IEEE Workshop on Nonlinear Circuit Networks
December 7-8, 2018

Difference in Synchronization of Chaotic Circuit
Network due to Difference in Coupling Direction
Akari Oura, Kyohei Fujii, Shuhei Hashimoto, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Eng., Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email:{oura, fujii,s-hashimoto, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract—In this study, we investigate the synchronization phenomenona of the complex network in which the chaotic circuits
are coupled with one direction. We observe the synchronizationby using the voltage difference between the nodes expressed by
the chaos circuit. Moreover, we also observe the synchronization
phenomena when the coupling strength is changed.

II. N ETWORK MODEL
Figure 1 shows basic chaotic circuit. In this study, chaotic
circuits are applied to nodes of the network. A proposed
network model is shown in Fig. 2 and Fig. 3. The network
consists of a chaotic circuit connected in one direction. Eachnode distance and coupling strength are equal.

I. I NTRODUCTION
Synchronization phenomenon is one of familiar phenomena
existing in nature. It is fundamentally studied everywhere
in all fields such as electricity, machinery, biological systems and so on. Synchronization phenomena are observed
everywhere. Synchronization phenomena are the most familiar
phenomena. For example, flashing firefly, crowing tree frogs,
beating rhythm of the heart. Chaos is actually used for trafficsystems and weather forecasts. Chaos is applied to biology,
medical science and engineering. Also, mathematical models
have been proposed for complex networks and attracting
attention. Actually, we observe various networks such as theInternet, transportation network, cranial nerve cells, humanrelationships. Recently, many scientists study synchronization
phenomena of chaotic circuits. Especially, synchronization ofchaos is very interesting phenomena. Additionally, coupledsystems of chaotic elements produce many kinds of complexity
phenomena such as clustering, chaos propagation and so on.
Studying synchronization of chaos is expected useful in avariety of fields.
In this study, we investigate the synchronization phenomenon of a complex network connected by unidirectional
coupling of coupled chaotic circuits with one-way coupling. Inaddition, we observe and compare the synchronization when
the nodes of the network model to be used are connected in
the reverse direction and when they are connected in bothdirections. Each node is applied chaotic circuit and connectedwith resistance via buffer. By using computer simulations,
we observe interesting synchronization phenomena in the
proposed chaotic circuits network.

Fig. 1.

Fig. 2.

Fig. 3.

- 19 -Circuit model.

Connecting nodes.

Network model (n=10).

First, the circuit equations are given as follows:din= v2ndt1 ∑dv1n(v1n − v1k )C1= gv1n − idn −dt
Rk∈C
n
dv
 C2 2n = −in + idn ,dt(n = 1, 2, 3, ..., 10),L(1)

we approximate the i − v characteristics of the nonlinear
resistor consisting of the diodes, Gd (v1n − v2n − a) (v1n − v2n > a)0(|v1n − v2n | ≤ a)idn =(2)Gd (v1n − v2n + a) (v1n − v2n < −a).
By using the parameters and variables as follows:√C2axn , v1n = ayn , v2n = azn
in =L√dC2t = LC2 τ, “ · ” =, α=dτC1√√√LLL
1β=Gd , γ =g, δ =.
C2C2R C2
The normalized circuit equations are given as follows: ẋ = znẏ = αγyn − αβf (yn − zn ) − αδf (yn − yk )ż = βf (yn − zn ) − xn .

Fig. 4. Phase difference (CC1-CC2, CC1-CC3, CC1-CC4, CC1-CC5, CC1CC6, CC1-CC7, CC1-CC8,CC1-CC9, and CC1-CC10). δ = 0.22.

Figure 5 shows the voltage difference. We did not observe
difference of voltage in CC1-CC2 to CC5. However, weobserved difference of voltage in CC1-CC6 to CC10. Weobserved that the voltage difference is smaller in CC1-CC6
than in CC1-CC9 to CC10. We anticipate CC6 to CC8 may
be synchronized as time goes on.

(3)

(4)

Where the nonlinear function corresponding to the characteristics of the nonlinear resistor of the diodes and are describedas follows:(yn − zn > 1)
 yn − zn − 10(|yn − zn | ≤ 1)f (yn − zn ) =(5)yn − zn + 1(yn − zn < −1).
By computer simulations, we observe the synchronization
by calculating the voltage and surveying the voltage difference.
In addition, we observe the synchronization in the case where
the nodes of the network model in Fig.3 are connected in thereverse direction and in the case where they are connectedin both directions. We observe the difference by comparing
the synchronization when the node is connected in the reverse
direction and when it is connected in both directions.

