NOLTA 2006
11-14 september
Bologna - ITALY

Pattern Dynamics of Phase Synchronization
in a Family of Coupled Several One-Dimensional Chaotic Maps
Masahiro Wada† and Yoshifumi Nishio‡†Department of Information Science and Systems Engineering, Konan University
8–9–1 Okamoto, Higashinada-ku, Kobe 658-8501, Japan‡Department of Electrical and Electronic Engineering, Tokushima University
2–1 Minami-Josanjima, Tokushima 770-8506, Japan,
Email: wada–m@konan-u.ac.jp, nishio@ee.tokushima-u.ac.jp
Abstract— In this study, phase synchronization behavior and control of its patterns in coupled chaotic maps areinvestigated. There are many types of chaotic map, thencoupled chaotic systems yield wide variety of complex
phenomena and further it is shown possibility to several engineering applications. The chaotic maps which have been
governed by n-th power polynomial or sinusoidal functions
is properly selected as a chaotic cell, then each chaotic map
is connected to neighbors as a ring array or network structure. Several phase synchronization patterns and its control
method are shown.
1. Introduction
We have now interests how to various patterns in nature were cleated. Coupled chaotic systems attract many
researchers’ attention as a good model which can realize
the complicated phenomena in the natural world, and further its dynamics can yield a wide variety of complex andstrange phenomena. The coupled systems existing in nature exhibit great variety of phenomena such as complex
mechanisms for all of the systems in the universe. Thesephenomena can be found in a metabolic network, a humansociety, the process of a life, self organization of neuron, abiological system, an ecological system and so many nonlinear systems. Among the studies on such coupled systems, many interesting researches relevant to the spatiotemporal chaos phenomena on the coupled chaotic systems
have been studied until now, e.g. mathematical model inone- or two-dimensional network investigated earnestly by
Kaneko [1]-[4], and found in physical circuit model [5].
The construction of multi–agent system on the coupledcubic map system has been reported [6]. Moreover, research of complicated phenomena and emergent property
in the coupled cubic maps on 2-dimensional network system has been also reported [7]. The studies of coupled map
lattice(CML), globally coupled maps(GCM) and so manystudies concerned with such complex systems provided us
tremendous interesting phenomena. This is an interesting
report that phase dynamics are controlled due to change its
parameter themselves. We had also reported the researchon spatio-temporal phase patterns in coupled maps using a
fifth-power function [8][9], in which it has been carried out
in the unique case. However many coupled chaotic systems
have wide variety of features and moreover its dynamics isalso expected to be applied much engineering applications,

there are many problems which should be solved in large
scale coupled network systems by their complexity.
In this study, spatio-temporal chaotic behavior in coupled chaotic maps is investigated from the point of viewin more faithful natural world. The chaotic map which
has been governed by n-th power polynomial or sinusoidal
functions is properly selected as a chaotic cell. We consider the model which chaotic cells are mutually connected
to neighbors as a ring structure (i.e. CML type) by arbitrary coupling strength. Then, we show some phenomena
which spatio-temporal chaos, complex behavior and several phase patterns can be found in the proposed coupledsystems. Furthermore, its control dynamics are realized bychanging a perturbation parameter.
2. Model Description
Chaotic maps are generally used for several approaches
to investigate chaotic phenomena on coupled chaotic systems. Especially, the logistic map and the other types ofchaotic maps such as a cut map, a circle map, a tent map,a cubic map are well known and popular. Obviously, it isnecessary to have a lot of equilibrium points with the complex phenomena that corresponds to the natural world. Let
us consider two types of chaotic map. Firstly, the chaotic
map from an n-th power polynomial function written as follows.
f (x) =nX

ai x i + ϕ(1)i=1where ai is a parameter which can determine for theirchaotic feature, further ϕ is a new parameter for perturbation as a small variable value. The parameter ϕ should benormally set as zero. If it is needed to adopt the map withrespect to the origin, odd-numbered coefficients ai are only
set suitable values in (1). In other words, even-numberedcoefficients are set as all zero. Then, we can easily confirm
that it generates chaos in this function. The some diagramsof the function (1) are shown in Fig. 1 with some equilibrium points.
Secondary, the chaotic map from a sinusoidal function
written as follows.

