A Multi-Agent System and State Controlof Coupled Chaotic MapsyMasahiro WADA , Kazumasa HIRAIy andzYoshifumi NISHIO

yDept. of Information Science and Systems Engineering, Konan University, Kobe 658-8501, Japanwada-m@konan-u.ac.jp, khirai@mbox.kyoto-inet.or.jp
zDept. of Electrical and Electronic Engineering, Tokushima University, Tokushima 770-8506, Japannishio@ee.tokushima-u.ac.jp

Abstract| Many types of coupled chaotic systems had been studied in the wide elds and some
applications had been proposed, so several chaotic behavior are very important in the universe. Emergentbehavior and construct of a multi-agent system incoupled simple chaotic maps are investigated in thisstudy.
I. Introduction

The coupled systems existing in nature exhibit great
variety of phenomena such as complex mechanisms forall of the systems in the natural elds or in the universe. These phenomena can be seen in an information
network, a human society, the process of a life, self organization of neuron, a biological system, an ecological system and so many nonlinear systems. Thus theconcept of the process of natural selection is very important to be realized some extended application byusing nonlinear features. In fact unsolved problems innatural elds are still remained, especially the processof evolution in a human being and human intelligence.
We have considered that almost phenomena are equivalently described as coupled nonlinear systems. Hitherto, many types of coupled chaotic systems have beenstudied as one of the most interesting phenomena in
the natural elds[1]-[3]. Recently, many types of coupled chaotic systems have been investigated by many
researchers and then a lot of phenomena were reported
in many papers. We have studied several chaotic circuits coupled by one element for applying the engineering systems, and reported some featured phenomena.
Further, cellular automata(CA)[4] and cellular neural network(CNN)[5] has been also studied in a large
eld scienti cally. The studies of coupled map lattice(CML), globally coupled maps(GCM), generalizedCA(GCA)[6] and so many studies provided us tremendous interesting phenomena and in uenced some applications. Recently, a multi-agent system constructedby simple coupled chaotic maps have been reported
and shown validity of the model[7][8]. The past works

had been almost treated only simple coupled systems
or simpli ed models, and the total number of systems
is small. However, Ichikawa reported that chaos generates in a simple CA on any conditions[9]. It was aninteresting system with emergent property. The term
emergence leads us to the world of a novel approach forseveral applications. It is important to investigate several phenomena in simple coupled chaotic systems. In
the past study, both CML and GCM type models wereusually used as coupled chaotic systems. In this study,
several coupled types are proposed and phenomena in
the coupled chaotic systems are investigated. Furthermore, a multi-agent system constructed by the coupledsystems are also investigated.
II. System description

Chaotic maps are generally used for several approaches
to investigate chaotic phenomena on coupled chaoticsystems. Especially, the logistic map and the othertypes of chaotic maps such as a cut map, a circle
map, a tent map, a cubic map are well known andpopular. CML and GCM are the special models coupled by thus chaotic maps. Let us consider a network of chaotic maps which is placed spatially on twodimensional plane. Let a local subscript number ofeach cell de ne as

23ci01;j 01 ci01;j ci01;j +1
4 ci;j 01
ci;jci;j+1 5ci+1;j 01 ci+1;j ci+1;j+1

Firstly we consider simple chaotic map of a subsystem in the coupled network. The subsystem considered here is very simple whose equation is presented
as followsx(t+1) = f (x(t) ) = x(t)(1 0 x2(t))(1)

