2005 International Symposium on Nonlinear
Theory and its Applications (NOLTA2005)
Bruges, Belgium, October 18-21, 2005

Spatio-temporal Phase Patterns
in Coupled Chaotic Maps with Parameter Deviations
Masahiro Wada† , Kiyoto Kitatsuji† and Yoshifumi Nishio‡†Department of Information Science and Systems Engineering, Konan University,‡Department of Electrical and Electronic Engineering, Tokushima University
Email: wada–m@konan-u.ac.jp, mn424002@center.konan-u.ac.jp, nishio@ee.tokushima-u.ac.jp
Abstract
In this study, spatio-temporal chaotic behavior in coupled chaotic maps with parameter deviations is shown. Thechaotic map which has been governed by a fifth-power
polynomial function is properly selected as a chaotic cell.
We consider a simple model which each cell has parameterswith slight error margin are connected to four neighbors byarbitrary coupling strength. Analytical bifurcation diagramand Lyapunov exponent of the chaotic map are investigatedrigorously. Several phase patterns of spatio-temporal chaos
are shown.
1. Introduction
It is interested in various pattern formations how to create in the natural world very much in a broad sense. 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 canyield a wide variety of complex and strange phenomena.
The coupled systems existing in nature exhibit great variety of phenomena such as complex mechanisms for allof the systems in the universe. These phenomena can befound in a metabolic network, a human society, the process of a life, self organization of neuron, a biological system, an ecological system and so many nonlinear systems.
Among the studies on such coupled systems, many interesting researches relevant to the spatio-temporal chaos phenomena on the coupled chaotic systems have been studied until now, e.g. mathematical model in one- or twodimensional network investigated earnestly by Kaneko[1][4], and found in physical circuit model[5]. Moreover, research of complicated phenomena and emergent property
in the coupled cubic maps on two-dimensional network
system has been also reported[6]. The studies of coupled
map lattice(CML), globally coupled maps(GCM) and somany studies concerned with such complex systems provided us tremendous interesting phenomena. We had reported the research on spatio-temporal phase patterns incoupled maps, using a fifth-power function[7], in which it
has been carried out in the unique case which parameters of
the chaotic map and coupling strength are all the same settings. However many coupled chaotic systems have wide
variety of features and moreover its dynamics is also expected to be applied much engineering applications, there
are many problems which should be solved in large scale

Figure 1: Chaotic map model by a fifth-power polynomial
function for a=5.2, b=10.0 and c=4.1.
coupled network systems by their complexity.
In this study, spatio-temporal chaotic behavior in coupled chaotic maps with parameter deviations is investigatedfrom the point of view in more faithful natural world. The
chaotic map which has been governed by a fifth-power
polynomial function is properly selected as a chaotic cell.
We consider the model which chaotic cells are mutuallyconnected to some neighbors for spreading on the twodimensional space by arbitrary coupling strength. Then,
we show some phenomena which spatio-temporal chaos,complex behavior and several phase patterns can be foundin the proposed coupled systems.
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. Let us consider afifth-power polynomial function as a chaotic subsystem ineach cell written as follows.
f (x) = ax5 − bx3 + cx,

a, b, c > 0(1)where a, b and c are parameters which can determine for
their chaotic feature. We can easily confirm that it generates chaos in this function. The function (1) is shown

178

Bifurcation diagram for b=10.0

14Fixed Point
12xn(a)Divergence

10aAsymmetry86(a)Symmetry(b)

4

λDivergence

233.5

c44.5

Figure 2: Bifurcation diagram for b = 10.0.
in Fig. 1 with some equilibrium points. In general, moreplural patterns can be expected to be generated comparedwith [6], because many equilibrium points can be taken ina higher dimensional map.
From (1), it can be calculated rigorously several bifurcation conditions and boundary region. For instance, a necessary and suﬃcient condition for having fixed points off (x∗ ) = x∗ from (1) except the origin can be calculated
exactly as
b2 − 4a(c − 1) > 0 .

xn(b)

(2)

λWe can only confirm that the system diverges in this setting. Since this system (1) is formed with respect to theorigin, variable x moves around neither plus or minus region symmetrically by setting of the parameter. Similarly,this condition can be calculated exactly as
b2 − 4ac < 0 .

