Comparison between Cross-Coupled Chaotic Circuits
and Direct-Coupled Chaotic Circuits
Yumiko Uchitani and Yoshifumi Nishio
Department of Electronic and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, 770-8506 JAPAN
Email: {uchitani, nishio}@ee.tokushima-u.ac.jp
Abstract
Studies on chaos synchronization in coupled chaoticcircuits are extensively carried out in various ﬁelds. In
this study, we compare the synchronization behaviors
of cross-coupled chaotic circuits and direct-coupledchaotic circuits.
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 phenomena of
coupled chaotic circuits are extensively carried out invarious ﬁelds [1]-[9].
In our past studies [10][11], two simple chaotic circuits cross-coupled by inductors are investigated. As
a result, we could observe interesting state transition
phenomena.
In this study, we compare the synchronization behaviors of cross-coupled chaotic circuits and directcoupled chaotic circuits in order to know how thestructure of the coupling aﬀects the observed phenomena.

Figure 1: Cross-coupled chaotic circuits.

the normalized circuit equations are given as follows.
2. Circuit model
Figure 1 shows the cross-coupled circuit model. In
the circuit, simple chaotic circuits [12][13] are crosscoupled via inductors L2 .
By using the following variables and the parameters,√
L1 i11
v11v12, y1 =, z1 =, x1 =

CVVV


√ 2

L1 i21v21v22
x2 =, y2 =, z2 =,CVVV2√√
L1 i12L1 i22(1)w=,w=,12CVC
22 V√√C2L1L1α=,β=G,
γ=g,

CCC
122
√L
 δ = 1 , t = L1 C2 τ,L2ẋ1 = z1






ẋ2 = z2ẏ1 = α{γy1 − w1 − βf (y1 − z1 )} ẏ2 = α{γy2 − w2 − βf (y2 − z2 )}ż1 = β f (y1 − z1 ) + w2 − x1ż2 = β f (y2 − z2 ) + w1 − x2ẇ1 = δ(y1 − z2 )ẇ2 = δ(y2 − z1 )(2)

While, Fig. 2 shows the direct-coupled circuit model.
Equation (3) indicates the normalized circuit equa-- 27 -

2.5

0-2.5

0

2.5 -2.5(1a)02.5(1b)2.0

y1 0-2.0
2.0

y2 0-2.0
0

Figure 2: Direct-coupled chaotic circuits.

τ 1200(1c)tions of the direct-coupled model.
ẋ1 = z1ẋ2 = z2ẏ1 = α{γy1 − w1 − βf (y1 − z1 )} ẏ2 = α{γy2 + w1 − βf (y2 − z2 )}ż1 = β f (y1 − z1 ) − w2 − x1






ż2 = β f (y2 − z2 ) + w2 − x2





ẇ1 = δ(y1 − y2 )ẇ2 = δ(z1 − z2 )

2.5

0

(3)
-2.5

0

2.5 -2.5

(2a)

0

2.5

(2b)

2.0

y1 0

where f are the nonlinear functions corresponding to
the v − i characteristics of the nonlinear resistors consisting of the diodes and are assumed to be describedby the following 3-segment piecewise-linear functions:(y1 − z1 > 1)

 y1 − z1 − 10(|y1 − z1 | ≤ 1)
(4)
f (y1 − z1 ) =
y1 − z1 + 1(y1 − z1 < −1)(y2 − z2 > 1)

 y2 − z2 − 10(|y2 − z2 | ≤ 1)
(5)
f (y2 − z2 ) =y2 − z2 + 1(y2 − z2 < −1).

-2.0
2.0

y2 0-2.00

τ 1200(2c)

Figure 3: State transition phenomenon around anti-phasesynchronization (computer calculated result). α = 2.0, β =4.0, γ = 0.1, and δ = 0.0014. (1) Cross-coupled chaoticcircuits. (2) Direct-coupled chaotic circuits. (a) Attractor
on y1 − z1 plane. (b) Attractor on y1 − y2 plane. (c) Timewaveform.

3. State transition phenomenon
From the cross-coupled chaotic circuits, we could
observe interesting state transition phenomenon. Figure 3(1) shows computer calculated results of Eq. (2)- 28 -

with the Runge-Kutta method. While, Fig. 3(2) shows
the computer simulated result for the case of thedirect-coupled chaotic circuits. In both cases, the twocircuits exhibited chaos but almost synchronized inanti-phase in the sense that the attractors were almostin the quadrant II or IV on the y1 − y2 plane. The behaviors of the circuits are very interesting because thesolutions on the yi − zi planes seem to be attracted
to the ﬁxed points located at around (yi , zi )=(±1.2,0). However, after converging to the ﬁxed points thesolution abruptly moves toward the other ﬁxed point.
When one circuit switches to/from the positive region
from/to the negative region in this way, the other follows the transition after a few instants.

