2011 International Workshop on Nonlinear Maps and their Applications
NOMA'11, Evora, Portugal, September 15-16, 2011

Clustering Phenomena Obtained from CoupledRulkov MapsYuji Takamaru, Hiroshige Kataoka, Yoko Uwate and Yoshifumi Nishio
Department of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, 770-8506, Japan
Email: {takamaru, hiroshige, uwate, nishio}@ee.tokushima-u.ac.jp
3

Abstract— In this study, we investigate clustering phenomena
observed in coupled map lattice (CML). A two dimensional map
which produces spiking-bursting behavior like real biologicalneurons is used for CML. By using computer simulations, weconfirm that the data are separated to several groups when adiﬀerent parameter which depends on the data information isset for each map.

2Xn+1

1I. Introduction

II. Rulkov Map
In recent years, the two dimensional map with the interesting phenomena was proposed by Rulkov [4]-[6] (see Fig. 1).
This map produces spiking-bursting neural behavior.
The expression of the Rulkov map is described as following:==f (xn , xn−1 , yn )yn − µ(xn + 1) + µσ + µσn ,Pk-1

Coupled map lattice (CML) is well known as mathematicalmodels in discrete-time system. CML has proposed by Kaneko[1], to use as general models for the complex high-dimensional
dynamics, such as biological systems, networks in DNA,economic activities, neural networks, and evolutions.
In recent years, we often deal with huge amounts of data.
It is useful to partition such data sets into clusters of items
that share similar properties. The clustering algorithms using
the CML have proposed [2], [3]. Here, the logistic map waschosen as the simple chaotic map.
In this study, we consider a two dimensional map (Rulkovmap) which produces spiking-bursting behavior like real biological neurons [4]-[6]. First, we investigate basic synchronization phenomena when two identical Rulkov maps arecoupled. In this case, the in-phase synchronization with same
amplitude can be observed. Next, the control parameter of one
side map is varied. We confirm the in-phase synchronization
with diﬀerent amplitude depending on value of the control
parameter. We focus on the amplitude of map for clustering
algorithm. By using computer simulations, we confirm that thedata are separated to several groups when a diﬀerent parameter
which depends on the data information is set for each map.

xn+1
yn+10(1)where x is the fast and y is the slow dynamical variable. µ= 10−3 and α and σ denotes the control parameters of themaps. And parameters g shows the coupling weight between

Xpu-2-3-3

Xps

-2-1

0
Xn

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Fig. 1. Rulkov map. The dashed line illustrates a super stable cycle Pk .
The stable and unstable fixed points of the map are indicated by x ps and x pu ,respectively.

the maps. The nonlinear function f (xn , xn−1 , yn ) is describedas following:α/(1 − xn + u), (xn ≤ 0)f (xn , xn−1 , yn ) = 
α + u,(0 < xn < α + u and xn−1 ≤ 0)−1,(xn ≥ α + u or xn−1 > 0)(2)
σ is set to random value, and α is the parameter to control
the fast variable.
III. Two Coupled Rulkov Maps
First, we consider the basic synchronization phenomena
when two Rulkov maps are coupled as chain topology. The
equation of the chain of Rulkov maps is described as:
xi,n+1 = f (xi,n , xi,n−1 , yi,n )1+ g(xi+1,n − 2xi,n + xi−1,n )
2
yi,n+1 = yi,n − µ(xi,n + 1) + µσ + µσi,n1+ g(xi+1,n − 2xi,n + xi−1,n )(3)
2
where parameter g denotes the coupling weight of the between
the maps. Each parameters of x1 , x2 sets individually. When
α1 and α2 are equal. The waveform of each maps are shownin Fig. 2. The in-phase synchronziation with same amplitude
is observed.

- 88 -Next, we vary the control parameter as follows; α1 = 10.0
and α2 = 20.0. The waveforms of each maps are shown inFig. 3. From this figure, we can see that two maps havediﬀerent amplitude with same phase. Namely, we observe thein-phase synchronization with diﬀerent amplitude dependingon value of the control parameter. We focus on the diﬀerent
amplitude of map for clustering algorithm.
IV. Three Coupled Maps with Distance Information
x1 (α1 = 10.0).

Next, we consider the case that three coupled maps placedon a two dimension area including the positional information.
Furthermore, from synchronization results we investigate theclustering phenomena by using the positional information. The
alignment of three maps is shown in Fig. 4.

x2 (α2 = 10.0).
Fig. 2.

Two coupled maps with same parameters.

Fig. 4.

x1 (α1 = 10.0).

The alignment of three maps.

