Physica A 188 (1992) 210-216 North-Holland m a

Controlling low-dimensional chaos by proportional feedback

Bo Peng, Valery Petrov and Kenneth Showalter 1 Department of Chemistry, West Virginia University, Morgantown, WV 26506-6045, USA

Low-dimensional chaos can be controlled by proportional feedback to one or more of the system constraints in a simple scheme based on 1D maps. The stabilized system exhibits periodic behavior even though the corresponding autonomous system remains within the parameter range of chaotic behavior. We examine the effects of random noise and delay between sampling and subsequent control perturbation. Providing these factors are small, the simple control scheme is stable.

1. Introduction

Although the long-term behavior of a chaotic system is unpredictable, the behavior in the local vicinity of a fixed point on the Poincar6 section, corresponding to an unstable periodic orbit, can easily be predicted. Ott, Grebogi and Yorke ( O G Y ) [1] have utilized this local predictability in a method for stabilizing periodic orbits in a chaotic system through deliberate per turbat ions of a dynamical parameter . The variation in the paramete r can be sufficiently small that the corresponding autonomous system remains in the chaos region throughout the controlling process.

For systems exhibiting low-dimensional chaos, the O G Y method can be reduced to a very simple scheme [2, 3]. The algorithm takes advantage of the local predictability around unstable fixed points of 1D maps. Control is achieved by proportionally varying a dynamical parameter each iteration according to the deviation of the system state from the fixed point to be stabilized. The simplicity of proport ional-feedback control may provide signifi- cant advantages in settings where targeting procedures must be minimized. The map-based algorithm has been applied to stabilize unstable periodic orbits in a simple three-variable chemical model [2, 4] and a four-variable biological model [3, 5].

In this study we examine the effects of two inevitable destabilizing factors in

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B. Peng et al, I Controlling low-dimensional chaos by feedback 211

practical applications of the control scheme: noise and delay between sampling and the controlling perturbation. We present the control scheme in section 2, and in section 3 we examine the effects of noise and delay. Advantages and limitations of the method are considered in section 4.

2. Map-based control scheme

Consider the linear 1D map of x n vs. x ,+ k,

x , , + k = f ( x ~ - x s ) + x~ . (1)

The fixed point x s is unstable if ]f[ > 1. However , the unstable point can be stabilized by shifting the fixed point from x~ to x'~ according to the current system state [2, 3]:

' f ( X n - X ~ ) + x s (2) X ~ = f - - 1

Replacing x s in eq. (1) by x's of eq. (2) yields x,÷~ = x~. Hence, the system is directed to the original fixed point on the next iteration from any point in the domain of the linear map.

The fixed point is shifted by varying a system parameter , with the variation proport ional to the deviation x n - x~. Providing the latter is small, the shift is proport ional to the parameter change. A chaotic system is bound to visit the vicinity of each fixed point as it evolves in time, and the desired fixed point x S is stabilized by switching on control when the system falls within the linear region of x S. The necessary variation of the parameter p is determined according to

f (x,, - .,,-J - xn - X s 5P = ( f - 1) dxJdp g (3)

The proportionali ty factor g may be determined according to the horizontal shift of the map at x S on varying the control parameter p (see fig. 3, ref. [2]). Thus, the control scheme samples the difference between the system state and the particular unstable point to be stabilized and proportionally varies a control parameter according to the difference.

The map-based control scheme can be applied to continuous chaotic systems exhibiting effectively 1D dynamics around the desired unstable point with an appropriate selection of the monitored variable, control parameter and Poin- car6 section [2, 3]. We consider the chaotic behavior of the three-variable chemical model

212 B. Peng et al. / Controlling low-dimensional chaos by feedback

d__a_a= d~- /Z(K + y ) - a - a/3 2 , (4)

o- : a + - / 3 , ( 5 )

6 ~ = / 3 - 3 , , (6)

where a , /3 , and 3' are dependen t variables, r is dimensionless t ime, a n d / , , K, o- and 6 are parameters [4]. Per iod- l , period-2 and period-4 limit cycles are

readily stabilized by moni to r ing /3 on the Poincard section and propor t ional ly

varying /, [2].