2.50-2.50

2.5 -2.5(1a)02.5(1b)2.0

4. Comparison of synchronization behaviors

y1 0In this section, we compare the synchronization behaviors of the cross-coupled chaotic circuits and directcoupled chaotic circuits.
First, for the cross-coupled chaotic circuits, we canobserve similar state transition phenomenon aroundin-phase synchronization as Fig. 4(1) and quadraturephase synchronization as Fig. 4(2). However, as far as
we carried out computer simulations, we cannot observe the state transition phenomena around in-phaseor quadrature-phase for the case of the direct-coupledchaotic circuits.
Secondly, we notice that the sojourn times between
the state transitions are diﬀerent from Figs. 3(1c) and3(2c). Figure 5 shows how the sojourn times aroundanti-phase synchronization change as the coupling parameter changes. The horizontal axis is coupling parameter δ and the vertical axis is the average lengthsof the transitions in τ . The curve of circles shows theaverage period of the state transitions of y1 for the
cross-coupled chaotic circuits. The curve of crosses
shows the average period of the state transitions of
y1 for the direct-coupled chaotic circuits. From thisﬁgure, we can see that the sojourn time of the directcoupled chaotic circuits is almost half of that of thecross-coupled chaotic circuits.

-2.0
2.0

y2 0
-2.00

τ 1200(1c)

2.50

-2.50

2.5 -2.5(2a)02.5(2b)2.0

y1 0-2.0
2.0

y2 0-2.00

τ 1200(2c)5. Conclusions
Figure 4: State transition phenomenon observed fromIn this study, we have compared the synchronization
behaviors of the cross-coupled chaotic circuits and thedirect-coupled chaotic circuits in order to know how
the structure of the coupling aﬀected the observed phenomena. By computer simulations, we conﬁrmed two
diﬀerences. The ﬁrst is that the state transition phenomenon of the cross-coupled chaotic circuits appears
around anti-phase, in-phase, and quadrature-phasesynchronizations, while that of the direct-coupledchaotic circuits appears only around anti-phase synchronization. The second is that the sojourn time ofcross-coupled chaotic circuits (computer calculated result).
α = 2.0, β = 4.0, γ = 0.1, and δ = 0.0014. (1) Around inphase synchronization. (2) Around quadrature-phase synchronization. (a) Attractor on y1 − z1 plane. (b) Attractor
on y1 − y2 plane. (c) Time waveform.

- 29 -τ
600

500

400

300

200

100

00.0008δ
0.001

0.00120.00140.00160.00180.002

Figure 5: Sojourn time between state transitions versus coupling parameter (anti-phase synchronization). α = 2.0,
β = 4.0, γ = 0.1, and δ = 0.0014. Circle: Cross-coupled. Cross: Direct-coupled.

the direct-coupled chaotic circuits is almost half of thatof the cross-coupled chaotic circuits. Clarifying themechanism of these diﬀerences is our future work.

[7] Y. Nishio and A. Ushida, “Chaotic Wandering and its
Analysis in Simple Coupled Chaotic Circuits,” IEICE
Trans. Fundamentals, vol. E85-A, no. 1, pp. 248-255,
2002.

Acknowledgments
This work was partly supported by Yazaki Memorial
Foundation for Science and Technology.
References[1] N. Platt, E.A. Spiegel and C. Tresser, “On-Oﬀ Intermittency: A Mechanism for Bursting,” Phys. Rev.
Lett., vol. 70, no. 3, pp. 279-282, 1993.

[8] G. Abramson, V.M. Kenkre and A.R. Bishop, “Analytic Solutions for Nonlinear Waves in Coupled Reacting Systems,” Physica A: Statistical Mecanics andits Applications, vol. 305, no. 3-4, pp. 427-436, 2002.
[9] 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.

[2] P. Ashwin, J. Buescu and I. Stewart, “Bubbling of Attractors and Synchronization of Chaotic Oscillators,”
Phys. Lett. A,193, pp. 126-139, 1994.

[10] Y. Uchitani, R. Imabayashi and Y. Nishio, “State
Transition Phenomenon in Cross-Coupled ChaoticCircuits,” Proc. of NOLTA’07, pp. 397-400, Sep.
2007.

[3] E. Ott and J.C. Sommerer, “Blowout Bifurcations:
the Occurrence of Riddled Basins and On-Oﬀ Intermittency,” Phys. Lett. A,199, pp. 39-47, 1994.

[11] Y. Uchitani and Y. Nishio, “Investigation of State
Transition Phenomena in Cross-Coupled Chaotic Circuits,” Proc. of ISCAS’08, pp. 2394-2397, May. 2008.

[4] Y. Nishio and A. Ushida, “Spatio-Temporal Chaos
in Simple Coupled Chaotic Circuits,” IEEE Trans.
Circuits Syst. I, vol. 42, no. 10, pp. 678-686, 1995.

[12] M. Shinriki, M. Yamamoto and S. Mori, “Multimode Oscillations in a Modiﬁed van der Pol Oscillator
Containing a Positive Nonlinear Conductance,” Proc.
IEEE, vol. 69, pp. 394-395, 1981.

[5] N.F. Rul’kov and M.M. Sushchik, “Robustness ofSynchronized Chaotic Oscillations,” Int. J. Bifurcation and Chaos, vol. 7, no. 3, pp. 625-643, 1997.
[6] M. Wada, Y. Nishio and A. Ushida, “Analysis of Bifurcation Phenomena in Two Chaotic Circuits Coupled by an Inductor,” IEICE Trans. Fundamentals,
vol. E80-A, no. 5, pp. 869-875, 1997.

[13] N. Inaba, T. Saito and S. Mori, “Chaotic Phenomena
in a Circuit with a Negative Resistance and an Ideal
Switch of Diodes,” Trans. of IEICE, vol. E70, no. 8,
pp. 744–754, 1987.

- 30 -