IEEE Workshop on Nonlinear Circuit Networks
December 8-9, 2023

Synchronization Phenomena in Chaotic Circuits in
Unidirectional Coupling
Kohei Suzue, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan
Email: c612001195@tokushima-u.ac.jp , uwate, nishio@ee.tokushima-u.ac.jp
Abstract—In this study, we investigate the variation of synchronization phenomena with the coupling strength between thecircuits when the chaotic circuits, called Nishio-Inaba circuits, arecoupled in a unidirectional manner. As a result, when the couplingstrength is equal in two pairs, the synchronization rate is higher
in the pair with equal parameters. On the other hand, when thecoupling strength of the pair with different parameters was twicethat of the pair with the same parameters, the synchronization
rate of the two pairs became equal from a certain value.

I. I NTRODUCTION
Synchronization phenomena are numerous in nature. Thisphenomenon is exemplified by the metronome and the collective firing of neurons, the nerve cells of the brain. In
recent years, research on AI(Artificial Intelligence) has been
conducted extensively, and a variety of high-performance AI
products have been commercialized. However, current AI uses
linear circuit models to generate images and sort out similar
information, whereas we humans actually use nonlinear andaperiodic signals. Therefore, we believe that we can make AI
more similar to the human brain by using chaotic circuitsthat have the same characteristics as biological signals. In
this study, we investigate the synchronization phenomenon of
chaotic circuits as a basic research. The chaotic circuit usedin this study is the Nishio-Inaba circuit published in 1989 [1].
In a previous study, the synchronization phenomenon ofchaotic circuits in networks coupled by unidirectional coupling
was investigated using a chaotic circuit different from theNishio-Inaba circuit [2][3]. In that study, synchronization
phenomena of unidirectional chaotic circuits coupled in otherthan ring topology were investigated.
In this study, the synchronization phenomena of chaotic
circuits connected by unidirectional coupling from two chaotic
circuits were investigated using the Nishio-Inaba circuit. Inorder to investigate the synchronization phenomena in moredetail, three nodes were used to investigate the synchronization
phenomena with each node in the case where one node isaffected by two nodes. The synchronization rates at each parameter were investigated when the coupling strength between
the circuits was varied.

Fig. 1.

Circuit Model.

The circuit equations in Fig. 1 are given as follows:
di1

L1
= v + ri1 ,

dτ


di2
= v − vd ,
L2
dτ


 C dv = −i1 − i2dτ(1)

For Eq. (1) vd represents the voltage at the diode portionof Fig. 1, and its characteristic equation is defined as follows:
vd =rd2

i2 +VV− i2 −rdrd
(2)

The normalization parameters for normalizing Eq. (1) areas follows:

II. C IRCUIT AND S YSTEM M ODELr
i1 =CV xn ,L1r

√C,
L1
1
γ= ,R

α=r

L1 C
V yn , v = V zn ,L2
√
L1
L1 C
β=, δ = rd,
L2L2pdt = L1 Cτ, ” · ” =dτi2 =Figure 1 shows a schematic of the Nishio-Inaba circuit
used in this study.
The normalized equation for this circuit is given as Eq. (3).

-5-However, one count is defined as the solution of the circuit
equation by the Runge-Kutta method on the simulation.

ẋi = αxi + zi ,






 y˙i = zi − f (yi ),nXz˙=−x−βy−γi j(zi − zj ),
iii




i,j=0



f (yi ) = 2δ yi + 1δ − yi − 1δ(3)

Figure 2 shows the model of the system used in this study.
In this figure, a single Nishio-Inaba circuit is defined as asingle node.

Fig. 2.

III. S IMULATION R ESULTS
In this section, we describe the change in the synchronization rate for each value of α when the ratio of couplingstrengths is changed.
In this study, parameters other than α in the chaotic circuit
were set to values of β = 3.0 and δ = 470.The counts weretaken as all counts from 40000 to 100000 counts, except forcounts 0 to 39999 counts, which were expected to be transient.

System Model.

In this study, the value of α a parameter indicating thedegree of chaos in the system shown in Fig. 2, is set as follows.
Case1.

α1 = 0.425,

α2 = 0.415

Case2.

α1 = 0.420,

α2 = 0.410

Case3.

α1 = 0.425,

α2 = 0.405

Case4.

