2008 IEEE Workshop on
Nonlinear Circuit Networks, Tokushima, Dec 5-6, 2008

Coexistence of Many Different
Synchronization States in Coupled Chaotic Circuits
Yumiko Uchitani and Yoshifumi Nishio
Department of Electrical and Electronic Engineering, Tokushima University
Email: {uchitani, nishio}@ee.tokushima-u.ac.jp

I. I NTRODUCTION
Synchronization phenomena in complex systems are very
good models to describe various higher-dimensional nonlinear
phenomena in the field of natural science. Studies on synchronization phenomena of coupled chaotic circuits are extensively
carried out in various fields [1][2]. We consider that it isvery important to investigate the phenomena related withchaos synchronization to realize future engineering application
utilizing chaos.
In our past studies, two simple chaotic circuits cross-coupled
by inductors are investigated. As a result, we could observe
interesting state transition phenomena [3][4]. In particular,
we noticed that small oscillations between transitions frompositive region to negative region tend to be synchronized inanti-phase in spite of the synchronization of the transitions.
In this study, we investigate coexistence of many differentsynchronization states. The computer simulation results areverified by real circuit experiments.
II. C IRCUIT M ODELThe normalized circuit equations are given as follows [3][4].
ẋ1 = z1





ẋ2 = z2ẏ1 = α{γy1 − w1 − βf (y1 − z1 )} ẏ2 = α{γy2 − w2 − βf (y2 − z2 )}(1)
 ż1 = β f (y1 − z1 ) + w2 − x1 ż2 = β f (y2 − z2 ) + w1 − x2ẇ1 = δ(y1 − z2 )ẇ2 = δ(y2 − z1 )where f are the nonlinear functions corresponding to the v −i characteristics of the nonlinear resistors consisting of thediodes and are assumed to be described by the following 3segment piecewise-linear functions:(yk − zk > 1)
 yk − z k − 10(|yk − zk | ≤ 1)
(2)
f (yk − zk ) =
yk − z k + 1(yk − zk < −1)
III. I NVESTIGATION OF SMALL OSCILLATIONSIn this study, we pay our attentions on the coexistence ofmany different synchronization states. We can observe manydifferent synchronization states.
Figure 2 shows some examples of different synchronization
states obtained for the same parameter values. Figure 3 shows
two examples of the different synchronization states observed
from real circuit experiments.
Figure 5 shows two examples of the different synchronization states observed from real circuit experiments.
IV. C ONCLUSIONSIn this study, we have investigated coexistence of manydifferent synchronization states.from simple chaotic circuits
cross-coupled by inductors. By computer simulations andcircuit experiments, we confirmed that there are many different
synchronization states coexist.
R EFERENCES
Fig. 1.

Circuit model.

Figure 1 shows the circuit model [3][4]. In the circuit,
two Shinriki-Mori chaotic circuits [5][6] are cross-coupled via
inductors L2 .

[1] G. Abramson,V.M. Kenkre and A.R. Bishop, “Analytic Solutions
for Nonlinear Waves in Coupled Reacting Systems,” Physica A,
vol. 305, no. 3-4, pp. 427-436, 2002.
[2] I. Belykh, M. Hasler, M. Lauret and H. Nijmeijer, “Synchronization and Graph Topology,” Int. J. Bifurcation and Chaos,vol. 15, no. 11, pp. 3423-3433, 2005.

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2.5

2.5

2.0

y1

-2.0
2.0

0
y20

0

2.5

y10

2.5

-2.5

0

2.5

0

2.5

2.5

0
y2

τ

(b)

-2.5

500

0
-2.0
2.0

0(a)

τ 500

2.0

y1

0

-2.0
0

0

0
-2.0

500

500

0

y20

-2.0
2.0

y2

τ

-2.0
2.0

0

τ

0

2.0

0

2.0

0

2.5

y10

y1

0

-2.0

2.5

-2.0

2.5

-2.5

-2.5

500

-2.0
2.0

y2-2.5

τ

2.0

00

y20

2.5

0-2.0
2.00-2.0-2.5

2.0y1002.5(a)0-2.00(b)τ500

Fig. 2. Some examples of different synchronization states. (computer simulation results). α = 2.0, β = 4.0, γ = 0.1, and δ = 0.0014.
(a) Attractor on y1 − y2 plane. (b) Timewaveform.

(1)

(a)

(b)

(2)

(a)

(b)

Fig. 3. Two examples of different synchronization states. (circuit experimental results). L 1 = 10.56mH, L2 = 1.28H, C1 =33.3nF, C2 =49.5nF

and g=515mS. (a) Attractor on v11 − v12 plane. Horizontal and vertical: 5 V/div. (b) Time waveform v11 and v21 . Horizontal 0.5 ms/div
and vertical: 5 V/div.

[3] Y. Uchitani, R. Imabayashi and Y. Nishio, “State Transition
Phenomenon in Cross-Coupled Chaotic Circuits,” Proc. ofNOLTA’07, pp. 397-400, Sep. 2007.
[4] Y. Uchitani and Y. Nishio, “Investigation of State Transition Phenomena in Cross-Coupled Chaotic Circuits,” Proc. of ISCAS’08,
pp. 2394-2397, May. 2008.
[5] M. Shinriki, M. Yamamoto and S. Mori, “Multimode Oscillations in a Modified van der Pol Oscillator Containing a Positive

Nonlinear Conductance,” Proc. of IEEE, vol. 69, pp. 394-395,
1981.
[6] N. Inaba, T. Saito and S. Mori, “Chaotic Phenomena in a Circuitwith a Negative Resistance and an Ideal Switch of Diodes,”Trans. of IEICE, vol. E70, no. 8, pp. 744-754, 1987.

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