531

f (x) = ec|x| sin(ax) + bx + ϕ(2)x(a)(a)(b)

λ- a1

(c)

(d)

x(b)
(e)
(f)
Figure 1: Several chaotic maps by n-th power polynomial
and sinusoidal functions for ϕ = 0. Setting parameters:(a) n = 3, a3 = −2.75, a1 = 2.75, (b) n = 5, a5 = 5.50,
a3 = −10.0, a1 = 4.17, (c) n = 7, a7 = −2.85, a5 = 11.6,
a3 = −14.6, a1 = 5.80, (d) n = 9, a9 = 2.08, a7 = −13.15,
a5 = 27.4, a3 = −21.3, a1 = 5.46, (e) sin(10x) + 0.5x, (f)e−0.9|x| sin(10x) + 0.9x, and the others are all 0.
where a, b and c are parameters which can determine for
their chaotic feature. Especially the parameter b is an important factor in order to suppress its divergence. The somediagrams of the function (2) are also shown in Fig. 1. From(1) and (2), it can be calculated rigorously several bifurcation conditions and boundary region.
In order to evaluate the function (1), Lyapunov exponent
can be calculated as follows.
λ = lim

N→∞NXlog
k=1d f (xk )dx

λ- bFigure 2: Bifurcation diagram and Lyapunov exponent: (a)changing a1 for a7 = −3.10, a5 = 11.5 and a3 = −12.4 incase of n = 7, (b) changing b for a = 10.0 and c = −0.90 incase of using sinusoidal function (2).
is seen. Therefore chaotic maps possessing several equilibrium points can yield various wide interesting behavior.
3. Phase Patterns in Coupled Chaotic Maps

(3)

Lyapunov exponent is a very important measurement often
used to show the existence of chaos. Some Lyapunov exponents with bifurcation diagram by changing one parameterare shown in Fig. 2. These are typical results which canbe obtained from computer calculation. In case of using
polynomial functions, period doubling and tangent bifurcation can be confirmed. On the other hand, in case of using sinusoidal functions, complicated bifurcation property
can be confirmed even if the result of Lyapunov exponent

In this section, we consider a coupled chaotic system that
one of these maps as a chaotic subsystem in each cell. It
can be considered easily that coupled chaotic systems havewide variety of phase patterns. The term “spatio-temporal”
is extensively used for irregular dynamical behavior observed from large scale complex systems of the relevant
to both time and space. In this study, in order to confirm
spatio-temporal chaos or phase patterns in the faithful natural world, consider a coupled model of the chaotic maps
which are connected to neighbors on a ring array structure
as shown in Fig. 3. Each chaotic cell is connected to neigh-532

ε /2

ε /2

ε /2

C

C

ε /2
CC

ε /2
C

ε /2
CFigure 3: Coupled chaotic system as a ring array or coupledlattice structure.
(a)bors by arbitrary coupling strength ε. The total system byCML is represented as
xk (t+1) = (1−ε) f xk (t)ε (4)+ f xk−1 (t) + f xk+1 (t) ,2(k = 1, 2, · · · , N)
where t is an iteration, k is an index number of the cellwhich follows the cyclic rule, and N is a size of coupledcell number, respectively.
Some numerical simulation results of model (4) for N =50 are shown in Fig. 4 with coupling strength ε = 0.30. The
initial condition for each cell is given as xk (0) ∈ [0.49, 0.51]
uniformly. The figure indicates a grade of synchronization state for phase difference, with gray scale colors beand black
which correspond to syntween whitechronous and asynchronous state, respectively. Hereby thesynchronous state with gray scale colors in 100 steps is displayed. A lot of interesting phenomena were confirmed
though all the results can not be represented more here.

