2022 International Symposium on Nonlinear Theory and Its Applications,NOLTA2022, Virtual, December 12-15, 2022

Clustering Using Chaos Synchronization with Learning Algorithm
Yoko Uwate†, Martin Schule ‡, Thomas Ott ‡ and Yoshifumi Nishio††Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami Josanjima, Tokushima, Japan
Tel: +81-88-656-7662, Fax: +81-88-656-7471
Email: {uwate, nishio}@ee.tokushima-u.ac.jp‡Institute of Applied Simulation, Zurich University of Applid Sciences
Einsiedlerstrasse 31a, 8820 Waedenswil, Switzerland
Email: {scli, thomas.ott}@zhaw.ch
Abstract—
In our previous studies, we have proposed the method ofclustering using synchronization of coupled chaotic circuits
networks. It was conﬁrmed that applied learning algorithm
like Hebbian learning is eﬀective for advanced clustering
when the data patterns of clusters are overlapped.
In this study, we investigate the eﬀect of couplingstrength of whole network for clustering performance.
Also, the eﬀect of the learning parameters is investigated.
From computer simulations results, we conﬁrm that clustering with learning algorithm has no eﬀect for the clustering error.
1. Introduction
Some observed phenomena from coupled oscillatory
networks can be a good models to express the physical phenomena in our living life. Therefore many researchers investigate models using coupled nonlinear circuits and observe several interesting phenomena.
In our previous studies [1]-[4], we have investigatedclustering in two-dimensional networks of complex chaoticcircuits, where the coupling strength reﬂects the network
distance information. We showed that circuits that are arranged close to each other can achieve phase synchronization, whereas coupled circuits located far away from eachother cannot be synchronized. We also observed that thesenetworks of coupled chaotic circuits can be split into diﬀerent synchronized groups, thus revealing a clustering phenomenon. To achieve more complex and advanced clustering, we proposed a clustering method based on a new
chaotic coupled circuit network applying learning rule. Fora more complex clustering example, we consider a circuit
layout with overlapping clusters. By using computer simulations, we conﬁrmed that chaotic circuit network with thelearning rule is more eﬀective than the standard chaotic circuit network.
In this study, we investigate the eﬀect of the scaling parameter of the coupling strength for the whole network. We

conﬁrm that in the case of the method without learning rule,the number of miss edges between the two clusters are increased with the scaling parameter of the coupling strength.
While, in the case of the method with learning rule, there
are no aﬀection of miss edges.
2. Chaotic Circuits Model
The chaotic circuit model and obtained chaos attractor
are shown in Fig. 1. The chaotic circuit has been investigated in the literature [5]-[7]. This circuit consists ofthree memory elements, one linear negative resistance element, and one nonlinear resistance element consisting of
two diodes. The negative resistance is realized using thelinear region of a negative impedance converter made froman operational ampliﬁer.

(a) Chaotic circuit.

(b) Chaotic attractor.

Figure 1: Chaotic circuit model.
The approximate I − V characteristics of the nonlinear
resistance element are indicated by the following equation,
where the parameter rd is the slope of the nonlinear resistance.

ORCID iDs First Author:0000-0002-2992-8852, Second Author:
0000-0002-9379-4887, Third Author:
0000-0003-1635-7816, Fourth
0000-0002-0247-0001
Author:vd (i2 ) =!
VVrd
i2 +− i2 −.
2rdrd

By changing the variables, such thatr
i1 =CV x; i2 =L1√

L1C
Vy; v = Vz;
L2This work is licensed under a Creative Commons Attribution NonCommercial, No Derivatives 4.0 License.

