2010 International Workshop on Nonlinear Circuits,Communication and Signal Processing
NCSP'10, Waikiki, Hawaii, March 3-5, 2010

Synchronization Phenomenon in Small World Network of Coupled Cubic Maps
Yumiko Uchitani, Hironori Kumeno and Yoshifumi Nishio
Tokushima University
2-1,Minami-Josanjima, Tokushima, Japan
Phone:+81-88-656-7470
Email: {uchitani, kumeno, nishio}@ee.tokushima-u.ac.jp

xn+1

Abstract
The small world network model is a simple model of thestructure of social networks, which possesses characteristics
of both regular coupling and random coupling. To structure
the small world networks, each map is rewired to other maps
chosen at random with a probability. In this study, we investigate synchronization phenomena of small world networkmodel of one hundred coupled cubic maps. As a result, weconfirmed that various wave propagations were observed bychanging the probability.

xn+1

xn

(a)

xn(b)

1. Introduction
Synchronization phenomena in complex systems are very
good models to describe various higher-dimensional nonlinear phenomena in the field of natural science. Studies onchaos synchronization in coupled chaotic circuits are extensively carried out in various fields [1][2]. On the other hand,studies of network model are very important, because they
help us to understand the basic features and requirements ofvarious systems. In real life, social networks are known to be
not completely random network nor completely local. Thiswas modeled in an interesting work as the small world network model [3].
In our past studies [4], we have investigated synchronization phenomena in coupled cubic maps. The cubic map has
two attractors located symmetrically with respect to the origin and they merge as increasing a control parameter. By
computer simulations, we observe interesting state transition
phenomenon. As a result, the two maps are synchronized forthe parameters giving periodic solutions.
In this study, we propose small world network model ofcoupled cubic maps. Then, we investigate synchronization
phenomena in this model.
2. Coupled cubic map
In our past studies, we investigated two coupled cubicmaps. Figure 1 shows return maps of the cubic maps.
The cubic map is described by the following equation:x(n+1) = ax3(n) − (a + 1)x(n) ,Figure 1: Cubic map. (a) a = −3.451. (b) a = −3.70.

where n is an iteration and a is a parameter which determines
chaotic features. Figure 1(a) shows an asymmetric 3 periodic
attractor, and Fig. 1(b) shows a symmetric 6 periodic attractor.
The coupled cubic maps are described by the following
equation:3 g(x(n) ) = ax(n) − (a + 1)x(n)x1(n+1) = (1 − )g(x1(n) ) + (g(x2(n) ))(2)
x2(n+1) = (1 − )g(x2(n) ) + (g(x1(n) )),where  is a coupling parameter. Figure 2 shows examplesof time waveforms in two coupled cubic maps. Figure 2(a)shows the time waveform obtained with the parameters giving the periodic solution in Fig. 1(a) and a negative couplingstrength. While, Fig. 2(b) shows the time waveform obtainedwith the parameters giving the periodic solution in Fig. 1(b)and a positive coupling strength. From the coupled cubicmaps, we confirmed that the two maps are synchronized forthe parameters giving periodic solutions.
3. Small world network model

In this section, we explain a small world network model ofcoupledcubic maps. The network topology of small world(1)
is located between regular graph and random graph. Watts- 49 -

1.0

x1

the regular coupling.
g(x(n) ) = ax3(n) − (a + 1)x(n) xi(n+1) = (1 − )g(xi(n) ) + /4(g(xi−2(n) )+ g(xi−1(n) ) + g(xi+1(n) ) + g(xi+2(n) ))(i = 1, 2, 3, ..., 100)

0-1.0
1.0

x2 0-1.0n20000

(a)

(3)

1.0

x14. Synchronization phenomenon

0-1.0
1.0

x2 0-1.0n20000(b)Figure 2: Time waveform of two coupled cubic maps. (a) α =−3.4505 and  = −0.062. (b) α = −3.70 and  = 0.0375.