(3)However, even if in case of the condition (3), chaotic oscillation is not symmetry with respect to the origin except in
special case, e.g. a = 8.0, b = 10.0 and c = 3.0, if satisfied
as follows.
f ( f (s1 )) > 0 .

(4)

Since this equation can not be calculated explicitly, the bifurcation curve is obtained numerically by computer calculation.
Furthermore, we can also obtain the conditions that solution emanates to infinity by calculating from the following
evaluation. The values of [ f  (x) = 0, x > 0] at coordinate
si while in the condition (2),√
3b − 9b2 − 20ac
s1 =,10a(5)√
3b + 9b2 − 20ac
,
s2 =
10a

- a
Figure 3: Bifurcation diagram and Lyapunov exponents bychanging a for fixed b = 10.0. (a) c = 4.1 , (b) c = 3.5 .
then, solve f (s1 ) > p1 and f (s2 ) < p2 the followingb + b2 − 4a(c − 1),f (s1 ) >2ab + b2 − 4a(c − 1),f (s2 ) < −2a(6)we can obtain the parameter of divergence point numerically. For example, by calculating for the settings b = 10.0and c = 4.1 as a = 6.39953 and 4.65828, respectively. Bifurcation diagram is partially illustrated in Fig. 2 by changing both a and c for fixed b = 10.0. Moreover, some analytical bifurcation curve can also be obtained such as tangent
bifurcation, fixed point and so on.
In order to evaluate the function (1), Lyapunov exponent
can be calculated as follows.
N
d f (xk ) 
 .
log (7)λ = limN→∞dt k=1179

an equation for a description model of the entire systemthat each cell is connected to 1-neighbor Ξ1 (i j) with 4 cellsby cyclic rule as follows.
 ε  
xi j (t+1) = (1−ε) f xi j (t) +f xkl (t)4 kl∈Ξ

(8)1(a-1)

(a-2)

(b-1)
(b-2)
Figure 4: Snapshot of chaotic map with attractor: (a) coexist with fixed point and divergence for a = 9.6, b = 10.0 andc = 3.5, (b) coexist with chaotic attractor and divergence for
a = 5.0, b = 10.0 and c = 3.1.
Lyapunov exponents with bifurcation diagram by increasing a gradually for fixed b = 10.0 and c = 4.1 are shownin Fig. 3 (a). In case of this parameter setting, we can easily calculate parameter in which symmetry collapses is as
a > 250/41. We can confirm that solution diverges fora > 6.39953. Figure 3 (b) also shows bifurcation diagramand Lyapunov exponents for c = 3.5 across the parameter aas shown in Fig. 2. Some snapshots of attractor obtained forspecial cases of coexistence from (1) are shown in Fig. 4;
converge to equilibrium point and divergence for (a) a =9.6, b = 10.0 and c = 3.5, attract to chaotic orbit and divergence for (b) a = 5.0, b = 10.0 and c = 3.1. It can be
confirmed that solution coexists with an equilibrium pointand divergence to infinity, with chaos and divergence, respectively. As we can see, several modes of chaos, limit
cycle and periodic window can be seen by changing theparameters.
3. Spatio-temporal Chaos in Coupled Chaotic Maps
It can be considered easily that coupled chaotic systems
have wide variety of phase patterns. The term “spatiotemporal” is extensively used for irregular dynamical behavior observed from large scale complex systems of therelevant to both time and space. In this study, in order toconfirm spatio-temporal chaos or several phase patterns in
the faithful natural world, we consider a network modelof the chaotic maps placed spatially on two-dimensional
space are connected to neighbors by arbitrary couplingstrength ε. Generally it is able to be considered a coupledmodel which each cell is connected r-neighbor cells Ξr (i j)by coupling strength ε. In order to investigate simply relevant to time and spatial transition, we define and consider