Fig. 5. Voltage difference(CC1-CC2, CC1-CC3, CC1-CC4, CC1-CC5, CC1CC6, CC1-CC7, CC1-CC8,CC1-CC9, and CC1-CC10). δ = 0.22.

We observe synchronization between CC8 and other nodes.
Figures 6 show the phase difference. We observe did not observe in-phase state in CC1-CC6 to CC10. We have observedthat the difference between CC9 and CC10 is particularlylarge.

III. S IMULATION RESULTS
In this study, we fix parameters as α = 0.5, β = 20, γ =0.5 and δ = 0.22 on all circuits. The simulartion results areshown in Figs. 4 to 11. Figure 4 show the phase difference.
We observe in-phase state, in CC1-CC2 to CC5. However, wedid not observe inphase state in CC1-CC6 to CC10.

- 20 -Fig. 6. Phase difference (CC8-CC1,CC8-CC2, CC8-CC3, CC8-CC4, CC8CC5, CC8-CC6, CC8-CC7, CC8-CC8,CC8-CC9, and CC8-CC10). δ = 0.22.

We observed the synchronization when the nodes of thenetwork model of Fig. 3 are connected in the reverse direction.
Figures 7 show the phase difference. We understood that CC8
is not synchronized with any nodes.

Fig. 9. Voltage difference(CC8-CC1, CC8-CC2, CC8-CC3, CC8-CC4, CC8CC5, CC8-CC6, CC8-CC7, CC8-CC8, CC8-CC9, and CC8-CC10). δ = 0.22.

Fig. 7. Phase difference (CC8-CC1, CC8-CC2, CC8-CC3, CC8-CC4, CC8CC5, CC8-CC6, CC8-CC7, CC8-CC8, CC8-CC9, and CC8-CC10). δ = 0.22.

where nodes are connected in the direction of Fig. 3. Moreover,Fig. 10 shows that the voltage difference from CC8 to CC10
is larger than the voltage difference between CC1 and CC2.
We understood that the magnitude of the voltage differencediffers between the case of the network model of Fig. 3 andthe case of connecting in the same direction of the nodes.

We observed the synchronization when the nodes of thenetwork model of Fig. 3 are connected in both directions.
Figures 8 show the phase difference.We observed that thephase difference is similar to the result in Fig. 6 for thenetwork in Fig. 3.

Fig. 8. Phase difference (CC8-CC1, CC8-CC2, CC8-CC3, CC8-CC4, CC8CC5, CC8-CC6, CC8-CC7, CC8-CC8, CC8-CC9, and CC8-CC10). δ = 0.22.

Figure 9 shows the voltage difference. We observed thatCC9 and CC10 are larger in voltage difference than CC1 toCC7.
Figure 10 shows the voltage difference when the nodes ofthe network model of Fig. 3 are connected in both directions.
Figures 8 and Fig. 10. show that the voltage difference issmaller when connecting in both directions than in the caseFig. 10. Voltage difference(CC8-CC1, CC8-CC2, CC8-CC3, CC8-CC4, CC8CC5, CC8-CC6, CC8-CC7, CC8-CC8, CC8-CC9, and CC8-CC10). δ = 0.22.

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Here, we observed the phase difference when the node ofthe network model of Fig. 3 was connected in both directions.
Figure 11 shows the phase difference. Comparing Fig. 11 withFig. 4, we understood that CC1 has a phase difference fromCC2 to CC5. Also, we observed that the phase difference
between CC9 and CC10 is larger than the phase difference
from CC2 to CC8.

Fig. 11. Voltage difference(CC1-CC2, CC1-CC3, CC1-CC4, CC1-CC5, CC1CC6, CC1-CC7, CC1-CC8, CC1-CC9, and CC1-CC10). δ = 0.22.

IV. C ONCLUSIONS
We studied synchronization phenomena of coupled chaotic
circuit with one-way coupling. Synchronization phenomena
was observed certain part of the circuit with completely inphase state. Also, we understand that the synchronization
phenomenon changes greatly by changing the direction connecting the networks. In this study, we used ten chaotic
circuits. For the future works, we will use more numberof chaotic circuits. Also, we will focus on other nodes andobserve synchronization phenomena. By using more chaotic
circuits, we expect to observe complicated phenomena such asclustering. Also, we expect to observe chaos synchronization.
Also, it is expected that differences in characteristics will beobserved by changing network connection.
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Synchronization Phenomena of Coupled Chaotic Circuit
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20th Conference on Electrical Relations Association Shikoku Branch
Conference Presentation paper, no. 1-23, p. 23, Sep. 2013.
[4] M. Inoue and H. Hata, Foundations and development of chaos science ,1999, pp.171199.

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