J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)263–266
000–000

Control of Chaos by Linear and Nonlinear Feedback Methods
SAITO Takanori, ITOH Sanae-I.1 and YAGI Masatoshi1
Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Fukuoka 816-8580, Japan
1
Research Institute for Applied Mechanics, Kyushu University, Fukuoka 816-8580, Japan(Received: 9 December 2003 / Accepted: 2 June 2004)

Abstract
The chaotic orbits in the Rossler system are controlled into a periodic cycle by two methods; the delayed feedback
controlling method which continues to control chaos by self-controlling feedback, and a control system including online trained neural network controller. It is found that (1) stabilization of chaotic orbits by the former method depends
on the initial conditions and the gain parameters and that (2) the linear neural controller fails to control chaotic orbits
in the Rossler system and choice of the threshold function (nonlinear function) is found to be essential for the second
method.
Keywords:
chaos, delayed feedback controlling method, neural controller

1. Introduction

In this paper, both methods are tested to the Rossler
system to check these methods are applicable to not only amap system but also to coupled ordinary differential equations
with a few degrees of freedom. Performance of these control
methods is compared with each other. It is found that thecontrol of chaos by the Pyragas method is strongly affectedby the gain of perturbation and is sensitive to the initialconditions for the period three cycle. We also apply the NNC
to the Rossler system. It is found that the original linear NNC
fails to control the Rossler system and is only available for amap system. It is concluded that the nonlinear threshold
function is essential to control a chaotic orbit in the Rossler
system and that the NNC with the hyperbolic tangent function
shows the best performance. This paper is organized asfollows. In Sec. 2, the method of delayed feedback control isapplied to the Rossler system, and the sensitivity ofparameters on the controlling chaos is examined. In Sec. 3,
the NNC is applied to the Rossler system. The dependenceof the threshold function is examined. The chaotic orbit iscontrolled by using the NNC with the hyperbolic tangent
function as the threshold function. In Sec. 4, we summarize
the results.

Chaotic phenomena are frequently observed in plasma
discharges and in the fundamental plasma experiments [1].
Recent researches extend not only to identify but also tocontrol chaos itself [2]. In fusion plasmas, the control ofplasma turbulence is a critical issue to attain the self-ignitionconditions. Therefore, techniques of controlling chaos might
be useful even for fusion plasmas. To establish controlling
method of plasma turbulence, we will investigate thecontrolling of chaos as the first step. During the initial phaseof developing turbulence, the pichfork bifurcation issometimes observed in experiments, which is described by asimple model with a few degrees of freedom [3]. Therefore,
we expect these controlling methods are usufull to control
chaos in the initial phase of developing turbulence.
The original controlling method of chaos called OGY
method is developed by Ott, Grebogi and Yorke. It utilizes
the existence of UFP (unstable fixed point) embedded within
the chaotic attractor [4]. The chaotic orbit is stabilized byapplying small perturbations into the system. However, thismethod requires the knowledge of the location of UFP so thatthe location should be tracked in advance by the linear
prediction. Later, Pyragas proposed the delayed feedback
controlling method which does not require the knowledge ofthe location of UFP or unstable periodic orbit (UPO) bymodifying the OGY method [5]. The alternative method ofcontrolling chaos is given by aplication of the neural network
controller (NNC). On-line trained linear neural controller isproposed by Konishi and Kokame, and is applied to control
the chaotic orbit in the two dimensional map system such asan Henon map [6].

2. Chaos control by delayed feedbackmethod
The Rossler system with a perturbation F(t) is given by©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: tsaito@riam.kyushu-u.ac.jp1263

Saito T. et al., Control of Chaos by Linear and Nonlinear Feedback Methods

found that τ = 5.90, 11.75, 17.50 correspond to the periodone, two, three, respectively. Other local minimum correspondto the UFPs.
Next, we numerically find k that gives the local minimum
values of D2(t) while fixing the period, τ obtained in Fig. 2.
Figure 3 shows the dependence of 〈D2(t)〉 on k in the caseswith τ = 5.90 (thick line), τ = 11.75 (dotted line) and τ =17.50 (dotted dashed line). It is seen that k = 0.18, k = 0.12
and k = 0.06 correspons to the local minimum for period one,
two and three cycles, respectively. Figure 4 shows period-one
cycle of the Rossler system. Two cases with (τ = 5.90, k =0.18) and ( τ = 5.90, k = 0.8) are shown for the initialcondition of x0 = y0 = z0 = 5.5. The thick line corresponds tothe case with k = 0.18 and the dotted line to the case with =0.8. It is shown that the choice of k is important to stabilize
UPO.
We investigate the sensitivity of controlling chaos on
initial conditions for each period τ with an optimized valueof k. It is found that the control of chaos on the period-one
cycle (τ = 5.90) or two cycle (τ = 11.75) does not depend on
the initial condition. However, the control of chaos on the⎧ dx = – y – z⎪ dt⎪⎪ dy⎨ = x + 0.2 + F(t ), F(t ) = k[ y(t – τ ) – y(t )] = k[ D(t )] (t > 50) (1)
⎪ dt
⎪ dz
⎪⎩ dt = 0.2 + z( x – 5.7) , F(t ) = 0 (t > 50)