– 552 –(1)

rr

√pC
L1L1C= α;= β; rd= δ; t = L1CτL1L2L22(2)1.5The normalized circuit equations can be described by

0.5y

 dx


= αx + zdτ dy= z − f (y)dτ dz = −x − βydτ

1(3)

-0.5

-1-1.5"cluster1.dat""cluster2.dat""cluster3.dat"

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

-2-2

(4)

The following equations represent the circuit equations
when all the chaotic circuits are coupled globally with eachother (all-to-all coupling).
 dxi
= αxi + zidτ
dyi
= zi + f (y) dτ(5)NXdziγi j (zi − z j )= −xi − βyi −
dτ

j=1

(i, j = 1, 2, · · ·, N)
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 piecewiselinear function. In this study, the value of γi j reﬂects thedistance between the circuits in an inverse manner, as described using the following equation:

γi j =g.
(di j )20-1.5-1-0.5

0x

0.5

11.5

2

Figure 2: Circuit arrangement with overlapped clusters(N=100, the number of clusters: 3, red points: 30 (cluster1), green points: 30 (cluster-2), blue points: 40 (cluster-3)).
synchronization of chaotic circuit networks. The Hebbianrule states 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 synchronizedchaotic circuits is made stronger, and the coupling between
the un-synchronized chaotic 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 connectedwith coupling 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.
• (check synchronization:) In order to check whether
two nodes are alike, we calculate the synchronization
ratio for every pair of oscillators. If the synchronization ratio is larger than 80, 60, and 50%, the corresponding coupling strength becomes stronger with
∆γ = 0.00001. And, if the synchronization ratio issmaller than 20, 40, and 49the corresponding couplingstrength becomes weaker with ∆γ = −0.00001.

(6)Here, di j denotes the Euclidean distance between the i −th circuit and the j−th circuit, while g is a scaling parameterthat determines the coupling strengths.

In order to analyze the synchronization ratio, we deﬁne a synchronization state as

3. Clustering Arrangement and Learning
The circuit arrangement with overlapped clusters isshown in Fig. 2. This time, we consider a more diﬃcultcase: a circuit layout with overlapping clusters. The number of clusters is set to three, and the chaotic circuits are
randomly placed according to a normal distribution fromthe center of each cluster.
The purpose of clustering for this circuit layout is not todivide it into three clusters, but to extract as many nodes aspossible from the central part of the cluster.
In this study, we use clustering method that applies Hebbian rule [8] as well as the determination of clusters by|xk − xn | < 0.3(k ∈ S n )[step-3] Step-2 is repeated until 10 iterations are reached(H = 10).
[step-4] At the ﬁnal state (G = 10), we check the synchronization ratio for every pair of oscillators.
4. Simulation Results
Figures 3 and 4 show the example of clustering resultsof chaotic circuit network with/without learning rule. The– 553 –ber of high-synchronized edges are increased with the scaling parameter g.

800

Non-hebb
Hebb-aHebb-b
Hebb-c700

600

Number of edges

lines in the ﬁgure represent edges between circuits withhigh synchronization rates. In the method with learning
rule, there are many lines within clusters. In contrast, themethod without learning rule has many lines between clusters, therefore it is diﬃcult to determine the clusters. Theseresults show that changing the weights of the couplings, it
is eﬀective for clustering.
21.5

500

400

300

12000.5

y10000
Total-0.5

Clusters

MissCluster-1

Cluster-2

Cluster-3(a) g = 0.00006.

-1
800"cluster1.dat""cluster2.dat""cluster3.dat""edge_hebb.dat"-1.5

Non-hebb
Hebb-aHebb-b
Hebb-c700-2-1.5-1-0.5

0x0.5

1

1.5

2600

Number of edges-2

Figure 3: Example of clustering result with learning rule(g = 0.00008).

500

400

300

200

100

20
TotalClusters

Miss

Cluster-1

Cluster-2

Cluster-3

1.5(b) g = 0.00008.

