2007 International Symposium on Nonlinear Theory and itsApplications
NOLTA'07, Vancouver, Canada, September 16-19, 2007

State Transition Phenomenon in Cross-Coupled Chaotic Circuits
Yumiko Uchitani, Ryo Imabayashi and Yoshifumi Nishio
Department of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, 770-8506 JAPAN
Email: {uchitani, bayashi, nishio}@ee.tokushima-u.ac.jp
Abstract—Studies on chaos synchronization incoupled chaotic circuits are extensively carried out invarious fields. In this study, two simple chaotic circuits cross-coupled by inductors are investigated. Interesting state transition phenomenon around chaos
synchronization is observed by computer simulations
and circuit experiments.

1. Introduction
Synchronization phenomena in complex systems
are very good models to describe various higherdimensional nonlinear phenomena in the field of natural science. Studies on synchronization phenomenaof coupled chaotic circuits are extensively carried out
in various fields [1]-[10]. We consider that it is very
important to investigate the phenomena related withchaos synchronization to realize future engineering application utilizing chaos.
In this study, two Shinriki-Mori chaotic circuits [11][12] cross-coupled by inductors are investigated. We observe the generation of interesting statetransition phenomenon around chaos synchronization.
Computer simulations and circuit experiments are carried out to investigate the phenomenon in detail.

2. Circuit Model
Figure 1 shows the circuit model. In the circuit,
two Shinriki-Mori chaotic circuits are cross-coupled via
inductors L2 .
First, we approximate the v − i characteristics ofthe nonlinear resistors consisting of the diodes by thefollowing 3-segment piecewise-linear functions.
 G(v11 − v12 − V ) (v11 − v12 > V )0(|v11 − v12 | ≤ V )id1 =


G(v11 − v12 + V ) (v11 − v12 < −V )G(v21 − v22 − V ) (v21 − v22 > V )0(|v21 − v22 | ≤ V )id2 =


G(v21 − v22 + V ) (v21 − v22 < −V )

(1)

(2)

- 397 -Figure 1: Circuit model.

The circuit equations are described as follows.
di11

 L1 dt = v12



di12

L1= v22dt
dv11C1= −id1 − i21 + gv11
dt


dv21


= −id2 − i22 + gv21
 C1dt C dv12 = i + i − id122112

dt




dv22



C2= id2 + i12 − i21

dt




di21


= v11 − v22L2


dt





 L di22 = v − v
21221dt(3)

By using the following variables and the parameters,

rC2



i11 =V x1 , v11 = V y1 , v12 = V z1 ,Lr 1C2
i21 =V x2 , v21 = V y2 , v22 = V z2 ,

L

r 1r

C2C2i=Vw,i=V w2 ,12122
L
L1
1
rr



C2L1L1

α=,
β=G,
γ=g,


C
C
C
122

p
L
 δ = 1 , t = L1 C2 τ,
L2

2.50(4)-2.50(a)2.5 -2.50(b)

2.5

3.0

y1 0-3.0
3.0

y2 0

the normalized circuit equations are given as follows.
ẋ1 = z1
ẋ2 = z2 ẏ1 = α{γy1 − w1 − βf (y1 − z1 )} ẏ2 = α{γy2 − w2 − βf (y2 − z2 )}(5)ż1 = β f (y1 − z1 ) + w2 − x1





ż2 = β f (y2 − z2 ) + w1 − x2





 ẇ1 = δ(y1 − z2 )ẇ2 = δ(y2 − z1 )-3.00300

τFigure 2: State transition phenomenon around inphase synchronization (computer calculated result).
α = 1.5, β = 5.0, γ = 0.2, and δ = 0.005. (a) Attractor on y1 − z1 plane. (b) Attractor on y1 − y2plane. (c) Time waveform.
(c)

2.50

where f are the nonlinear functions corresponding to

the v − i characteristics of the nonlinear resistors andare described as follows.
 y1 − z1 − 10f (y1 − z1 ) =
y1 − z1 + 1
 y2 − z2 − 10
f (y2 − z2 ) =y2 − z2 + 1(y1 − z1 > 1)(|y1 − z1 | ≤ 1)(y1 − z1 < −1)(y2 − z2 > 1)(|y2 − z2 | ≤ 1)(y2 − z2 < −1).