In this section, we investigate synchronization phenomenon in small world network of coupled cubic maps. Figure 4 shows the computer simulated results. In the figure, thered (bright) areas represent that xi are positive. While, theblue (dark) areas represent that xi are negative. We can see
various wave propagations by changing the probability p. In
every case, the coupled maps produce complex patterns, however the maps do not synchronize stably.
Figure 5 shows distribution of the frequencies of sojourntime. The sojourn time is the number of iteration during x1
is continuously in the positive or negative region. The horizontal axis indicates the sojourn time, and the vertical axis
indicates the frequency. From these figures, we can see thatthe sojourn time becomes longer as increasing the probability
p except the case of p = 0.05. Additionally, the frequencies
has the peaks of the sojourn time which are around 5, under
all conditions.
5. Conclusions

(a)

(b)

(c)

Figure 3: A small world graph. (a) Regular coupling. (b) Small
world network. (c) Random coupling.

In this study, we investigated synchronization phenomenon
in the small world network of one hundred coupled cubicmaps. By computer simulations, we confirmed that various
wave propagations were observed by changing probability
p. In every case, the coupled maps produce complex patterns, however the maps do not synchronize stably. Then,the sojourn time becomes longer as increasing the probability
p. Investigating the mechanism of this synchronization phenomenon is our future research.
References

proposed a method to structure small world network [3]. The
small world graph in Fig. 3 shows that the small world network can be obtained by small world couplings between theregular graph and the random graph. In this study, we consider one hundred coupled cubic maps like Fig. 3. Each map
is coupled to the first neighbor maps and the second neighbors
as shown in Fig. 3(a). To structure the small world network,
each coupling is rewired to other maps chosen at random witha probability p. In the case of p = 0, any couplings are notrewired. On the other hand, in the case of p = 1, all the couplings are rewired as shown in Fig. 3(c) (random coupling).
Figure 3(b) shows the small world network. The followings
are the equations of one hundred cubic maps for the case of[1] G. Abramson,V.M. Kenkre and A.R. Bishop, “Analytic Solutions for Nonlinear Waves in Coupled Reacting Systems,” Physica A, vol. 305, no. 3-4, pp. 427-436, 2002.
[2] I. Belykh, M. Hasler, M. Lauret and H. Nijmeijer, “Synchronization and Graph Topology,” Int. J. Bifurcation and Chaos,vol. 15, no. 11, pp. 3423-3433, 2005.
[3] D.J. Watts and S.H. Strogatz, “Collective Dynamics of SmallWorld Networks,” Nature, vol. 393, pp. 440-442, Jun. 1998.
[4] Y. Uchitani and Y. Nishio, “Investigation of Complicated Phenomenon in Coupled Cubic Maps,” Proc. of NOMA’09, pp. 9194, Sep. 2009.

- 50 -  

(b) p = 0.025.
x1 

x1000(c) p = 0.05.

   (a) p = 0.

       (e) p = 0.46.

60000(d) p = 0.085.

 n 

(f) p = 1.

Figure 4: Time waveform of coupled cubic maps. α = −3.4505 and  = −0.07. (a) p = 0. (b) p = 0.025. (c) p = 0.05. (d) p = 0.085.
(e) p = 0.46. (f) p = 1.

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2500

2000

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Frequency

Frequency

2500

1500

1500

1000

1000

500

500

00110100

1000

10000

1000001e+06

110100

Sojourn time

10000

1000001e+06100000

1e+06

1000001e+06(b) p = 0.025.

2500

2500

2000

2000

Frequency

Frequency(a) p = 0.

1500

1500

1000

1000

500

500

00110100

1000

10000

1000001e+06

110100

Sojourn time

1000

10000

Sojourn time(c) p = 0.05.

(d) p = 0.085.

2500

2500

2000

2000

Frequency

Frequency

1000

Sojourn time

1500

1500

1000

1000

500

500

00110100

1000

10000

100000

1e+06

Sojourn time110100

1000

10000

Sojourn time(e) p = 0.46.

(f) p = 1.

Figure 5: Distribution of the frequencies of sojourn time. α = −3.4505 and  = −0.07. (a) p = 0. (b) p = 0.025. (c) p = 0.05.
(d) p = 0.085. (e) p = 0.46. (f) p = 1.

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