whereis a parameter which can determine thestrength of nonlinear characteristic. We can easilycon rm that chaos generates in this subsystem. It iscase of coupling for 4 neighbors(Neumann type) and8 neighbors(Moore type) can be considered. For example, while the subsystems are coupled locally in thecase that 4 neighbors subsystems are coupled to one
by the weight " as shown in Fig. 2, then the stateof a subsystem at any placed on the total system is
described as followsxi;j(t+1) = ki;jFigure 1: A mathematical cubic map model for one
subsystem for = 2:6.
εεCCεεCεCεεεCε+we assumeki;j +εεCεFigure 2: A model of coupled network subsystems by4 neighbors.
well known that we can con rm a crisis of chaos if
is greater than around 2.6, which it is obtained fromFig. 1. Hence the chaotic map can obtain both positive and negativevalue while is over 2.6. If > 3,q
2pi.e. 3 3 > 1+ , we can also con rm that the system
diverges.
Secondly, we propose some types of structure of coupled chaotic systems, in which a chaotic subsystem arecoupled each other in a local group. Generally the coupled types are roughly divided into two categories, alocally coupled network and a globally coupled network. A schematic model of the coupled systems isshown in Fig. 2 in the case of locally coupled network.
Now we consider the locally coupled network becauseof the reason for state-control of each subsystem inthe later part. The adopted structures in this study
are divided into three connected types. In the normalcase, subsystems placed on the edge of the coupled network are connected only to the local subsystems, thatis, the edge of network is free, not connected. In theother cases, subsystems placed on the edge are connected to the subsystem on the opposite side, just likeas the structure of a tube or a pipe, and subsystems
placed on the edge are all connected to the opposite
side of the coupled network, just like as the surface
of a ring or a doughnut shape. Throughout this paper, we usually treat above three types of locally coupled systems. Then, the subsystems are distributed on
the two-dimensional place, that is, the subsystems areplaced on a plane and then all the subsystems are connected to 4, 8 or more local subsystems. Generally, thexi;j (t) 1 0 x2i;j (t)Xm;nXm;n(2)km;n xm;n(t)km;n = 1;(3)

where if it is coupled by 4 neighbors then km;n = "=4,hence ki;j = 1 0 ", the rst and second subscripted letters fi; j g of the variable x mean identical numbers
at each position, the third letter t means iteration
time respectively, and " corresponds to the strengthof a coupling gain for neighborhoods. The subscripted
number follows the cyclic rule. It is to be noted that if" = 0, it means that the systems are not coupled eachother, the xed points of each subsystem can calculate
easily.
A characteristic of this coupled system is of using
the state in the time one step before. Therefore theparameter can be chosen larger than 3, provided that" is enough large(< 1), because the value of (1 0") in the equation of the total system works to besuppressed its divergence.
III. Computer simulation

The computer simulation in those systems is shown.
The case of the normal type coupled system which
the subsystems are placed on 2-dimensional plane ismainly treated hereafter. Now we choose coupledchaotic maps in case of 100 times 100 scale network.
Some illustrated gures are shown. Some waveforms of<the subsystems (Ci;i ; 1 <= i = 100) with respect to theiteration of the time step are shown in Fig. 3. It isshown that each subsystem oscillates randomly in therst half of the time step while " is set to be 0 in thisinterval and they converge to the xed point when "is set to be 0.3 after 300 time steps. We can con rmthat all or almost subsystems are stabilized and they
converge to the xed point or periodic oscillation. The
initial conditions for all subsystems are set as an uniform random number between -1 and 1. Figure 4 (a)and (b) are illustrated the spatial pattern of each stateand whitein the same parameter ofwith black
Fig. 3 at 100 and 500 time steps respectively. The
black(white) painted square means that the state ispositive(negative). Figure 5 is also illustrated several
spatial pattern results in which (a) and (b) are the caseof " = 0:1 and 0:4 for = 2:6, (c) is the case of " = 0:1

0! iterations
Figure 3: Several waveforms of the subsystems (Ci;i ;
1<=i<= 100) in the total system without/with couplingfor = 2:6 when the all subsystems are coupled at 300
time steps after.