where y (t – τ ) is delayed output signal and k is the gain of
perturbation. This system has a chaotic attractor. Figure 1
shows the x – y phase portrait of the Rossler system without
perturbation (k = 0), where the chaotic attractor is clearlyseen.
The delayed time τ corresponds to the period of the UPO
or UFP, when F(t) = 0, i.e., D2(t) = [(y (t) – y (t – τ )]2 = 0.
From numerical results with different τ, we will estimate avalue of τ which gives the local minimum of D2(t) for k =0.2. Figure 2 shows the dependence of the time averaged
perturbation 〈D2(t)〉 on τ for k = 0.2. The time average isperformed during the time interval t = 150 – 300. 〈D2(t)〉 is
calculated for each value of τ with different initial conditions
such as x 0 = y 0 = z 0 = 0.5, 1.0, 1.5,…, 10 and thecorresponding 20 values of 〈D2(t)〉 for each τ are shown. It is< D 2 (t) >101050.1
00.001

Y-510-5-10

10-7-15-10-50

51010-9

150X

510
τ

1520

Fig. 2 Dependence of 〈D 2(t )〉 on τ.

Fig. 1 Chaotic attractor of the Rossler system.

D2 (t)

10100

105
k=0.18

1Y 0

τ=11.75k=0.80.1
τ=5.90-50.01
τ=17.50-10-100.001
00.2

0.4

0.60.81-5051015XkFig. 3 Dependence of D 2(t ) on k.

Fig. 42264

Period-one cycle of the Rossler system for τ =5.9 andk = 0.18 and 0.8.

Saito T. et al., Control of Chaos by Linear and Nonlinear Feedback Methods
Spectrum104

101000control

5100

uncontroly 0101-50.1-10-10-50

5100.011500.1

0.2

xFig. 5the x-y phase portrait of the Rossler system controledby adding a perturbation.

Wij ( k + 1) = Wij ( k ) + ∆Wij ( k ),

θ i ( k + 1) = θ i ( k ) + ∆θ i ( k ),∆Wij ( k ) = –η∆θ i ( k ) = –η

(2)

∑j =1

∂E ( k ) ∂ U i ( n k ).
∂ U i ( n k ) ∂θ i ( k )(6)

E( k ) = kC EC ( k ) + kU EU ( k ),(7)1
EC = [ X (tk + 1) – X (tk )]T [ X (tk + 1) – X (tk )],2(8)withandEU =(3)the watcher passes a control signal from the NNC to thechaotic system, and then the weights of connection from theinput neuron to output neuron in the NNC are updated byback propagation method. Initial values of the weights are
determined within the range of –0.01 to 0.01 by a random
number. ε is assumed to be a small positive value. The NNCis given byOi (tk ) =∂E ( k ) ∂ U i ( n k ),∂Ui (nk ) ∂Wij ( k )η is the learning rate and the error is defined bywhere X(t) is the state vector, F denotes chaotic system andU(t) is the control signal. When the orbits X(t) and X(t – τ)of the chaotic system without control satisfyN(5)whereIn this section, we use neural network controller tocontrol the chaotic system, i.e., the Rossler system. Figure 7
shows the block diagram of NNC. Here the NNC consists of
two layers such as input and output layers [5]. The following
chaotic system is considered:Ui (tk ) = f (Oi (tk ) + θ i ( k ))0.5

where j is the number of input neuron, i, the number of output
neuron and k is the number of times in which the NNC istrained (i.e., the number of times the watcher operates). Wij is
the weight of output neuron, and Oi (tk ) is the signal whichgoes into neuron of the output layer through the weight fromith neuron of the input layer, and θi (k) is the bias of the ith
neuron of the output layer. The weights and the biases are
updated by using

3. Chaos control by neural controller|| X (t ) – X (t – τ ) || < ε

0.4

Fig. 6 Dependence of power spectrum on frequency.

period three cycle (τ = 17.5) depends on the choice of theinitial condition.
Figure 5 shows the x – y phase portrait of the Rossler
system with τ = 11.75 for t >150. It is successfully controlled
to the period-two cycle. Figure 6 shows the dependence ofthe power spectrum on the frequency with/without control forτ = 11.75. The peak is observed at f = 0.085 for the case withcontrol, which implies period-two cycle ( f = 1/τ). It shouldbe mentioned that the chaotic orbits of the Rossler system
also converge to UFPs, which correspond to τ = 3.30 and τ =9.30. In these cases, the value of k weakly affects thestabilization of UFP compared with stabilization of UPO.

