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

Synchronization of Chaos Networks by Changing Layout Distribution
Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: {uwate, nishio}@ee.tokushima-u.ac.jp
Abstract

model when it is applied to three different circuit distribution
layouts. In the case of the proposed model, it was found thatIn this study, we investigate the synchronization rate of it is possible to extract the central part by using synchronizachaotic circuits for different parameters of the Gaussian dis- tion information of the circuit distribution even if the circuit
tribution of the circuit arrangement when one clustering is distribution changes.
used. We analyze the effectiveness of the proposed clusteringby chaotic circuit network for any circuit layout distributions.
2. Chaotic Circuits Model
1. Introduction

In this section, we explain the chaotic circuit model. Figure
1 shows the chaotic circuit model that has been investigatedin the literature [8]-[10]. This circuit consists of three memory elements, one linear negative resistance element, and one
nonlinear resistance element consisting of two diodes. The
negative resistance is realized using the linear region of a negative impedance converter made from an operational amplifier.

Chaotic systems are very simple, but they produce surprisingly complex signals. Therefore, it seems impossible fortwo chaotic systems to synchronize, but they can synchronize
if the two systems exchange information in an appropriate
way. The fundamentals of chaotic synchronization have beenstudied with regard to the geometry and stability of synchronization [1], [2]. Examples of applications using chaos synchronization include secure communication and image encryption [3], [4].
It is known that chaotic synchronization is also observedin coupled systems of chaotic circuits realized in electroniccircuits. In the study of coupled chaotic circuit networks,
chaotic circuits are in many cases mathematical models. We
believe it is important to investigate the phenomena observed
from chaos in electronic circuits for engineering applications.
Our research group has proposed a clustering method using(a) Chaotic circuit.
(b) Chaotic attractor.
synchronization of chaotic circuit networks [5]. As an advanced version of this method, we have proposed a method
Figure 1: Chaotic circuit model.
that enables more difficult clustering in which the coupling
between chaotic circuits varies with the synchronization rate,
and we have confirmed the effectiveness of this method [6],The approximate I − V characteristics of the nonlinear[7]. By using computer simulations, we confirmed the effec- resistance element are indicated by the following equation,
tiveness of the clustering accuracy of the proposed method. where the parameter rd is the slope of the nonlinear resisA discussion of the results shows that when the circuit ar- tance.
rangement is generated to follow a Gaussian distribution, therdVV
clustering accuracy depends on the parameters of the Gausvd (i2 ) =.
(1)
− i2 −
i2 +2rdrd
sian distribution.
In this study, we focus on the synchronization performance
By changing the variables, such thatof the proposed model (coupled chaotic circuits network withr√learning) by using the several types of circuit layout distribuC
L1 CV x; i2 =V y; v = V z;
i1 =tions. We compare the proposed model and the conventional
L1L22022023 RISP International Workshop on Nonlinear Circuits, Communications and Signal
Processing (RISP NCSP 2023), Feb. 28 - Mar. 3, 2023

r

√
pC
L1 C
L1= α;= β; rd= δ; t = L1 Cτ
L1L2L2(2)The normalzied equations represent the circuit equations
when all the chaotic circuits are coupled globally with eachother (all-to-all coupling).
dxi
= αxi + zidτdyi= zi + f (y)dτ(3)NXdziγij (zi − zj )= −xi − βyi −dτ

j=1


(i, j = 1, 2, · · ·, N )(4)Y

where f (y) is described as follows:δ
11f (y) =y+.
− y−2
δδ

Figure 2: Gaussian distribution depending on σ.
2

2

1.5

1.5

1

1

0.5

0.5

0-0.5

In the computer simulations, we set the parameters to be
α = 0.460, β = 3.0 and δ = 470. The characteristic function
f (y) can be described as a three-segment piecewise-linearfunction. In this study, the value of γij reflects the distance
between the circuits in an inverse manner, as described using
the following equation:

0Y

r-0.5

-1

-1-1.5-1.5-2-2-2-1.5-1-0.500.5

11.5

2-2-1.5-1-0.5

X

00.5

11.5

2

X

(a) σ=0.3.

(b) σ=0.5.

2

1.5

1

Y

0.5

γij =g(dij )20-0.5

.

(5)

-1

-1.5

-2-2-1.5-1-0.500.5

11.5

2XHere, dij denotes the Euclidean distance between the i−thcircuit and the j − th circuit, while g is a scaling parameterthat determines the coupling strengths.

(c) σ=0.7.
Figure 3: Circuit Arrangement.

3. Circuit Arrangement and Learning Process
In our previous studies, we have confirmed that the proposed method which is coupled chaotic network with learning has a high ability to extract cluster centers. Therefore, we
consider three types of circuit arrangements with parameterσ generated by a Gaussian distribution. Figure 2 shows threetypes of Gaussian distributions when the parameter σ is set to0.3, 0.5 and 0.7. The three types of circuit arrangements areshown in Fig. 3. As the value of σ increases, the circuits are
spread out and arranged in a two-dimensional plane.
Next, we explain about learning process of coupled chaotic
circuits networks. We have proposed a new clustering methodthat applies Hebbian rule by changing the coupling strength
depending on the synchronization rate. The Hebbian rulestates that synapses, which connect neurons become more efficient when neurons fire repeatedly, and less efficient when
they do not fire for long periods of time. We apply this Hebbian rule to chaotic circuit synchronization. In other words,

the coupling between the synchronized chaotic circuits ismade stronger, and the coupling between the un-synchronizedchaotic circuits is made weaker. The Hebbian rule is applied
to the chaotic circuits network as following steps.
[step-1] At the initial state, all nodes are fully connected withcoupling strengths depending on distance.
[step-2] After a transient phase, we apply two rules for asequence of generations. Each generation has length τh =10, 000.