where t is a number of iteration in each cell, {i, j} is anindex number of cell.
Some numerical simulation results on such model (8) as
size of 50×50 cells on two-dimensional space are shown in
Fig. 5 for the parameter fixed as a = 5.45, b = 10.0 and c= 4.1. The initial condition for each cell is given at random between −1.0 and 1.0. The figure indicates a gradeof synchronization state for phase diﬀerence with a nearby
average, which black and white colors correspond tosynchronous and asynchronous states, respectively. Hereby
the synchronous states are displayed with 100 steps grayscale colors. Figure 5 shows some snapshots of phase patterns as time increasing for a = 5.45, b = 10.0, c = 4.1 and ε= 0.25. The pattern at time t = 1 means initial state, which
black region almost occupied visibly. However, as time
passed, the entire state becomes asynchronously. Finally,some part only remains synchronous state at t = 1000. In
this case, the phase pattern behaves almost asynchronous
well, however some parts produce unique patterns while
keeping synchronous state as a cluster.
On the other hand, we now consider the situation thatsome chaotic cells have slight error margin as comparedwith the other cells in order to investigate from the point ofview in more faithful natural world. Figure 6 shows somenumerical simulation results at time t = 1000 for giving
the maximum parameter deviations with 0.1%, 0.5%, 1%
and 10% to only a, respectively. The initial conditions areputting as all the same values in each result. Hence it has
just diﬀerent parameters a with slight error margin. We
can confirm that some results could be obtained as diﬀerent
phenomena. In order to investigate the influence on synchronization depending on the parameters, a synchronous
ratio was calculated. Results of the ratio of synchronous
states with 5% maximum parameter deviations of a areshown in Fig. 7. We can confirm that a synchronous ratio is a high average according to the area of asymmetricregion as shown in Fig. 2. Similar results can be confirmed
when there is no error margin of the parameters. These
results are some typical examples obtained by numericalsimulation. From some numerical results, it has been confirmed that a slight error margin does not exert or influenceon the entire system too much. However, the phenomena
such the mutation can be also confirmed with parameter
deviations in some situations. Therefore, some phenomenaof spatio-temporal chaos or complex behavior of several
phase patterns can be found in these systems.
4. Conclusions
In this study, a fifth-power function for using as a chaotic
maps was proposed, further rigorous analysis, bifurcation

180

t=1

t=10

t=100

t=200

(a)

(b)

(c)(d)
Figure 6: Some other phase patterns obtained from numerical simulation at time t = 1000 as fixed parameters b =10.0 and c = 4.1 with maximum parameter deviations forthe value of a = 5.45, (a) 0.1%, (b) 0.5%, (c) 1%, (d) 10% .

t=500
t=1000Figure 5: Some snapshots of numerical simulation results
depended by time to two-dimensional space for fixed parameters a = 5.45, b = 10.0 and c = 4.1 without parameter
deviations. ε = 0.25, t: number of iteration. Number ofcells is 50 × 50.

1ratio

0.8
0.6
0.4
0.2

diagram and Lyapunov exponents were shown. Some illustrated computer simulation results of spatio-temporalchaotic behavior and several phase patterns in coupledchaotic maps have been shown. We conclude that the supposed or similar coupled chaotic systems can be regarded
as a good model for realizing complex phenomena in theuniverse concerned with self organization, mechanisms ofpattern formation and so on. However some studies of pattern dynamics and the mechanism of clustering phenomena
in such complex phenomena and many works have beenleft.

01231083.5

c644.5

a4Figure 7: Ratio of synchronous states by changing the parameters a and c on coupled chaotic maps for b = 10.0 and
e = 0.25 with 5% maximum parameter deviations of a attime t = 1000.

Acknowledgment
This work was partly supported by The Hirao Taro Foundation of the Konan University Association for Academic
Research and by GRANT-IN-AID for “Open ResearchCenter” project from MEXT, Japan.
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Physica D, vol. 34, pp. 1–41, 1989.

181[3] K. Kaneko, “Spatiotemporal Chaos in One- and TwoDimensional Coupled Map Lattices,” Physica D, vol. 37,
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[5] Y. Nishio and A. Ushida, “Spatio-Temporal Chaos in Simple Coupled Chaotic Circuits,” IEEE Trans. on Circuit andSystems-I, vol. 42, no. 10, pp. 678–686, Oct. 1995.
[6] M. Wada, K. Hirai and Y. Nishio, “A Multi-Agent System and State Control of Coupled Chaotic Maps,” Proc. ofNOLTA’01, vol.1, pp. 211–214, 2001.
[7] K. Kitatsuji and M. Wada, “Spatio-temporal Chaos and Several Phase Patterns in Coupled Chaotic Maps using FifthPower Function,” Proc. of NCSP’05, pp. 113–116, 2005.