In this s tudy we focus on the period-2 limit cycle of the system; fig. 1 shows the strange at t ractor and the stabilized periodic orbit. Fig. 2 illustrates the

stabilizing effect of the control algori thm in the second-re turn map. Ideally, the

segment a round the fixed point (fi`` = 77.0) would be a horizontal straight line

according to fi,,+2 =/3s- The curvature of the map around/3., indicates that the control range extends beyond the linear region of the fixed point [3].

3. Effects of noise and delay

We have considered the effects of small errors, uncontrol lable per turbat ions , and nonlineari ty on the map-based scheme [3]. Such imperfect ions are an

°6. ii ~ ....." • "~

log fl 1~ o ~Olog Fig. 1, Strange attractor (...) and period-2 unstable limit cycle (- ) of the three-variable chemical model, eqs. (4)-(6). The limit cycle is stabilized by varying the parameter p, according to the value of /3,, as prescribed by eq, (3). Parameter values: K = 65, cr = 5 × 10 3, 6 =2 x 10 2, p, = 0.154. Limit cycle period: % = 0.125.

B. Peng et al. / Controlling low-dimensional chaos by feedback 213

q O

q ~ .... ~.~

O- 00

I

q O / t"q

I I I

~.0.0 40.0 60.0 80.0 100.0

9o Fig. 2. Second-return map of the chemical system during control, constructed from the attractor in fig. 1 with the Poincard section defined by 3' = 15, ~, > 0. Values of/3,, near the period-2 fixed point at higher /3 are monitored and IX is varied accordingly. The control range is /3s--6 and (/3~, g) = (77.0, -1.5 x 104).

i n t e g r a l pa r t o f a n y a p p l i c a t i o n in a prac t ica l se t t ing , a n d we n o w c o n s i d e r

r a n d o m p e r t u r b a t i o n s o n /3 a f te r each s a m p l i n g a c c o r d i n g to a G a u s s i a n

d i s t r i b u t i o n . T h e a d d e d no i se m i g h t be c o n s i d e r e d , for e x a m p l e , to be the

r e su l t o f c o n c e n t r a t i o n f l uc tua t i ons in an o p e n , s t i r red reac to r . T h e resu l t o f

s t ab i l i z i ng p e r i o d - 2 wi th the n o i s e is s h o w n in fig. 3, w h e r e the cha rac t e r i s t i c

q C)

O ". " O- "'-" " '"'"

.'.•..'. • q

d - ~r~-:~'~,

d c~ I I I I

I0.0 50.0 90.0 150.0 170.0 210.0 250.0

7-

Fig. 3. Values of /3 on the Poincar6 section as a function of time during control of period-2, illustrating the effect of random noise. Noise is imposed on /3 directly after applying controlling perturbation. The control range is /3s + 2 and standard deviation of the Gaussian noise is 0.23.

214 B. Peng et al. / Controlling low-dimensional chaos by feedback

width of the distribution is scaled according to the control range. The system is stabilized for most of the calculation, although there are small random fluctuations at the two period-2 fixed points. An asymmetry in the amplitude of

the fluctuations is apparent , with the larger amplitude at the fixed point where noise is added. The asymmetry is qualitatively the same when the noise is

imposed at the low fls fixed point, indicating that the difference is due to the intrinsic dynamics of the autonomous system.

The stabilization is sometimes interrupted by bursts of chaotic behavior due to the occasional occurrence of large-amplitude noise. The system quickly returns to the control range, however, and is again stabilized until the next occurrence of large-amplitude noise. The bursts of chaos become more fre- quent as the characteristic width of the noise distribution is increased or as the control range is decreased. A similar response to imposed Gaussian noise was found with the O G Y scheme [1].