The control parameter α is fixed with distance information.
We determine the parameters according to the distance fromthe reference point. x1 and x2 are placed with close distance,then the control parameter α is fixed as α1 =α2 =10.0. And x3
is placed with far distance form x1 and x2 , then the control
parameter α3 is set to α3 =20.0. The time waveforms obtained
form each map are shown in Fig. 5.
From these results, we can see that the amplitude of third
map has diﬀerent value from first and second maps. The
maximum values of the amplitude of each map are x1 = 4.78,
x2 = 4.78 and x3 = 12.15. In this case, we can cluster three
coupled maps into two groups between x1 , x2 and x3 by using
the value of the amplitude (see Tab. 1).
TABLE IGroup of three coupled maps.
Mapx1
x2x3

x2 (α2 = 20.0).
Fig. 3.

Two coupled maps with diﬀerent parameters.

- 89 -Amplitude
4.78
4.78
12.15

Group
112

x1 (α1 =10.0).

Fig. 6.

The alignment of ten maps.

x2 (α2 =10.0).

x3 (α3 =20.0).
Fig. 5.

(a) Group 1 (x1 , x2 , x3 , x4 , x5 )Three coupled maps

V. Application for General Networks
Finally, we consider the case of general networks. Ten maps
are placed randomly on the two-dimensional are including
the positional information. k nearest neighbors is used as theway of coupling between the maps. In this coupling method,
the number of k maps which are placed with k-th closest
distance from the reference map are connected. In this study,
the parameter of k is set to k = 2. The figure of alignment ofcoupled ten maps is shown in Fig. 6.
We fix first map (x1 ) as the standard map and the parameterα1 is set to α1 = 10. The other parameters of α sets by using
the following equation.
αi =(20 − 10) · didmax(b) Group 2 (x6 , x7 )(4)

The parameter of di is the distance from the standard pointand dmax is the most farthest distance from the standard point.
So the other parameters α of the maps are set to α2 = 10.67,
α3 = 10, α4 = 12.33, α5 = 10, α6 = 15.66, α7 = 15.33,
α8 = 16.34, α9 = 20, and α10 = 19. Simulation results areshown in Fig. 7.

- 90 -(c) Group 3 (x8 , x9 , x10 )Fig. 7.

General Networks.

The maximum values of the time waveforms of ten maps
are x1 = 4.71, x2 = 5.18, x3 = 4.71, x4 = 6.35, x5 = 4.71,
x6 = 8.80, x7 = 8.55, x8 = 9.31, x9 = 12.12, and x10 = 11.35.
In this case, we can cluster into three groups, group 1 is x1 ,x2 , x3 , x4 and x5 , group 2 is x6 and x7 , group 3 is x8 , x9 andx10 (see Tab. 2).
TABLE IIGroup of ten coupled maps
Mapx1
x2
x3
x4x5x6x7x8x9x1 0

Amplitude
4.71
5.18
4.71
6.35
4.71
8.80
8.55
9.3112.12
11.35

Group
111
1
1222
33

VI. Conclusions
In this study, we have investigated clustering phenomena
observed in CML. We used the Rulkov map which produces
spiking-bursting behavior like real biological neurons. By
using computer simulations, we confirmed that the data areseparated to several groups when a diﬀerent parameter which
depends on the data information is set for each map. In ourfuture works, we study large-scale CML, and propose eﬃcientalgorithm for clustering by using synchronization mechanism.
References[1] K. Kaneko, “Spatiotemporal Intermittency in Coupled Map Lattice”, Prog.
Theor. Phys., vol75, no.5, pp.1033-1044, 1985.
[2] L. Angelini, F. De Carlo, C. Marangi, M. Pellicoro, and S. Stramaglia,“Clustering Data by Inhomogeneous Chaotic Map Lattice”, PhysicalReview Letters, vol. 85, no. 3, pp. 554-557, 2000.
[3] T. Ott, M. Christen and R. Stoop, “An Unbiased Clustering Algorithm
Based on Self-organization Processes in Spiking Neural Networks”, Proc.
of NDES’06, p. 143-146, 2006.
[4] N. F. Rulkov, I. Timofeev, and M. Bazhenov, “Oscillations in LargeScale Cortical Networks: Map-Based Model”, Journal of Computational
Neuroscience vol. 17 pp. 203-223, 2004.
[5] N. F. Rulkov, “Modeling of Spiking-Bursting Neural Behavior using Twodimensional Map”, Physical Rev., E, vol. 65 041922, 2002.
[6] A. L. Shilnikov and N. F. Rulkov “Origin of Chaos in a Two-Dimensional
Map Modeling Spikin-Bursting Neural Activity”, International Journal ofBifurcation and Chaos, vol. 13, No. 11 (2003) pp. 3325-3340.

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