Clustering in Globally Coupled Chaotic Circuitswith Changing Weights
Yoko Uwate and Yoshifumi Nishio
Dept. of E. E. Eng., Tokushima University
2-1 Minami-Josanjima Tokushima
770-8506 Japan
Email: {uwate, nishio}@ee.tokushima-u.ac.jp
Abstract— In this study, we focus on clustering phenomena in a
network composed of coupled chaotic circuits. In this circuit
network, the coupling strength is reflected by the distanceinformation when the chaotic circuits are placed in a twodimensional space. We propose a method that applies Hebbianrule to synchronization. Namely, the coupling weights are changed
depending on synchronization states. By using the computer
simulations, it is confirmed that the proposed methods is more
effective than the standard method for clustering.

II.

CIRCUIT MODEL AND CIRCUIT ARRANGEMENTS

In this section, we explain the chaotic circuit model. Figure
1 shows the chaotic circuit model that has been investigated inthe literature. 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.

Keywords; synchronization; coupled circuits; clustering

I.

INTRODUCTION

In recent years, we need to deal with increasing amounts of
information in our daily lives. In order to structure and analyze
such data, it is effective to divide each data into clusters. The
goal of a clustering algorithm is to find data clusters that consistof similar elements. Previously, many clustering studies havebeen performed using discrete time models, such as coupled map
lattices (CMLs) and self-organizing maps (SOMs). However,few studies of clustering have been performed using acontinuous time model. Therefore, this study focuses on thestudy of clustering phenomena in continuous-time models using
actual electronic circuits.

Fig. 1 Chaotic circuit.

The following equations represent the circuit equations
when all the chaotic circuits are coupled globally with eachother (globally coupling).

Coupled chaotic circuits can be realized using electroniccircuits and various interesting phenomena can be observed inthese circuits. In recent years, many studies have reported on
application of the clustering and synchronization phenomenathat can be observed in coupled chaotic circuits to naturalsciences. The reason for this interest is that the characteristics ofthe chaos phenomena observed in coupled chaotic circuits also
exist in real life, in phenomena such as human behavior,emotions and heartbeats. Coupled chaotic circuits thus have thepotential to be applied to a variety of different fields. In ourprevious studies, we proposed several types of clustering
methods [1], [2] and recently we proposed a new clustering
method based on new chaotic coupled circuit network applying
Hebbian rule [3]. By using computer simulations, we confirmedthat chaotic circuit network with changing weights is more
effective than the conventional chaotic circuit network.
However, the circuit layout was so complex that it was difficult
to evaluate the accuracy of the clustering.

In the computer simulations, we set the parameters ofcircuit system to be0.460,
3.0 and470. The
characteristic function
can be described as a threesegment piecewise-linear function. In this study, the value ofreflects the distance between the circuits in an inverse
manner, as the following equation:

In this study, we focus on the accuracy of clustering by using
coupled chaotic circuits with Hebbian rule.

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clusters. In contrast, the conventional model has many lines
between clusters, therefore it is difficult to determine theclusters. These results show that changing the weights of thecouplings, it is effective for clustering.

Here,denotes the Euclidean distance between -th
circuit and -th circuit, where is a scaling parameter that
determines the coupling strengths.
Figure 2 shows the circuit arrangement using 100 chaotic
circuits. There are three clusters.

Figure 4 shows the number of edges with highsynchronization rates in each category. It can be seen that theconventional model has more synchronized edges in all
categories. However, the number of miss categories (edges
between clusters) is also high, and the accuracy of the clusters is
not good. In contrast, the proposed model has fewer edges in themiss category, which can be said to be good for clustering
accuracy.

Fig. 2 Circuit arrangement. (N=100, the number of clusters:
3, red points: 50 (cluster-1), green points: 30 (cluster-2), blue
points: 20 (cluster-3)).

III.

PROPOSED ALGORITHM

In this study, we propose a new clustering method thatapplies Hebbian rule as well as the determination of clusters bysynchronization of chaotic circuit networks. 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 Hebbianrule to chaotic circuit synchronization. In other words, thecoupling between the synchronized chaotic circuits is madestronger, and the coupling between the unsynchronized chaotic
circuits is made weaker. The Hebbian rule is applied to thechaotic circuits network as following steps.

(a) Proposed model.
(b) Conventional model.
Fig. 3 Examples of clustering results.

[step-1] At the initial state, all edges are fully connected withthe coupling strength depending on distance.
Fig. 4 Number of edges.

[step-2] After a transient phase, we apply two rules for asequence of generations.

V.

(check synchronization:) In order to check whether twonodes are alike, we calculate the synchronization ratio forevery pair of oscillators. If the synchronization ratio islarger than 60%, the corresponding coupling strength
becomes stronger with ∆0.01 . While thesynchronization ratio is smaller, the coupling strength
becomes weaker with ∆0.01.

In this study, we focus on a clustering method based on a
new chaotic coupled circuit network applying Hebbian rule inorder to achieve more complex and advanced clustering. By
using computer simulations, we confirmed that chaotic circuit
network with the Hebbian rule is more effective than theconventional chaotic circuit network for clustering.

[step-3] step-2 is repeated until 10 iterations are reached.

REFERENCES[step-4] At the final state, we check the synchronization ratio
for ever pair of chaotic circuits.
IV.

[1]

CLUSTERING RESULTS[2]

Figure 3 shows one example of clustering results obtainedby the proposed and conventional models. The lines in the figure
represent edges between circuits with high synchronization
rates. In the proposed model, there are many lines within

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CONCLUSION

[3]

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K. Ago, Y. Uwate and Y. Nishio, “Investigation of Clustering in Complex
Networks Using Coupled Chaotic Circuits,” Proc. of IFAT’15, pp. 101102, 2015.
K. Oi, Y. Uwate and Y. Nishio, “Synchronization and Clustering inCoupled Parametrically Excited Oscillators with Small Mismatch,” Proc.
of ISCAS’15, pp. 910-913, 2015.
Y. Uwate, T. Ott and Y. Nishio, “Synchronization and Clustering ofChaotic Circuit Networks with Hebbian Rule,” Proc. of NCSP’22, pp.
373-376, 2022.

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