In a practical application of the control scheme, there may be a significant delay between sampling the system state and application of the control per turbat ion, and we now consider the effects of such delays. To simulate delay in a typical experimental setting, we sample the system and then switch on the perturbat ion after time 7a, as shown in fig. 4. There is also a delay in terminating the perturbation, corresponding to switching on the next perturba- tion. Factors such as variation of the local features of the limit cycle and the position of the Poincar6 section determine the effect of the delay on the control. In general, the system can be controlled if the delay is small compared to the period of the targeted unstable limit cycle. For example, the period-2

. . . . . . .

÷ I

L . . . . . . . 1

I

% -r

Fig. 4. Schematic drawing of delayed control perturbation. Measurement of Aft,, = fl,, -/3~ shown by dashed line ( - - - - - - ) ; adjustment of /x resulting in A/3~ shown by solid line ( ). Delay between sampling and perturbation is r d. Control range is between dot-dash lines ( - - - - - . - - ) at top and bottom.

B. Peng et al. / Controlling low-dimensional chaos by feedback 215

limit cycle is stabilized by the algorithm providing the delay time is smaller than 14% of the period.

4. Discussion

Continuous systems exhibiting low-dimensional chaos can be controlled by simple proportional feedback using the map-based algorithm presented here. The variation in parameter value necessary for control can be very small, depending on the desired control range, and subsequent variations for main- taining stabilization rapidly decay to extremely small values. Hence, the corresponding autonomous system remains in the chaotic region even though the controlled system exhibits periodic behavior.

In most cases, control may be accomplished by measuring the deviation of one variable from its fixed-point value and proportionally varying one parame- ter. In some cases, however, the one-parameter scheme may be inadequate due to the incompleteness of discrete maps in describing transient behavior of continuous systems. The difficulty can be overcome by proportionally varying several parameters in a manner analogous to the single-parameter scheme [3].

The most attractive feature of the proportional-feedback scheme is its simplicity, making it especially amenable to potential applications. In addition to the three-variable chemical model considered here, the algorithm has been applied to a four-variable biological model for diffusion-induced chaos [3]. Although the multiple-parameter variation was used for the more complex system, it was also possible to achieve control with the single-parameter scheme. We have also applied the single-parameter scheme to the Lorenz [6] and R6ssler [7] systems, and readily stabilized their corresponding period-1 unstable orbits. In the Lorenz system, control was achieved by monitoring the maximum in the variable z and varying the parameter R (parameters: or = 10, R = 28, b = 2.6666). In the R6ssler system, the variable y was monitored on the Poincar6 section defined by x = 2.0, k > 0 and the control perturbation was applied to the parameter c (parameters: a = 0.2, b--0.2, c = 5.7).

In this study, we have demonstrated that control is possible with the simple map-based scheme in the presence of random noise and with delayed response. As either the amplitude of the noise or the extent of the delay is increased, a threshold is reached where the algorithm fails. When these factors are relative- ly small, however, the control scheme is robust.

Acknowledgements

We thank the National Science Foundation (Grants No. CHE-8920664 and INT-8822786) and WV-EPSCoR for financial support of this work.

216 B. Peng et al. / Controlling low-dimensional chaos by feedback

References

[1] E. Ott, C. Grebogi and J.A. Yorke, Phys. Rev. Lett. 64 (1990) 1196. [2] B. Peng, V. Petrov and K. Showalter, J. Phys. Chem. 95 (1991) 4957. [3] V. Petrov, B. Peng and K. Showalter, J. Chem. Phys. 96 (1992) 7506. [4] B. Peng, S.K. Scott and K. Showalter, J. Phys. Chem. 94 (1990) 5243. [5] I. Lengyel and I.R. Epstein, Chaos 1 (1991) 69. [6] E.N. Lorenz, J. Atmos. Sci. 20 (1963) 130. [7] O.E. R6ssler, Phys. Lett. A 57 (1976) 397.