Effect of Lateral Connection on Synchronization
Phenomena in Chaotic Circuits Coupledwith Non-Uniform Coupling Strength
Yuki Matsubara, Yuki Ishikawa, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering
Tokushima University
2-1 Minami-Josanjima, Tokushima 7708506, Japan
E-mail: {matsubara, ishikawa, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract— Synchronization phenomena in coupled oscillators havechange in synchronization rate when lateral connection is added
to chaotic circuits coupled with non-uniform coupling strength. As
a result, it is confirmed that the change in synchronization rate
differs depending on the location where the lateral connection isadded.
Keywords; Synchronization, Chaotic Circuits

I.
INTRODUCTION
Synchronization is a phenomenon in which oscillatorsinteract with each other and their oscillations are aligned. Thisphenomenon has been investigated in a wide range of research
fields, including human brain networks [1] and communication
networks [2]. Synchronization phenomenon is also observed incoupled oscillators. In particular, an interesting phenomenon
called chaos synchronization, in which the behavior is alignedby coupling, can be observed even in systems with complex
behavior such as chaotic circuits. Therefore, research on coupledchaotic circuits has been active. Previous studies have often
focused on the topology and coupling strength of bidirectionally connected networks with uniform couplingstrength [3],[4]. However, there is little study of the couplingstrength in chaotic circuits with non-uniform coupling strength.
Therefore, in this study, we confirmed the change in thesynchronization phenomena when lateral connections are added
to a chaotic circuit coupled with non-uniform coupling strength.
We also confirmed the comparison of the synchronization
phenomena when the location where the lateral connection
added is changed.
II.

NETWORK MODEL

Figure 1 shows the chaotic circuit used in this study. The
chaotic circuit consisted of a negative resistor, an inductor,
two capacitors, and a bidirectional diode. The chaotic circuit iscalled the Mori-Shinriki circuit. Figure 2 shows the networkmodel proposed in this study. This mode is a network modelwith non-uniform coupling strength, as the coupling strength
differs between the black and red arrows.

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Figure 1. Circuit model.

Figure 2. Network model.
The normalized circuit equations for the circuit in Fig.1 areshown as follows.
𝑥̇ = 𝑧𝑛{𝑦̇ = 𝛼𝛾𝑦𝑛 − 𝛼𝛽𝑓(𝑦𝑛 − 𝑧𝑛 ) − 𝛼𝛿𝑓(𝑦𝑛 − 𝑦𝑘 )(1))
𝑧̇ = 𝛽𝑓(𝑦𝑛 − 𝑧𝑛 − 𝑥𝑛
The normalized characteristic equations of the bidirectional
diode are shown as follows.
𝑦𝑛 − 𝑧𝑛 − 1 (𝑦𝑛 − 𝑧𝑛 > 1)(|𝑦𝑛 − 𝑧𝑛 | ≤ 1)𝑓(𝑦𝑛 − 𝑧𝑛 ) = {0𝑦𝑛 − 𝑧𝑛 + 1 (𝑦𝑛 − 𝑧𝑛 < 1)(2)

𝛿 shows the coupling strength between the circuits.

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III.

RESULT

In this study, the synchronization phenomenon is observedby computer simulation. The parameters are 𝛼 = 0.5, 𝛽 = 20, and
𝛾 = 0.5. We set 𝛿 = 0.70 for the black arrows and 𝛿 = 0.35 forthe red arrows in Fig. 2

confirmed that the synchronization rate increases when lateralconnections are added at locations (8-9) where there are moreconnections.

A. Case of adding lateral connections to the different layers
It is confirmed the change in synchronization rate for eachconnection when lateral connections are added to different
layers: circuit 2 and circuit 3, circuit 4 and circuit 5, circuit 8 andcircuit 9, and circuit 12 and circuit 13. Figure 3(a) shows thechange in synchronization rate when a lateral connection( 𝛿 =0.35) is added. Figure 3(b) shows the change insynchronization rate when a lateral connection (𝛿=0.70) is added.

(a) lateral connection (𝛿=0.35).

(b) lateral connection (𝛿=0.70).

(a) lateral connection (𝛿=0.35).

Figure 4. The change in synchronization rate when a lateralconnection is added.
IV.

(b) lateral connection (𝛿=0.70).
Figure 3. The change in synchronization rate when a lateralconnection is added.
From Fig. 3, it is confirmed that the synchronization rateincreases by adding lateral connections in circuits 2 and 3, circuit
4 and 5, and circuit 8 and 9. In contrast, it is confirmed that thesynchronization rate decreases by adding lateral connections incircuit 12 and circuit 13.

In this study, it is confirmed the change in synchronization
rate when lateral connection is added to chaotic circuits coupledwith non-uniform coupling strength. As a result, it is confirmedthat the change in synchronization rate varies depending on thelayer to which the lateral connection is added. In addition, evenin the same layer, the synchronization rate varies depending on
the location where the lateral connection is added. It isconsidered that the number of connections that the location
where the lateral connection is added influences thesynchronization rate.
REFERENCES[1]

B. Case of adding lateral connections to the same layer
It is confirmed the change in synchronization rate for eachconnection when lateral connections are added to same layers:
circuit 7 and circuit 8, circuit 8 and circuit 9, and circuit 9 andcircuit 10. Figure 4(a) shows the change in synchronization rate
when a lateral connection (𝛿=0.35) is added. Figure 4(b) shows
the change in synchronization rate when a lateral connection(𝛿=0.70) is added.

[2]

[3]

[4]

From Fig. 4, it is confirmed that the change insynchronization rate varies depending on the location where theconnections are added, even in the same layer. It also is979-8-3503-2703-8/23/$31.00 ©2023 IEEE

CONCLUSION

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D. Henao, M. Navarrete, M. Valderrama and M. L. V. Quyen,“Entrainment and synchronization of brain oscillations toauditory
stimulations,” Neurocience Resarch 156, pp. 271–278, July 2020.
A. Omri, M. Shaqfeh, A. Ali and H. Alnuweiri, “ Synchronization
Procedure in 5G NR Systems,” IEEE Access (Volume: 7), pp.41286–41295, Mar 2019.
K. Nakabai, Y. Uwate and Y. Nishio, “Synchronization in LadderCoupled Chaotic Circuits Including Ring Structures,” Proceedings ofIEEEInternational
Symposium
andSystems(ISCAS’19),
DOI:10.1109/ISCAS.2019.8702350 (5 pages), May 2019.
K. Ago, Y. Uwate and Y. Nishio, “Synchronization of Small-World
Coupled Chaotic Circuits with Parameter Mismatch,” Proceeding of RISPInternational Workshop on Nonlinear Circuits, Communications andSignal Processing (NCSP’16), PP.37–40, Mar 2016.

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