Physica A 188 (1992) 210-216
North-Holland
ma
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
m e t h o d for stabilizing periodic orbits in a chaotic system through deliberate
perturbations of a dynamical parameter. The variation in the p a r a m e t e 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 p a r a m e t e r each iteration
according to the deviation of the system state from the fixed point to be
stabilized. The simplicity of proportional-feedback control may provide significant advantages in settings where targeting procedures must be minimized. The
m a p - b a s e d 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
To whom correspondence should be addressed.
0378-4371/92/$05.00 © 1992- Elsevier Science Publishers B.V. All rights reserved
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 m e t h o d 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~
(1)
- x s ) + x~ .
T h e 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
(Xn-X~)+x
X~=f-- 1
s
(2)
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
proportional to the deviation x n - x~. Providing the latter is small, the shift is
proportional 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 Sis
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
5P = ( f -
f
(x,, - .,,-J - xn - X s
1) dxJdp
g
(3)
The proportionality 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 Poincar6 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-
(5)
: a +
-/3,
6 ~=/3-3,,
(6)
w h e r e a , / 3 , and 3' are d e p e n d e n t variables, r is dimensionless time, a n d / , , K, oand 6 are p a r a m e t e r s [4]. P e r i o d - l , period-2 and period-4 limit cycles are
readily stabilized by m o n i t o r i n g / 3 on the Poincard section and p r o p o r t i o n a l l y
varying /, [2].
In this study we focus on the period-2 limit cycle of the system; fig. 1 shows
the strange attractor and the stabilized periodic orbit. Fig. 2 illustrates the
stabilizing effect of the control algorithm in the second-return map. Ideally, the
s e g m e n t a r o u n d the fixed point (fi`` = 77.0) would be a horizontal straight line
according to fi,,+2 =/3s- The curvature of the m a p around/3., indicates that the
control range extends b e y o n d the linear region of the fixed point [3].
3. Effects of noise and delay
We have considered the effects of small errors, uncontrollable perturbations,
and nonlinearity on the m a p - b a s e d scheme [3]. Such imperfections are an
°6. ii
~
....."•"~
logfl
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
O00
~
....~.~
I
q
O
t"q /
~.0.0
I
I
I
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 p a r t o f a n y a p p l i c a t i o n in a p r a c t i c a l s e t t i n g , a n d w e 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 t e r e a c h 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 n o i s e m i g h t b e c o n s i d e r e d , for e x a m p l e , to b e t h e
r e s u l t o f c o n c e n t r a t i o n f l u c t u a t i o n s in a n o p e n , s t i r r e d r e a c t o r . T h e r e s u l t o f
s t a b i l i z i n g p e r i o d - 2 w i t h t h e n o i s e is s h o w n in fig. 3, w h e r e t h e c h a r a c t e r i s t i c
q
C)
O ". "
O- "'-""'"'"
.'.•..'.•
q
d - ~r~-:~'~,
dc~
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 r a n d o m
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 frequent 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
perturbation, 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 perturbation after time 7a, as shown in fig. 4. There is also a delay in
terminating the perturbation, corresponding to switching on the next perturbation. 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 c o m p a r e d
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 maintaining 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 parameter. 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 relatively 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.
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B. Peng et al. / Controlling low-dimensional chaos by feedback
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