

In the following computer calculations, we fix the parameter cv as 24.0 and ( 5 ) is calculated by using the RungeK u t t a method with step size At = 0.01.

III. IN A N D OPPOSITE-PHASESQUASI- SY N C € I RO NI Z AT1 0 N

(a)
-'&'I
(b)
IfFig. 2 Typical example of chaotic attractors observed
from the chaotic subcircuit. (a) Computer calculation for a = 24.0 and /3 = 0.295. (b) Circuit experiment fot L1 = 204.15?nH, L2 = 9.933mH, C =0.03425pF and T = 630R. Horizontal: 0.4mA/div.
Vertical: 1 \-/div.

At first, we approximate the i - U characteristics ofthe nonlinear resistor consisting of diodes by the following

1(4P(Xk+NYk) - zk -I',,3,Fig. 3 In and opposite-phases quasi-synchronization.
(a) Computer calculation for /3 = 0.29, y = 0.40 and
Aw = 0.0. (b) Circuit experiment fot T = 5700 and
R = 2.4k0. Horizontal and I'ertical: 0.4niA/div.

(2) is normalized as=:"*By changing the variables and parameters,;. ._ dX.k

Before we treat spatiotemporal chaos, we introdlice two
types of quasi-synchronizations of chaos in this section
and next one.
Fig. 3 shows an example of in and opposite-phases
quasi-synchronizations. In this case, each suhcircuit exhibits chaos as Fig. 2. But, two of four subcircuits arealmost synchronized a t the in-phase and the rest is almost synchronized to the two subcircuits with K phasedifference. Namely, phase difference with respect to thesubcircuit 1 is described as { T . 0, T } for the example inFig. 3. Though we omit other phase states in Fig. 3, there
coexist more two different Dhase states. n a m e h to. K . T Iand { T , T , 0).

Xk

I v . T W O PAIRS O F OPPOSITE-PHASESQUASI-SYNCHRONIZATION(4)

Fig. 4 shows a n example of two pairs of opposite-phases
quasi-synchronizations. In this case, subcircuits 1 and 2
are almost synchronized at the opposite-phase. Also subcircuits 3 and 4 are almost synchronized at the oppositephase. However, a pair of subcircuits 1-2 and the ot,her1473

Authorized licensed use limited to: Tokushima Univ. Downloaded on March 14,2010 at 06:51:18 EDT from IEEE Xplore. Restrictions apply.

pair of 3-4 are independent. ?I-e had coiisideied that
therr coexist more two different phase states. namely(1-3, 2-4) a n d {1-4. 2-3). However actually the combination of t h e decoupiing into two pairs is decided by theslight difference of real circuit elements and other combination states cannot be observed. Namely. this quasisynchronization is based on the asynirnetry of the couplinga n d cannot be generated in the case of conipletely symmetric coupling as A w = 0.0.'&'e consider that this phenomenon is deeply related with the clustering [12]. However we d o not call this phenomenon as spatiotemporalchaos because the spatial pattern is always the same.

Fig. 5 Spatio-temporal chaos:Self-switching of inand opposite-phases quasi-synchronizations. T =1x3630R and R = 1.3kR. Horizontal and Vertical: l.OmA/div.

-1,-3

1,-13

Fig. 4 Two-pairs of opposite-phases quasi-synchronization, (a) Computer calculation for 3 = 0.295,
y = 0.340 and A w = 0.01. ( b ) Circuit experiment
fot T = 580R and R = 850Q. Horizontal and Vertical: 0.4mA/div.

3

3V . SPATI OT E MP o R A L c H AOS
For large region in parameter space. we observed complex
chaotic motion. Namely, we observed that three phasestates of in and opposite-phases quasi-syncliroiiizatioiis
are switched automatically and randomly a s shown in Fig. 6 Spatio-temporal chaos: Self-switching of twopairs of opposite-phases quasi-synchronizations.
Fig. 5. The order of the appearance of three phase statesr = 630R and R = 960. Horizontal and Vertiis truly unpredictable. Further switching period is also
cal: l.Om.4/div.
chaotic, namely a state may be switched t o the next
state instantly and a state may be switched after about afew second. We also observed similar self-switching phex1
nomenon of three phase states of two pairs of oppositephases quasi-synchronizations as shown in Fig. 6. We cancall these phenomena as saptio-temporal chaos because
spatial pattern corresponding to tliree synchronization
states changes chaotically as time goes and it is causedby the local chaotic motion of subcircuits.
In order to investigate such phenomena. we define thePoincar6 section as i l = O where d q / d t > O and plot
the values of xi ( i = l , 2, 3, 4) 011 IC; - n. (n denotes the05oooO
number of iteration of the PoincarC map) plane when the--+ n: iteration
solution hits the Poincar6 section. Fig. 7 shows time seFig. 7 Spatio-temporal chaos: Self-switching of in andries of attractors corresponding to the self-switching of
opposite-phases quasi-synchronizations. p = 0.295,
in and opposite-phases quasi-synchronizations. For examy = 0.10 and Aw = 0.0.
ple, phase state of (. ,. 0) appears in the shaded area.

1474

Authorized licensed use limited to: Tokushima Univ. Downloaded on March 14,2010 at 06:51:18 EDT from IEEE Xplore. Restrictions apply.

