2023 RISP International Workshop on Nonlinear Circuits, Communications and Signal
Processing (RISP NCSP 2023), Feb. 28 - Mar. 3, 2023

Synchronization of Bi-Directionally Coupled Chaotic Circuitswith 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 770–8506, Japan
Email: {matsubara, ishikawa, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract
In this study, we investigate the synchronization of bidirectionally coupled chaotic circuits with non-uniform coupling strength in proposed model. First, computer simulationof two chaotic circuits is performed. Then, the number ofcircuits is increased to see how it affects the synchronization
phenomena. As a result, synchronization phenomena are also
observed when the coupling strength is non-uniform. The
number of circuits that are synchronized is related to the coupling strength that can keep the synchronization phenomena.

neously. Moreover, we also investigate the synchronization
phenomenon when the number of chaotic circuits is changed.
2. System model
Figure 1 shows the chaotic circuit used in this study. The
chaotic circuit consists of a negative resistor, two capacitors,
an inductor, and a bidirectional diode. The chaotic circuit iscalled the Mori-Shinriki circuit [6], [7].

1. Introduction
Synchronization phenomena are commonly observed innature, such as the resonance of a metronome, the luminescence of a firefly, or the beating of a heart. In recent years,
research has been conducted in various fields, such as synchronization of human brain networks [1] and communicaFigure 1: Circuit model.
tion networks [2]. Coupled oscillators are excellent for modeling nonlinear phenomena in nature, and synchronization is
The circuit equations of the chaotic circuit are shown as
one such nonlinear phenomenon. In particular, we can confollows:
firm an interesting phenomenon called chaos synchronization, in which chaotic circuits that emit complex behaviors
align their behaviors by coupling. Therefore, chaos synchrodin


= v2n
Lnization in coupled chaotic circuits is often studied [3]-[5].
dtAmong them, many studies have focused on the topology
dv1n1 ∑C1(v1n − v1k )= gv1n − idn −(1)and coupling strength of bi-directionally connected networks.
dtRk∈Cn
Those studies have shown that the coupling strength in bothdv
directions is equal. However, social networks such as human C2 2n = −in + idn ,dtrelationships do not always have equal bi-directional coupling(n = 1, 2, 3, 4),strength.
The i − v characteristic of nonlinear resistor consisting ofTherefore, in this study, we focused on the bi-directional
strength between chaotic circuits. We investigated the syn- the diodes is described as follows:
chronization phenomenon when the bi-directional coupling
strength is non-uniform, where the coupling strength in one
 Gd (v1n − v2n − a) (v1n − v2n > a)direction is fixed and the coupling strength in the other direc0(|v1n − v2n | ≤ a)idn =(2)tion is varied. In addition, a comparison is made with the case
Gd (v1n − v2n + a) (v1n − v2n < −a).
where the bi-directional coupling strength is varied simulta154

2023 RISP International Workshop on Nonlinear Circuits, Communications and SignalProcessing (RISP NCSP 2023), Feb. 28 - Mar. 3, 2023

By using the parameters and variables as follows:

Figures 2-4 show the system model used in this study. Thecoupling strength from other circuits to CC1 is varied, while
the coupling strength between other circuits is fixed at δ =0.22 for simulation.
In order to measure the synchronization phenomena quantitatively, we defined synchronization as in the followingequation.

√C2
in =axn , v1n = ayn , v2n = aznL√dC2t = LC2 τ, “ · ” =, α=dτC1√√√LL1
L
β=Gd , γ =g, δ =.
C2
C2
R C2|yi − yj | ≤ 0.01 (i ̸= j)(5)

The normalized circuit equations of the circuit are described as follows: ẋ = znẏ = αγyn − αβf (yn − zn ) − αδf (yn − yk )ż = βf (yn − zn ) − xn(3)

Where the nonlinear function corresponding to the i − vcharacteristic in Eq. (2) is described as follows:(yn − zn > 1)
 yn − zn − 10(|yn − zn | ≤ 1)f (yn − zn ) =yn − zn + 1 (yn − zn < −1).

Figure 2: Model A.

Figure 3: Model B.

Figure 4: Model C.

Figure 5: Voltage difference waveform.
(4)
The synchronization rate is expressed using the voltage difference waveform between circuits. When the voltage difference is within the reference value, it is considered synchronous, and when it is not within the reference value, it isconsidered asynchronous. The synchronization rate is thenindicated by the percentage of how far within the reference
value within a certain period of time.