X (t + 1) = F[ X (t )] + U (t ),0.3

Frequency

1U (tk )T U (tk ),2(9)We apply the NNC to the Rossler system:⎧ dx = – y – z + U (t )
1
⎪ dt
⎪⎪ dy
⎨ = x + 0.2 + U2 (t )
⎪ dt
⎪ dz
⎪⎩ dt = 0.2 + z( x – 5.7) + U3 (t ).

(10)

In this case, the NNC consists of three input neurons and three
output neurons. The three cases with different threshold
functions are investigated: (1) linear function f (x) = x and(4)Wij ( k ) X j (tk ),3265Saito T. et al., Control of Chaos by Linear and Nonlinear Feedback Methods

other parameters are ε = 1.0, η = 0.08, kC = 3.0 and kU = 1.0,(2) the sigmoid function f (x) = 1 +1e –x , (3) the hyperbolic–x
tangent function f (x) = 11 +– ee –x for period-one, two and three.
It is found that the case (1) fails for all cases; the case (2)fails in period-two and three cases for kC = 3.0, kU = 1.0.
However, the case (2) successfully controls period-three forkU = 4.2. The case (3) works for all cases without changing
parameters kC and kU. These results are shown in Figs. 8 and
9 for period-one and two cases. The dotted line corresponds
to the case with the linear function and the thick line to thecase with the hyperbolic tangent function. It is concluded thehyperbolic tangent function is the best candidate as thethreshold function of NNC for controlling the Rossler system.
The numbers of updating weights in the NNC is a good
indicator whether the NNC successfully controls the chaos,or not.
Finally, we extend the NNC with three layers andexamine the effect of numbers of neurons in hidden layer oncontrolling chaos. However, no advantage is found comparedwith the NNC with two layers. Therefore we conclude that
the NNC with two layers and hyperbolic tangent function as
the threshold function shows the best peroformance to control
the Rossler system.

4. Summary and discussion
X (t + 1)

Z– 1

Watcher
X (t) – X (t – τ )N+1
Z

The Pyragas method and the on-line trained NNC aretested for the Rossler system. It is found that the UPO of theRossler system is stabilized by the delayed feedback
controlling method, however, the controllability depends onthe choice of the initial condition and the gain of perturbation.
The NNC with linear threshold function works well for thecontrol of the chaos in map systems [6], however, it fails in
the application to the Rossler system as is shown in this paper.
The NNC is improved by changing the threshold function
such as the sigmoid function and hyperbolic tangent function.
We find that the NNC with hyperbolic tangent function shows
the best performace to control the Rossler system. We also
examine the NNC with three layers, however, no advantage
is found in comparison with the NNC with two layers for theRossler system. Once we find the suitable value of k, thenthe delayed feedback controlling method is faster than theNNC with hyperbolic tangent function, although it is notalways possible for the general case.
Fully developed plasma turbulence has a large degree offreedom, therefore, simple neural network system will notwork. As a future work, we will apply 3 layers neural network
controller to plasma turbulence.

–1X (t)<εU(t )

Neural Network
Controller(NNC)

Chaotic System

X (t + 1)- +Fig. 7 Block diagram of the control system.

Linear function
Hyperbolic tangent function

1050Y-5-10-15-10-5051015

Acknowledgements

XThis work is partly supported by the Grant-in-Aid for
Scientific Research of MEXT Japan, by the collaboration
programmes of National Institute for Fusion Science and of
the Research Institute for Applied Mechanics of Kyushu
University.

Fig. 8 Period-one cycle of the Rossler system with NNC.

Linear function
Hyperbolic tangent function

10References

5[1] P.Y. Cheung and A.Y. Wong, Phys. Rev. Lett. 59, 551(1987).
[2] T. Fukuyama, K. Taniguchi and Y. Kawai, Phys. Plasmas
9, 1570 (2002).
[3] P. Cvitanovic, Universality in Chaos (Adam Hilger Ltd,Bristol, 1984).
[4] E. Ott, C. Grebogi and J.A. Yorke, Phys. Rev. Lett. 64,1196 (1990).
[5] K. Pyragas, Phys. Lett. A 170, 421 (1992).
[6] K. Konishi and H. Kokame, Physica D 100, 423 (1997).

0Y-5-10-15-15-10-5051015X

Fig. 9 Period-two cycle of the Rossler system with NNC.

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