[n the figure we can confi mi that three phase states appear
chaotically. Fig. 8 shows time series of attractors corresponding t o the self-switching of two-pairs of oppositeIphases quasi-synchronizations. For example, phase stateo f {1-4, 2-3) appears in the shaded area. In the figure
we can confirrn that three phase states appear chaotically.

sion to large number of circuits case or another coupling
structures case.

X1

n: iteration
Self-switching of twopairs of opposite-phases quasi-synchronizations.
p = 0.295, y = 0.01 and A w = 0.0.
--+Fig. 8 Spatio-temporal chaos:

Because of the limited space, we cannot explain detailed
results on the effect of AIJ. But we confirmed that both ofthe appearing frequency and switching speed are deeply
related with the value of A w . We show one example inFig. 9.
1 1~~~~~~

030000-+ n: iteration
Fig. 9 Effect of the value of Au. /3 = 0.295, 7 = 0.10

J 0.005.
and A ~ =Further we confirmed that intermittency of one subcircuit sometimes causes bi-eakdown of a spatial pattern.
VI. CONCLUDINGREMARKSIn this study, we investigated four simple autonomouschaotic circuits coupled by one resistor. By carrying out
computer calculations and circuit experiments, we foundthat our very simple coupled circuit can exhibit spatiotemporal cliaos as well as two types of quasi-synchronizationsof cliaos in spite of that the nuniber of chaotic subcircuits
is only four. We wolud like t o emphmize again that thecircuit model in this article is the simplest real physicalsystems exhibiting spatiotemporal chaos.
Our future research is theoretical approach to spatiotemporal chaos including statistical s t u d y and exten-REFERENCES
Y. Ohmori, M. Nakagawa and T. Saito, “hlutual Coupling of Oscillators with Chaos and Period Doubling
Bifurcation,” Proc. of ISCAS’86, pp. 61-64, 1986.
L. M. Pecora and T. L. Carroll, ‘(Synchronization inChaotic Systems,” Phys. Rev. Lett.. Vol. 64, No. 8,
pp. 821-824, 1990.
M. J. Qgorzalek, “Taming Chaos - Part I: Synchronization,” IEEE Trans. Circuits Syst. I. Vol. 40. No.
10, pp. 693-699, Oct. 1993.
L. Kocarev, I<. S. Halle, K. Eckert, L. 0. Chua andU. Parlitz, “Experimental Demonstration of Secure
Communications via Chaotic Synchronization,” Int.
J. Bifurcation and Chaos, Vol. 2, No. 3, pp. 709-713,
Sep. 1992.
H. Dedieu, M. P. Kennedy and M. Hasler, ‘iChaos
Shift-Keying: Modulation and Demodulation of a
Chaotic Carrier Using Self-synchronizing Chua’s Circuits,” IEEE Trans. Circuits Syst. 11, Vol. 40, No. 10,
pp. 634-642, Oct. 1993.
Y. Nishio, K. Suzuki, S. Mori and A. Ushida, “Synchronization in Mutually Coupled Chaotic Circuits,”
Proc. of ECCTD’93, pp. 637-642, Sep. 1993.
Y. Nishio and A. Ushida, “Synchronization Phenomena in Chaotic Oscillators Coupled as a Ring,” Proc.
of NOLTA’93, pp. 613-616, Dec. 1993.
Y. Nishio and A. Ushida, “Multimode Chaos in Two
Coupled Chaotic Oscillators with Hard Nonlinearities,” Proc. ISCAS’94, Vol. 6, pp. 109-112, May 1994.
K. Kaneko, “Period-Doubling of Kink-Antikink Patterns, Quasiperiodicity in Antiferro-Like Structures
and Spatial Intermittency in Coupled Logistic Lattice,” Progress of Theoretical Physics, Vol. 72, No. 3,
pp. 480-486, Sep. 1984.
I<. Kaneko, “Spatiotemporal Intermittency in Coupled Map Lattices,” Progress of Theoretical Physics.
Vol. 74, No. 5, pp. 1033-1044, Nov. 1985.
I<. Kaneko, “Pattern Dynamics in SpatiotemporalChaos,” Physica D, Vol. 34, pp. 1-41, 1989.
I<. Kaneko, “Clustering, Coding, Switching, Hierarchical Ordering, and Control in a Network of ChaoticElements,” Physica D, Vol. 41, pp. 137-172. 1990.
V. I. Nekorkin and L. 0.Chua, “Spatial Disorder andWave Fronts in a Chain of Coupled Chua’s Circuits,”
Int. J. Bifurcation and Chaos, Vol. 3, No. 5, pp. 12811291, Oct. 1993.
A. L. Zheleznyak and L. 0. Chua, “Coexistence ofLow- and High-Dimensional Spatiotemporal Chaos ina Chain of Dissipatively Coupled Chua’s Circuits,“Int. J. Bifurcation and Chaos, Vol. 4,No. 3, pp. 639674, June 1994.
N.Inaba and S. Mori, “Chaotic Phenomena in Circuits with a Linear Negative Resistance and an IdealDiode,” Proc. of MWSCAS’88, pp. 211-214, Aug.
1988.

1475

Authorized licensed use limited to: Tokushima Univ. Downloaded on March 14,2010 at 06:51:18 EDT from IEEE Xplore. Restrictions apply.