-2.50(a)2.5 -2.50(b)

2.5

3.0

y1 0(6)-3.0
3.0

y2 0-3.0(7)

3. State Transition Phenomenon
From the circuit in Fig. 1, we can observe interesting state transition phenomenon around chaos synchronization.
Some examples of the phenomenon are shown inFigs. 2 and 3. These results are obtained by calculating Eq. (5) with the Runge-Kutta method. The
two circuits exhibit chaos but almost synchronized inin-phase in the sense that the attractor is almost in thequadrant I or III on the y1 − y2 plane. When one circuit switches to/from the positive region from/to thenegative region, the other follows the transition aftera few instants. The sojourn time between the statetransitions becomes longer as the coupling parameterδ decreases.

0(c)

τ300

Figure 3: State transition phenomenon around inphase synchronization (computer calculated result).
α = 1.5, β = 5.0, γ = 0.2, and δ = 0.008. (a) Attractor on y1 − z1 plane. (b) Attractor on y1 − y2plane. (c) Time waveform.

Figure 4 shows how the sojourn times change as
the coupling parameter changes. The curve of circles
shows the average period of the state transitions of y1 .
The curve of crosses shows the average time delay ofthe state transitions of y2 when the state transitionsof y1 are considered to be the reference.

- 398 -

τ

200

150

100

50

00.005

0.006

0.007

0.008

0.009

0.01

0.011

0.012

Figure 4: Sojourn time between state transitions versus coupling parameter (computer calculated result). α =2.605, β = 4.0, and γ = 0.1.

From this figure, we can confirm how the sojourntime between the state transitions changes according
to the change of the coupling parameter.
It is very interesting that we can also confirm thegeneration of the state transition around the antiphase synchronization for different set of parametervalues. An example of such modes are shown in Fig. 5.

2.5

3.0

In this study, we have investigated interesting statetransition phenomenon observed from two ShinrikiMori chaotic circuits cross-coupled by inductors.
Investigating the coexistence of the states and statistical analysis of the observed phenomena are ourimportant future work.

02.5 -2.5

(a)

0(b)

2.5

y1 0-3.0
3.0

Acknowledgments

y2 0-3.00

Because it is difficult to realize the very small coupling parameter like δ = 0.001, the circuit experiments
are carried out with relatively large δ.
The circuit experimental results are shown in Fig. 6.
The corresponding computer calculated results areshown in Fig. 7.
We can say that the both results agree well.
5. Conclusions

0-2.5

4. Circuit Experimental Results(c)τ300

Figure 5: State transition phenomenon around antiphase synchronization (computer calculated result).
α = 2.0, β = 4.0, γ = 0.1, and δ = 0.0014. (a)Attractor on y1 − z1 plane. (b) Attractor on y1 − y2plane. (c) Time waveform.

This work was partly supported by Yazaki Memorial
Foundation for Science and Technology.
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- 399 -2.50-2.5

0

2.5 -2.5

(a)

02.5

(b)

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(a)

0(c)

τ300

Figure 7: Computer calculated result corresponding toFig. 6. α = 1.5, β = 5.0, γ = 0.2, and δ = 0.015. (a)Attractor on y1 − z1 plane. (b) Attractor on y1 − y2plane. (c) Time waveform.

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(c)
Figure 6: Circuit experimental result. L1 = 9.93mH,
L2 = 648mH, C1 =32.8nF, C2 =49.5nF, and g=1.89mS.
(a) Attractor on v11 − v12 plane. Horizontal and vertical: 1 V/div. (b) Attractor on v11 − v21 plane. Horizontal and vertical: 1 V/div. (c) Time waveform v11
and v21 . Horizontal 0.5 ms/div and vertical: 2 V/div.

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