2009 IEEE Workshop on
Nonlinear Circuit Networks, Tokushima, Dec 11-12, 2009

Investigation of Synchronization Phenomenon in
Coupled Chaotic Circuits and Coupled Cubic Maps
Yumiko Uchitani

Yoshifumi Nishio

Dept. Electrical and Electronic Eng.,
Tokushima University
Email: uchitani@ee.tokushima-u.ac.jp

Dept. Electrical and Electronic Eng.,
Tokushima University
Email: nishio@ee.tokushima-u.ac.jp
ẋ1 = z1 ẋ = z22 ẏ = α{γy − w − βf (y − z )}11111 ẏ = α{γy − w − βf (y − z )}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. We consider that
it is very important to investigate the phenomena related withchaos synchronization to realize future engineering application
utilizing chaos.
In our past studies [1]-[3], two Shinriki-Mori circuits [4][5]cross-coupled by inductors are investigated. As a result, wecould observe several synchronization phenomena by changing
initial conditions On the other hand, the cubic map has twoattractors located symmetrically with respect to the origin andthey merge as increasing a control parameter. Such bifurcations are similar to those observed from Shinriki-Mori circuit.
In this study, we compare the synchronization behaviors ofcross-coupled chaotic circuits and the coupled chaotic maps.
II. C ROSS - COUPLED CHAOTIC CIRCUITS [1]-[3]In this section, we review the phenomena observed from thetwo cross-coupled chaotic circuits. Figure 1 shows the cross-2222ż1 = β f (y1 − z1 ) + w2 − x1ż2 = β f (y2 − z2 ) + w1 − x2ẇ1 = δ(y1 − z2 )ẇ2 = δ(y2 − z1 )From the circuit in Fig. 1, we could observe interestingstate transition phenomenon. Typical examples of the observed
phenomena are shown in Figs. 2. By computer simulations, weconfirmed that the circuits generated many different synchronization states, by changing initial condition.
2.0

y10
-2.0
2.0

00

y2-2.0-2.500

τ

5000

τ

5000

τ

500

2.5

2.5

2.0

y10
-2.0
2.0

00

y2-2.50-2.0

2.5

2.5

2.0

y10
-2.0
2.00
y2

Circuit model.
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coupled chaotic circuits. In this model, two simple autonomouschaotic circuits called as Shinriki-Mori circuit [4][5] are crosscoupled via inductors L2 . The normalized circuit equations aregiven as follows where δ is a coupling parameter.

(1)

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 − zk − 10(|yk − zk | ≤ 1)
(2)
f (yk − zk ) =yk − zk + 1(yk − zk < −1)

2.5

Fig. 1.

20(a)2.50-2.0(b)Fig. 2. Some examples of different synchronization states. (computersimulation results). α = 2.0, β = 4.0, γ = 0.1, and δ = 0.0014.
(a) Attractor on y1 − y2 plane. (b) Timewaveform.

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1.0

III. C UBIC MAP

x1

In this section, we consider two coupled cubic maps.
Figure 3 shows the cubic map.
xn+10-1.0
1.0

x2 0-1.0

xn+1n20000n20000n20000n20000(a)
1.0

x1
xn0-1.0
1.0

xnx2 0-1.0

(a)

(b)

(b)

1.0

x1

Fig. 3.

0-1.0
1.0

Cubic map. (a) a = −3.4505. (b) a = −3.70.

x2 0-1.0

The cubic map is described by the following equation:x(n+1) = ax3(n) − (a + 1)x(n)(c)
1.0(3)

where n is an iteration and a is a parameter which determines
the chaotic feature. We can easily confirm that the cubic mapgenerates various periodic solutions and chaos. Figure 3(a)shows asymmetric 3 periodic solution and Fig. 3(b) shows
symmetric 6 periodic solution.
The followings are the equations of the coupled cubic maps
where ² is a coupling parameter.
3 g(x(n) ) = ax(n) − (a + 1)x(n)x1(n+1) = (1 − ²)g(x1(n) ) + ²(g(x2(n) ))(4)
x2(n+1) = (1 − ²)g(x2(n) ) + ²(g(x1(n) ))Figure 4 shows examples of the computer simulated resultsof Eq. (4). Figure 4(a) shows the timewaveform obtained withthe parameters giving the periodic solution in Fig. 3(a) and anegative coupling strength. In the same way, Fig. 4(b) shows
the timewaveform obtained with the parameters giving theperiodic solution in Fig. 3(b) and a positive coupling strength.
From the coupled cubic maps, we can observe interesting statesynchronization phenomena. We could confirm that the twomaps synchronize for the parameters giving periodic solutions.
Similarly, we could confirm that the two maps synchronize forthe parameters near that giving periodic solutions as shown inFig. 4(c)(d). However, we do not understand the relationshipbetween the obtainable synchronization states and symmetry
property of the original solutions.
IV. C OMPARISON OF SYNCHRONIZATION BEHAVIORS
In this study, we compare the synchronization behaviors of
the cross-coupled chaotic circuits and coupled chaotic maps.
For the cross-coupled chaotic circuits, we can observe
several synchronization phenomena by changing initial conditions. On the other hand, for the coupled chaotic maps,
we can observe in-phase synchronization for the parameters
giving asymmetric periodic solutions and a negative couplingstrength. Further, we can observe anti-phase synchronization
for the parameters giving symmetric periodic solutions and ax10-1.0
1.0

x2 0-1.0(d)Timewaveform of coupled cubic maps. (a) α = −3.4505,
and ² = −0.062. (b) α = −3.70, and ² = 0.0375. (c) α = −3.40,
and ² = −0.087. (d) α = −3.61, and ² = 0.0045.
Fig. 4.

positive coupling strength. However, we do not understand therelationship between the obtainable synchronization states andsymmetry property of the original solutions.
V. C ONCLUSIONSIn this study, we compared the synchronization behaviors
of cross-coupled chaotic circuits and the coupled chaoticmaps. As a result, for the cross-coupled chaotic circuits, wecould observe several synchronization phenomena by changing
initial conditions. On the other hand, for the coupled chaoticmaps, the two maps synchronized for the parameters giving
periodic solutions. Clarifying the mechanism of the observed
phenomena is our future work.
R EFERENCES[1] Y. Uchitani, R. Imabayashi and Y. Nishio, “State Transition Phenomenon in Cross-Coupled Chaotic Circuits,” Proc. ofNOLTA’07, pp. 397-400, Sep. 2007.
[2] Y. Uchitani and Y. Nishio, “Investigation of State Transition Phenomena in Cross-Coupled Chaotic Circuits,” Proc. of ISCAS’08,
pp. 2394-2397, May. 2008.
[3] Y. Uchitani and Y. Nishio, “Synchronization of Small Oscillations in Cross-Coupled Chaotic Circuits,” Proc. of ISCAS’09,
pp. 2629-2632, May. 2009.
[4] M. Shinriki, M. Yamamoto and S. Mori, “Multimode Oscillations
in a Modified van der Pol Oscillator Containing a PositiveNonlinear Conductance,” Proc. IEEE, vol. 69, pp. 394-395, 1981.
[5] 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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