平成21年度電気関係学会四国支部連合大会

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18-14

Synchronization Phenomena in
Coupled Cubic Maps
Yumiko Uchitani
Yoshifumi Nishio(Tokushima University)
1. Introduction

1.0

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]. On the other hand,chaotic maps are generally used for several approaches toinvestigate chaotic phenomena on coupled chaotic systems.
Especially, a logistic map, a circle map, a tent map, a cubic
map are well known and popular. In this study, we investigate synchronization phenomena in coupled cubic maps.

x10-1.0
1.0

x2 0-1.0n20000

n20000(a)
1.0x1

2. Cubic map

0-1.0
1.0

xn+1

xn+1

x2 0-1.0

(b)
xn

xn

Figure 2: Timewaveform of coupled cubic maps. (a) α =−3.45043, and ² = −0.0477. (b) α = −3.69964, and ² =0.0508.

(b)

Figure 1: Cubic map. (a) α = −3.451. (b) α = −3.70.
x(n+1) = αx3n − (α + 1)xn(1)Figure 1 shows the cubic map Eq. (1) where n is an iteration, α is a parameter which determines the chaotic feature.
We can easily conﬁrm that it generates various periodical
solutions and chaos. Figure 1(a) shows asymmetric 3 periodic solution and Fig. 1(b) shows symmetric 6 periodic
solution.

Figure 3 shows how the sojourn times between the statetransitions change as the coupling parameter ² changes.
The horizontal axis is coupling parameter ² and the vertical axis is the average lengths of the iterations n. From thisﬁgure, we can see that the sojourn time between the transitions becomes longer as increasing the coupling parameter
².
5000

4500

4500

4000

4000

3500

3500

3000

iteration(a)3000

iteration

3. Coupled cubic maps

2500

2500

2000
2000

1500

In this study, we consider two coupled cubic maps. The
followings are the equations of coupled cubic maps where² is a coupling parameter.

{f (x(n) ) = αx3n − (α + 1)xn
x1(n+1) = (1 − ²)f (x1(n) ) + ²(f (x2(n) ))x2(n+1) = (1 − ²)f (x2(n) ) + ²(f (x1(n) ))

1500

1000

1000

500

500
0
-0.0485-0.0484-0.0483-0.0482-0.0481-0.048-0.0479-0.0478-0.0477-0.0476

epsilon(a)(2)

Figure 2 shows examples of the computer simulated resultsof Eq. (2). Figure 2(a) shows the timewaveform obtainedwith the parameter near that giving the periodic solution
in Fig. 1(a). In the same way, Fig. 2(b) shows the timewaveform obtained with the parameter near that giving
the periodic solution in Fig. 1(b). From the coupled cubic maps, we can observe interesting state synchronization
phenomena. The two maps synchronized in anti-phase inFig. 2(a) and synchronized in in-phase in Fig. 2(b). We
could conﬁrm that the two maps synchronize only for theparameters near that giving periodic solutions. However,
we do not understand the relationship between the obtainable synchronization states and symmetry property of theoriginal solutions.

00.05

0.05010.05020.05030.05040.05050.05060.05070.0508epsilon(b)

Figure 3: Sojourn time between state transitions. (a) α =−3.45043. (b) α = −3.69964.

4. Conclusions
In this study, we have investigated the synchronization
phenomena of two coupled cubic maps. We could observe
interesting synchronization phenomenon. We conﬁrmedthat the two maps synchronized for the parameters nearthat giving periodic solutions. And, the sojourn time between the transitions became longer as increasing the coupling parameter. Clarifying the mechanism is our futurework.

References[1] P. Ashwin, J. Buescu and I. Stewart, “Bubbling of Attractors and Synchronization of Chaotic Oscillators,” Phys.
Lett. A, 193, pp. 126-139, 1994.