1y0.5

800

Non-hebb
Hebb-aHebb-b
Hebb-c700

0600

Number of edges-0.5-1"cluster1.dat""cluster2.dat""cluster3.dat""edge_nohebb.dat"

-1.5

500

400300-2-2-1.5-1-0.5

0x0.5

11.5

2200

100

0TotalFigure 4: Example of clustering result without learning rule(g = 0.00008).
Figure 5 shows the number of edges with high synchronization rates in each category. It can be seen that themethod without learning rule has more synchronized edges
in all categories. However, the number of miss categories(edges between clusters) is also high, and the accuracy ofthe clusters is not good. In contrast, the method with learning rule has fewer edges in the miss category, which can besaid to be good for clustering accuracy.
Furthermore, in the case of the method without learning
rule, the number of miss edges between the two clusters
are increased with the scaling parameter of the couplingstrength. While, in the case of the method with learning
rule, there are no aﬀection of miss edges. Totally, the num-Clusters

Miss

Cluster-1

Cluster-2

Cluster-3(c) g = 0.00010.
Figure 5: Clustering results of number of edges.

Finally, we investigate the detection ratio of clusters bycounting the nodes which has edge within same cluster.
The simulation results are shown in Fig. 6. We conﬁrm thatthe detection ratio of cluster is increased when the scaling
parameter of the coupling strength has large value. However, it was diﬃcult to observe the any diﬀerences depending on the learning parameters. Therefore, we would liketo investigate the eﬀect of the learning parameters in detailas the future work.

– 554 –

5. Conclusion

1In this study, we used the clustering method based ona chaotic coupled circuit network applying Hebbian rulein order to achieve more complex and advanced clustering. We also investigated the eﬀect of coupling strengthof whole network for clustering performance. By using
computer simulations, we conﬁrmed that chaotic circuit
network with the learning rule is more eﬀective than thestandard algorithm.
In future work, we would like to study clustering phenomena in large-scale networks. Furthermore, we hope toapply the clustering methods developed here to data mining, image processing and other applications in real life situations.

Non-hebb
Hebb-aHebb-b
Hebb-c

0.95

Ratio

0.90.85

0.80.75

References
0.7c1c2c3[1] Y. Takamaru, H. Kataoka, Y. Uwate & Y. Nishio,“Clustering Phenomena in Complex Networks ofChaotic Circuits,” Proc. ISCAS’12, Vol. 22, pp. 320334, 2012.

(a) g = 0.00006.
1Non-hebb
Hebb-aHebb-b
Hebb-c0.95[2] Y. Takamaru, Y. Uwate, T. Ott & Y. Nishio, “Dependence of Clustering Patterns on Density of ChaoticCircuits in Networks,” RISP Journal of Signal Processing, vol. 17, no. 4, pp. 103-106, Jul. 2013.

Ratio

0.90.85

0.8[3] K. Ago, Y. Uwate & Y. Nishio, “Investigation of Clustering in Complex Networks Using Coupled ChaoticCircuits,” Proc. of IFAT’15, pp. 101-102, 2015.

0.75

0.7
c1

c2

c3

[4] K. Oi, Y. Uwate & Y. Nishio, “Synchronization andClustering in Coupled Parametrically Excited Oscillators with Small Mismatch,” Proc. of ISCAS’15, pp.
910-913, 2015.

(b) g = 0.00008.
1

Non-hebb
Hebb-aHebb-b
Hebb-c0.95[5] Y. Nishio, N. Inaba, S. Mori & T. Saito, “Rigorous
Analyses of Windows in a Symmetric Circuit,” IEEE
Transactions on Circuits and Systems, vol. 37, no. 4,
pp. 473-487, Apr. 1990.

Ratio

0.9

0.85

0.8

[6] C. Bonatto & J. A. C. Gallas, “Periodicity Hub andNested Spirals in the Phase Diagram of a Simple Resistive Circuit,” Phys. Rev. Lett., 101, 054101, Aug.
2008.

0.75

0.7
c1

c2

c3(c) g = 0.00010.
Figure 6: Clustering results of detection ratio of clusters.

[7] A. Koseska, E. Volkov, & J. Kurths, “Real-World
Existence and Origins of the Spiral Organizationof Shrimp-Shaped Domains,” Phys. Rev. Lett., 105,074102, Aug. 2010.
[8] D. O. Hebb, “Organization of Behavior,” 1949.

– 555 –