Analysis of Chaotic Circuit Networks
with One-Way Coupling
Akari Oura*, Kyohei Fujii*, Yoko Uwate*, and Yoshifumi Nishio*
dept. of Electrical and Electronic Engineering,Tokushima University
2-1 Minami-Japan, Tokushima 770-8506, Japan
Email: {oura, fujii, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract— In this study, we investigate the synchronization
phenomena of the complex network in which the chaotic circuits
are coupled with one direction. We observe the synchronization byusing the voltage difference between the nodes expressed by thechaos circuit. Moreover, we also observe the synchronization
phenomena when the coupling strength is changed.
Keywords; Complex network; Synchronization Phenomena;
Chaotic Circuit.

I.

INTRODUCTION

Synchronization phenomena are observed everywhere. For
example, ﬂashing ﬁreﬂy, crowing tree frogs, beating rhythm of
the heart. Chaos is applied to biology, medical science andengineering. 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 a varietyof ﬁelds.

Figure 2: Network model (n=10).
First, the circuit equations are given as follows:𝑑𝑖𝑛= 𝑣2𝑛
𝑑𝑡
1𝑑𝑣1𝑛
𝐶1
= 𝑔𝑣1𝑛 − 𝑖𝑑𝑛 − ∑(𝑣1𝑛 − 𝑣1𝑘 )
𝑅
𝑑𝑡
𝐶𝐶𝑛
𝑑𝑣2𝑛
𝐶
=−𝑖+𝑖𝑛𝑑𝑛{ 2 𝑑𝑡(n = 1,2,3, … ,10),𝐿

In this study, we investigate synchronization phenomena of
coupled chaotic circuits with one-way coupling. Each node isapplied chaotic circuit and connected with resistance via buffer.
By using computer simulations, we observe interesting
synchronization phenomena in the proposed chaotic circuits
network.
II.

we approximate the 𝑖 − 𝑣 characteristics of the nonlinear
resistor consisting of the diodes,

NETORK MODEL

𝐺𝑑 (𝑣1𝑛 − 𝑣2𝑛 − 𝑎)𝑖𝑑𝑛 = {0𝐺𝑑 (𝑣1𝑛 − 𝑣2𝑛 + 𝑎)

Figure 1 shows basic chaotic circuit. In this study, chaotic
circuits are applied to nodes of the network. A proposed networkmodel is shown in Fig. 2. The network consists of a chaotic
circuit connected in one direction.

(𝑣1𝑛 − 𝑣2𝑛 > 𝑎)(|𝑣1𝑛 − 𝑣2𝑛 | ≤ 𝑎)(𝑣1𝑛 − 𝑣2𝑛 < −𝑎).

By using the parameters and variables as follows:𝐶2
𝑖𝑛 = √ 𝑎𝑥𝑛 , 𝑣1𝑛 = 𝑎𝑦𝑛, 𝑣2𝑛 = 𝑎𝑧𝑛
𝐿t = √𝐿𝐶2 𝜏, "・" =𝐿
𝐿1 𝐿
𝛽 = √ 𝐺𝑑 , 𝛾 = √ 𝑔, 𝛿 = √ .
𝐶2
𝐶2
𝑅 𝐶2

Figure 1: Circuit model.

978-1-5386-7960-9/18/$31.00 ©2018 IEEE𝑑𝐶2,𝛼 =𝑑𝜏𝐶1

168

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The normalized circuit equations are given as follows:𝑥̇ = 𝑧𝑛{𝑦̇ = 𝛼𝛾𝑦𝑛 − 𝛼𝛽𝑓(𝑦𝑛 − 𝑧𝑛 ) − 𝛼𝛿𝑓(𝑦𝑘 − 𝑦𝑛 )𝑧̇ = 𝛽𝑓(𝑦𝑛 − 𝑧𝑛 ) − 𝑥𝑛 ,where the nonlinear function corresponding to the characteristicsof the nonlinear resistor of the diodes and are described asfollows:𝑦𝑛 − 𝑧𝑛 − 1()𝑓 𝑦𝑛 − 𝑧𝑛 = {0𝑦𝑛 − 𝑧𝑛 + 1(𝑦𝑛 − 𝑧𝑛 > 1)|
( 𝑦𝑛 − 𝑧𝑛 | ≤ 1)(𝑦𝑛 − 𝑧𝑛 < −1).

By computer simulations, we observe the synchronization by
calculating the voltage and observing the voltage difference.
III.

SIMULATION RESULTIn this study, we fix parameters as α = 0.5, β = 20, γ = 0.5
and δ = 0.22 on all circuits. The simulation results are shown inFigs. 3 to 5. Figures 3 and 4 show the phase difference.

Figure 5: Voltage difference. δ = 0.22.
IV. CONCLUSION
We studied synchronization phenomena of coupled chaotic
circuit with one-way coupling. Synchronization phenomena wasobserved certain part of the circuit with completely in-phasestate. In this study, we used ten chaotic circuits.

We observe completely in-phase state, in CC1-CC2 to CC5(Fig. 3). However, we did not observe in-phase state in CC1CC6 to CC10 (Fig. 4). We can say that there are two groups inthe chaotic network depending on the synchronization state.

For the future works, we will use more number of chaoticcircuits. By using more chaotic circuits, we expect to observecomplicated phenomena such as clustering. Also we expect toobserve chaos synchronization. Also, it is expected thatdifferences in characteristics will be observed by changing
network connection.

Figure 3: Phase difference (CC1-CC2, CC1-CC3, CC1CC4, and CC1-CC5). δ = 0.22.

ACKNOWLEDGMENT
This work was partly supported by JSPS Grant-in-Aid for
Scientific Research 16K06357.
REFERENCES[1]

Figure 4: Phase difference (CC1-CC6, CC1-CC7, CC1-CC8,CC1-CC9, and CC1-CC10). δ = 0.22.

[2]

[3]

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. We
observed that the voltage difference is smaller in CC1-CC6
than in CC1-CC9 to CC10.

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[4]

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