203• (Determination of synchronization:) In order to determine whether two nodes are alike, we calculate the synchronization ratio for every pair of oscillators. If the synchronization ratio is larger than 60%, the corresponding
coupling strength becomes stronger with ∆γ = 0.0001.
In order to analyze the synchronization ratio, we define2023 RISP International Workshop on Nonlinear Circuits, Communications and Signal
Processing (RISP NCSP 2023), Feb. 28 - Mar. 3, 2023

a synchronization state as

4000
Proposed model
Conventional model3500|zk − zn | < 0.3 (k ∈ Sn )Number of edges

3000[step-3] Step-2 is repeated until 100 iterations are reached(G = 100).
[step-4] At the final state (G = 100), we check the synchronization ratio for every pair of oscillators.
A flowchart of these steps is shown in Fig. 4.

2500
2000
1500
1000
500
00

102030

4050

60

7080

90100

GenerationStart(a) σ=0.3.
4000

G < 1003500

Proposed model
Conventional model

Number of edges

3000

Runge-Kutta

2500
2000
1500
1000
500

τ>10,000 &&τ%10,000==0Yes0No

01020405060

7080

90

100

Generation(b) σ=0.5.

1) Determination of synchronization
voltage differences: fabs(xi-xj) < 0.3
synchronization ratio > 60%

Applying
Learning

3040003500

Proposed model
Conventional model

3000
Number of edges2) Updating the weights
synchronization edge: γ +=0.0001
un-synchronization edge: γ -=0.0001
End2500
2000
150010003) Updating the generation
G++

500
0010

20

30

405060Generation

708090100(c) σ=0.7.

Figure 4: Flowchart of learning process.

Figure 5: Time evolution of number of synchronized edges.

4. Simulation Results
Figure 5 shows the simulation results of number of synchronized edges with generation. First, in the conventionalmodel, the number of synchronized edges is constant withina certain range of vibrations, even as the number of generators increases. In contrast, for σ = 0.3 in the proposed model,
the number of synchronized edges increases with the generations, peaking once around G = 40. Although it decreases
after the peak, the number of synchronized edges is significantly larger than in the conventional model. For σ = 0.5
and 0.7 of the proposed model, it can be seen that the number of synchronized edges also increases with the generators.
In both models, the number of synchronized edges is higher
when the value of σ is small. These results show that thenumber of synchronized edges is increased by updating theweights of the learning effects of the proposed model.

The results of displaying synchronized edges of the proposed and conventional models are shown in Figs. 6 and 7.
The results show that the proposed model has more synchronized edges where the density of circuits is higher. Thus, inthe case of σ=0.3, most nodes are synchronous. However,in the case of σ=0.7, nodes located away from the center areasynchronous. In the case of the proposed method, it can beseen that as σ increases, the range covered by the synchronization edges also expands. In the case of the conventional
method, when σ is set to 0.3, the majority of the circuit is synchronized. As σ increases, the number of synchronous edges
depends on the initial state and does not capture the characteristics of the cluster well.

204

22

2

2

1.5

1.5

1.5

1.5

11

1

1

0.5

0.5

0.5

0.5

-0.5

-0.5

00Y0Y0YY2023 RISP International Workshop on Nonlinear Circuits, Communications and Signal
Processing (RISP NCSP 2023), Feb. 28 - Mar. 3, 2023

-0.5

-0.5

-1-1-1-1-1.5

-1.5

-1.5

-1.5

-2-2-2-1.5-1-0.500.5

11.5

2-2-2-1.5

-1X

-0.5

00.5

11.5

2-1.5

-1-0.500.5

11.5

2-2-1.5-1X(b) σ=0.5.

(a) σ=0.3.

2

2

1.5

1.5

1

10.5

0.5

0-0.5

-0.5

00.5

11.5

2XYY

(a) σ=0.3.

-2-2X(b) σ=0.5.

0-0.5-1-1-1.5-1.5-2-2-2-1.5-1-0.5

00.5

1

1.5

2-2

X-1.5

-1-0.5

00.5

11.5

2X

(c) σ=0.7.

(c) σ=0.7.

Figure 6: Clustering results of the proposed model with learn- Figure 7: Clustering results the conventional model withouting.
learning.
5. Conclusion
In this study, we investigated synchronized edges of coupled chaotic circuits network using learning process. Bychanging the parameter of Gaussian distribution of circuit
layout, the proposed model with learning can extract the center of cluster. However, it was difficult to observe synchronization of the circuits far from the center. We would liketo study in the future a learning algorithm that synchronizes
even when the density of circuit layout is different. Furthermore, we hope to apply the clustering methods developed
here to data mining, image processing and other applications
in real life situations.
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