(b)

4. Pattern Dynamics and Control
Further, we attempt to control phase patterns of entire
coupled system to become synchronous state with additional swing of function or changing the coupling strength.
As shown in Fig. 4, it is confirmed that some parts are asynchronous state. Although all subsystem is the same, it isdifficult to perform to control entire system synchronously.
However control method should be simple as possible.
Therefore, we propose two simple methods below.

(c)(method 1)
εk = ε∗if |xk − xk+1 | < 0.5(method 2)(d)εk =(ε∗if |xk − xk+1 | < 0.5−0.1 if xk ≥ 0
ϕ=+0.1 if xk < 0In case of method 1, it is changed only coupling parameter εk in each cell. When the value of difference between
two neighbors is larger than 0.5, the coupling strength εkof the target cell changes to ε∗ . In case of method 2, weuse the condition of swing parameter ϕ in addition to themethod 1. Figure 5 shows a pattern obtained by method

Figure 4: Simulation results of phase synchronization state
in coupled chaotic maps as a ring array for N = 50: (a)n = 5, a5 = 6.20, a3 = −10.0, a1 = 4.10, (b) n = 7,
a7 = −2.85, a5 = 11.6, a3 = −14.6, a1 = 5.80, (c) n = 9,a9 = 2.08, a7 = −13.15, a5 = 27.4, a3 = −21.3, a1 = 5.46,
and (d) e−0.9|x| sin(10x) + 0.9x. Each result is performed inthe condition ε = 0.30.

533

Table 1: Increase ratio of synchronous state when the setting parameters are the same as figure 4.

n=5
n=7
n=9
e x sin

(a)

0.4
11.7
9.1
21.8
6.4

0.5
4.4
13.4
23.9
6.0

ε∗
0.6
6.7
14.5
25.3
2.5

0.7
9.0
15.0
28.9
-6.1

0.8
11.7
13.4
25.8
-12.1

[%]

Table 2: Increase ratio of synchronous state when the setting parameters are the same as figure 4 with swing parameter ϕ.
ε∗n=5n=7n=9e x sin(b)Figure 5: Simulation results of pattern dynamics with control method 2: (a) n = 7, a7 = −2.85, a5 = 11.6,
a3 = −14.6, a1 = 5.80, (b) transition of synchronous ratio. Each result is performed in the condition of change
parameters ε = 0.30 ↔ ε∗ = 0.50 and ϕ = ±0.1.
2 when the control is executed at t = 200. A lot of parts
are made synchronous can be confirmed. To clarify a synchronous ratio, the difference of the adjoined cells counted
the number of 0.2 or less and it was calculated. The meanvalues of the increase ratio before and after the control
when the initial condition is changed and tries it 1000 times
are shown in Table 1 and 2. The result that either was excellent was not obtained because two methods were carriedout by a specific parameter, and the control method should
be changed according to the characteristic of the chaotic
map. Therefore, it is necessary to investigate whether otherparameters are changed, and which control method is excellent though only control by the change in the couplingstrength and the swing parameter were used in this experiment.

0.3
-8.1
17.1
8.0
-0.1

0.4
-0.7
20.8
16.3
6.6

0.5
-0.9
20.8
17.3
6.0

0.6
0.8
34.1
17.6
2.4

0.7
4.3
22.8
18.7
-6.1

0.8
2.7
15.3
15.6
-12.2 [%]

Acknowledgment
This work was partly supported by GRANT-IN-AID for”Open Research Center” project from MEXT, Japan.

5. Conclusions
In this paper, some chaotic maps by n-th power polynomial or sinusoidal functions for using as a chaotic cellof coupled network have been proposed. Some illustrated
computer simulation results of spatio-temporal chaotic behavior and several phase patterns in coupled chaotic maps
have been shown. The simple method for controlling thepatterns has been proposed, and the effectiveness has been
investigated. We conclude that the supposed or similar coupled chaotic systems can be regarded as a good model for
realizing complex phenomena in the universe concernedwith self organization, mechanisms of pattern formationand so on. However some studies of pattern dynamics andthe mechanism of clustering phenomena in such complex
phenomena and many works have been left.
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