3. Simulation results
In this study, the synchronization phenomena are investigated by the computer simulations. The parameters as
α = 0.5, β = 20, γ = 0.5 on all circuits are fixed. Thesynchronization rate is averaged over five changes of the initial values for each circuit. Figure 6 shows the changes inthe synchronization phenomena in Model A. The couplingstrength from CC1 to CC2 is constant, and when the couplingstrength from CC2 to CC1 is varied, it is considered NonUniform, and when the coupling strength between CC1 andCC2 is both varied, it is considered Uniform. Figure 7 shows
the voltage difference between CC1 and CC2 of model A ateach coupling strength. The coupling strength from CC2 to
CC1 is varied. Figure 6 shows that the synchronization rate
remains close to 100[%] until the coupling strength of one
side becomes about half of the coupling strength of the otherside. In addition, the change in the synchronization phenomena for the Non-Uniform coupling strength is different fromthat for the Uniform coupling strength. The voltage difference waveforms in Fig. 7 show a clear distinction between
synchronous and asynchronous.
1552023 RISP International Workshop on Nonlinear Circuits, Communications and Signal
Processing (RISP NCSP 2023), Feb. 28 - Mar. 3, 2023

Figure 9: Voltage difference waveforms (Model B).
Figure 6: Relationship between synchronization rate andcoupling strength (Model A).

Figure 10 shows the change in synchronization phenomena in model C. The coupling strength from CC2, CC3, andCC4 to CC1 is varied. Figure 11 shows the voltage difference between CC1 and CC4 of Model C at each couplingstrength. Figure 12 compares the three proposed models with
non-uniform coupling strengths. from Fig. 10, Model C also
shows similar results to Model B. Moreover, compared toModel B, the coupling strength at which the synchronization
phenomena collapse is smaller in Fig. 12.

Figure 7: Voltage difference waveforms (Model A).
Figure 8 shows the change in synchronization phenomena inmodel B. The coupling strength from CC2 and CC3 to CC1
is varied. Figure 9 shows the voltage difference waveforms
between CC1 and CC3 of model B at each coupling strength.
From the Figs. 8 and 9, with uniform coupling strength, thesynchronization phenomena collapse from δ = 0.11. On theother hand, for non-uniform coupling strength, it is confirmedthat the synchronization phenomena collapse from δ = 0.05. Figure 10: Relationship between synchronization rate andWhen δ = 0.07, the voltage difference in Model A appears to coupling strength (Model C).
vary, but in Model B, no variation in the voltage difference isseen, considering that the number of circuits affects the synchronization phenomena.

Figure 11: Voltage difference waveforms (Model C).

Figure 8: Relationship between synchronization rate andcoupling strength (Model B).

1562023 RISP International Workshop on Nonlinear Circuits, Communications and Signal
Processing (RISP NCSP 2023), Feb. 28 - Mar. 3, 2023

tional Symposium on Circuits and Systems (ISCAS’19),
DOI:10.1109/ISCAS.2019.8702350 (5 pages), 2019.
[5] K. Ago, Y. Uwate and Y. Nishio, Influence of Local
Bridge on a Complex Network of Coupled Chaotic Circuits , Proceedings of International Symposium on Nonlinear Theory and its Applications (NOLTA’14), pp. 731734, 2014.

Figure 12: Comparison of synchronization rate for the threemodels.

[6] 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.
[7] N. Inaba, T. Saito and S. Mori, Chaotic Phenomena ina Circuit with a Negative Resistance and an Ideal Switchof Diodes , IEICE Trans., vol. E-70, pp. 744-754, 1987.

4. Conclusions
In this study, we investigate the synchronization of bidirectionally coupled chaotic circuits with non-uniform coupling strength. It is confirmed that the synchronization phenomena are kept even with non-uniform coupling strength.
Moreover, the non-uniform coupling strength showed thesynchronization phenomena even at weak coupling strengthcompared to when the coupling strength is weakened uniformly. It is also confirmed that by increasing the numberof circuits connected to CC1, the coupling strength at which
the synchronization phenomena collapse becomes smaller.
In the future, we would like to investigate the synchronization phenomena in system model with more meaningful directions by increasing the number of connection points with
non-uniform coupling strength. We would also like to checkthe synchronization phenomena when the connection state
between synchronized groups is changed.

References[1] M. G. Kitzbichler, M. L. Smith, S. R. Christensen andE. Bullmore, Broadband Criticality of Human Brain
Network Synchronization , PLoS Computational Biology (Volume: 5), pp. 1-13, 2009.
[2] A. Omri, M. Shaqfeh, A. Ali and H. Alnuweiri, Synchronization Procedure in 5G NR Systems , IEEE Access (Volume: 7), pp. 41286-41295, 2020.
[3] S. Masamura, T. Iwamoto, Y. Sugitani, K. Konishi and N.
Hara, Experimental investigation of amplitude death indelay-coupled double-scroll circuits with randomly timevarying network topology , Nonlinear Dyn (2020) 99,
pp. 3155-3168), 2019.
[4] K. Nakabai, T. Nara, Y. Uwate and Y. Nishio, Synchronization in Ladder-Coupled Chaotic Circuits Including Ring Structures , Proceedings of IEEE Interna157