(a)(b)
Figure 4: Simulation results illustrated to 2dimensional plane. These gures correspond to Fig. 3
at 100 and 500 time steps after respectively. (a) without coupling, the all subsystems oscillate separately,(b) with coupling, the all subsystems are coupled to 4
neighbors by ".
for = 2:8, (d) is the case of " = 0:4 for = 3:0 respectively. In these cases, we can con rm easily several
phenomena that stabilization of chaos, bifurcation tochaos via periodic oscillation, chaotic itinerancy andspatio-temporal chaos occur in this system. If " ischosen for a small value, the spatio-time transition of
the chaotic behavior can be observed. Several interesting chaotic behavior can be observed in these coupledchaotic maps, like a \game of life" as time growth,hence it has a transition of the xed point.
IV. Multi-agent system

In this part, we consider that state around the xed
point will be controlled by using a characteristic of thesubsystem. The construction of a multi-agent systemby the supposed coupled chaotic systems are investigated. For the purpose of some applications, we can

(a-1)

(a-2)

(a-3)

(b-1)

(b-2)

(b-3)

(c-1)

(c-2)

(c-3)

(d-1)
(d-2)
(d-3)
Figure 5: Several computer simulation results in thecase of 100 2 100 subsystems. (a) for = 2.6, " = 0:1,(b) for = 2.6, " = 0:4, (c) for = 2.8, " = 0:1, and(d) for = 3.0, " = 0:4. The gures from the left ofeach line, the time steps 1, 100 and 1000 are shown
respectively. The initial conditions are set randomly,
but all simulations are started at the same condition.
control the sequence of chaotic itinerancy for each subsystem in a large scale coupled chaotic systems by using state-control. Therefore it is dicult to know stateof all the subsystems and it is not for practical use in
the large scale coupled chaotic networks, we consider
that a subsystem can be controlled by itself as an autonomic control. On the other hand, the subsystems
can have the state either positive or negative values
for the origin. We investigate how to control the twostate which we suppose each state as an agent. Let
us consider one subcomponent which is constructedby an agent and coupled 4 neighbors, the parameter
and/or " of the agent are controlled by the stateof neighborhoods with response. As a feature of this
proposed system, we can know the state of the neighborhoods at one step before and we can put the stateof the other agents near to one agent when the coupling gain changes. By changing the coupling gain "or the strength of nonlinear characteristic , the stateof each subsystem can be controlled as a agent. In
this study, we only control the parameter " hereafter,
because the parameter is related with system char-acteristics closely. The system is divided roughly into
two state, that is randomly chaotic and two positivenegative state coexist stably as like shown in Fig. 3.
Namely, an agent controls the coupling gain " by itself
in spite of the other agents, because the agent knows
the state one step before. The concept of controllingparameter shows in the following.
1.it should be set as the agent behaves like the transition of chaos when the systems are uncoupled.
2.set appropriate " to get each agent to be stable.
3.make the coupling gain " of some agents to be week,then let an agent be the transition of chaos.
4.intensify " again when the system is to be expectedstate.
Repeat its procedure until the total system attain to
the expected state. Now the purpose of the all subsystems is to be the positive state. Here the coupling
gain is set to " = 0:4 when we want it to be stabilized,
it is set to " = 0:05 when we want it to be generated the transition. One of the simulation results isshown in Fig. 6 when the system is controlled by theproposed method. In this result, we added control sequence at 100 time steps after. The upper part of thisgure shows waveform of the subsystems from C1;1
to C100;100 . The lower part shows a number of agent
which attained to positive state in the total system.
We can con rm that agents almost attain to the purpose, but not all. If " is large, the probability into thepositive attractors increases. It is especially shownthat agents almost accomplish the purpose quickly.
The proposed systems which behave those phenomena are considered as one of models of a multi-agentsystem. If we can control the number of clusters of
the state, we respect of these features to be utilized
for several engineering applications.
V. Conclusions

In this study, chaotic phenomena in several typesof coupled chaotic systems have been investigatedand considered how to construct a multi-agent system. Some illustrated computer simulation results
have been shown. The construct of a multi-agent system with state-control for each subsystem has also
been shown. We conclude that the supposed coupledsystems whose several chaotic phenomena and multiagent systems are expected to be utilized for realization of a new type application, although study of themechanism of such phenomena and many works havebeen left.
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Figure 6: Transition of chaos and time evolution of
the total number of agent as a multi-agent system controlled by " = 0.4 and 0.05 for = 3.0. Transition of
the total number of agent in the positive state is also
shown.

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