α1 = 0.430,

α2 = 0.400

In this system, the ratio of coupling strength γ1 to γ2 wasinitially set to 1:1. Next, the ratio of coupling strength γ1 toγ2 was set to 2:1. Then, the value of coupling strength γ1
was varied from 0 3.0 in 0.5 increments, and where a moredetailed value was desired, it was varied in 0.1 increments.
The synchronization rate of the pair of chaotic circuits usedin each of these couplings was measured using Eq. (4).
To increase the reliability of the data, the results are variedin five different initial values and the average of these values
is taken.
Si =countsyni
× 100
countall(4)

In Eq. (4), Si is defined as the synchronization rate between
node 2 and node i, countsyni is the number of counts inwhich node 2 and node i were synchronized, and countall is
all counts.
In measuring the synchronization rate, we compared thevalue of z in Eq. (3) for each chaotic circuit; we counted
cases where the absolute difference in the value of z for apair of chaotic circuits was smaller than 0.1, and measured
the synchronization rate by finding the percentage from thetotal counts.

Fig. 3.

Case 1 ((a) γ1 : γ2 = 1 : 1(b) γ1 : γ2 = 2 : 1).

Figure 3 shows the simulation results of Case 1: in Case
1, node 1 is the chaotic solution, and the other nodes have
3-period solutions. Therefore, node 2 contains the chaoticsolution, and S1 is higher regardless of the ratio of thecoupling strength. Furthermore, S3 is less than 10 in almost
all cases.
Figure 4 shows the simulation results for Case 2; in Case2, all nodes have chaotic solutions. In the case of the ratio ofcoupling strength of 1:1, it can be seen that S3 is more than
5% larger than S1 because the parameters are the same. In thecase of the ratio of coupling strength of 2:1, S3 , which is thesynchronization rate between node 3 and node 2 with the same
parameters, is larger at low values of coupling strength, but
as the coupling strength increases, S1 and S3 become equal.
Further increasing the coupling strength, S1 becomes larger
than S3 .
Figure 5 shows the simulation results for Case 3; in Case3, all nodes have chaotic solutions. In the case of the ratio ofcoupling strength of 1:1, it can be seen that S3 is more than
10% larger than S1 because the parameters are the same. In
the case of the ratio of coupling strength of 2:1, S3 is larger-6-Fig. 4.

Case 2 ((a) γ1 : γ2 = 1 : 1(b) γ1 : γ2 = 2 : 1).

Fig. 6.

Case 4 ((a) γ1 : γ2 = 1 : 1(b) γ1 : γ2 = 2 : 1).

than 0.17, S1 is greater than S3 .
IV. C ONCLUSIONSIn this study, the Nishio-Inaba circuit was used to investigate the synchronization rate when chaotic circuits are
unidirectionally coupled. First, for the same coupling strength,
the synchronization rate was higher for pairs with the same
parameters unless the solution of the chaotic circuit was a 3period solution. Second, when the coupling strength of pairs
with different parameters was doubled, the synchronization
rate was higher for pairs with the same parameters as in
the same coupling strength case while the coupling strength
was small. However, as the coupling strength increased, thesynchronization rate remained constant regardless of the parameters. In the future, we would like to investigate how
the synchronization rate changes with coupling strength bychanging various parameters.
R EFERENCES
Fig. 5.

Case 3 ((a) γ1 : γ2 = 1 : 1(b) γ1 : γ2 = 2 : 1).

for lower values of coupling strength because the parameters
are the same, but as the coupling strength increases, thesynchronization rate approaches the same value.
Figure 6 shows the simulation results for Case 4; in Case4, all nodes have chaotic solutions. In the case of the ratio ofcoupling strength of 1:1, it can be seen that S3 is more than 5%larger than S1 because the parameters are the same. In the caseof the ratio of coupling strengths of 2:1, the synchronization
rate has the same value if the value of the coupling strength γ1
is greater than 0.1. When the coupling strength γ1 is greater-7-[1] Yoshifumi Nishio, Naohiko Inaba, Shinsaku Mori, Toshimichi Saito,“ Rigorous Analyses of Windows in a Symmetric Circuit ”, IEEEInternational Symposium on Circuits and Systems, vol. 3, pp. 21512154, May 1989.
[2] Akari Oura, Kyohei Fujii, Yoko Uwate, Yoshifumi Nishio, “Analysis ofChaotic Circuit Networks with One-Way Coupling”, International SoC
Design Conference, pp. 168-169, Nov. 2018.
[3] Yuki Matsubara, Yuki Ishikawa, Yoko Uwate, Yoshifumi Nishio,“ Synchronization of Bi-Directionally Coupled Chaotic Circuits with NonUniform Coupling Strength ”, RISP International Workshop on Nonlinear Circuits, Communications and Signal Processing, pp. 154-157, Feb.
2023.