The other parameters are same as in Fig. I. The attractor in this case is shown in figure 4.

§ 35. Development of Root-Finding Method Based on Genetic Algorisms

The other parameters are same as in Fig. I. The attractor in this case is shown in figure 4.

0.1 0.2 0.3 0.4 0.5 0.6

200 250 300 350 400 450 500 t

0.1 0.2 0.3 0.4

0.3 r--....----...,...- ...- ...

0.2 0.1

o

-0.1

0.8 0.6 0.4 y(t) ^0.2 0 -0.2 -0.4 -0.6

-1

-0.5 0 0.5

x(t)

0.6 r'I""I"...,---...

... T'""'I'...,l""I""I""..., 0.5

Fig.3. The time series rnta of x(t) with the relayed feedback. The controlling parameters are chosen as

A = OA,r = 2:Jr .

x(t)

Fig.2. The chaotic attractor in x - y phase space.

by

The standlrd 4

orrer Runge-Kutta method is not applicable for this case since we must evaluate the time relay term yet -r) in each step. We ampt the 4

orrer Adnns-Bashforth method in the predctor step and the 4

orrer Adnns-Moulton method in the corrector step as the numerical integration scheme.

Figure 1 shows the time series rnta of x(t) in the case with k::::; 0.05,B o = 0.045,B

= 0.16,A = O. For these parameters, the chaotic behavior is observed in the time series rnta Figure 2 shows the chaotic attrnctor

in x - y phase space for these parameters.

- = y dx dt

dy

-

_dt

-ky - x

B o ^{+ B}

^{cos t + A(y(t -r) - y(t))} Genetic algorithms (GAs) are rerivative-free optimization methoct based on the concepts of natural selection and evolutionary process. It finct significant applications in many areas. For examples, it is applied for the evolution of weighting functionl) of neural networks and is also applied for controlling chaos

Here we are interested in the application of GA for root fiming for the eigen function of the drift wave with maximum growth rate in huge parameter space. Also, the technicpe of controlling chaos via GA might be applicable for controlling plasma turbulence for the future.

Our strategy is as follows: in the first step of this research, we will start from the linear control method known as relayed feedJack methocf) and apply this method to Duffing eq.Iation to control chaos as an example. Then in the next step, we will investigate nonlinear control method such as neural network and GA for it and evaluate the merit and demerit of these methods.

The Duffing ecpation with feedJack term is given Yagi, M., Furuya, A., Hoh, 5.-1. (Kyushu Univ.)

Next, we tty to control the chaos. Figure 3 shows the time series chta of x(t) in the case of A::::; OA,r ::::; 2:Jr .

x(t)

FigA. The corresponding attractor (limit cycle) to figure 3.

References

1) D. Montana and L. Davis, in Proc. of International Joint Conf. on A.I., Morgan Kaufmann 1989.

2) E. Weeks andJ. Burgess, Phys. Rev. E 56 1531 (1997).

3) K. Pyragas, Phys. Lett. A 170421(1992).

4) Y. Veda, The Road to Chaos, Aerial Press (1992).

The control of chaos is successful in this case, however, to retermine appropriate A, we need the trial. In adition to this remerit, it is unknown which limit cycle will be stabilized if there are several limit cycle orbits.

The nonlinear controlling method of chaos is now reveloping. We will compare the results with those by delayed feedback method.

o

0.5

-0.5 x(t)

-1

150 200 250 300 350 400 450 500

t

Fig.I. The time series rnta of x(t) in the case with k = 0.05,B o ::::; ^0.045,B

= 0.16, A = O.

356