A mapbased algorithm for controlling lowdimensional chaosValery Petrov, Bo Peng, and Kenneth Showalter Citation: The Journal of Chemical Physics 96, 7506 (1992); doi: 10.1063/1.462402 View online: http://dx.doi.org/10.1063/1.462402 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/96/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Chaotic oscillations in a map-based model of neural activity Chaos 17, 043109 (2007); 10.1063/1.2795435 Mapbased neuron networks AIP Conf. Proc. 887, 69 (2007); 10.1063/1.2709587 Rules for controlling low-dimensional vocal fold models with muscle activation J. Acoust. Soc. Am. 112, 1064 (2002); 10.1121/1.1496080 Search for low-dimensional chaos in a reversed field pinch plasma Phys. Plasmas 6, 100 (1999); 10.1063/1.873264 A lowdimensional model for chaos in open fluid flows Phys. Fluids A 5, 1947 (1993); 10.1063/1.858821

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.120.242.61 On: Tue, 25 Nov 2014 12:19:27

A map-based algorithm for controlling low-dimensional chaos Valery Petrov, Bo Peng, and Kenneth St1owaltera )

Department o/Chemistry, West Virginia University, Morgantown, West Virginia 26506-6045

(Received 17 December 1991; accepted 5 February 1992)

A simple proportional-feedback algorithm for controlling chaos is presented. The scheme is a map-based variation ofa method recently proposed by Ott, Grebogi, and Yorke [Phys. Rev. Lett. 64, 1196 (1990) ] in which unstable periodic orbits embedded within a strange attractor are stabilized through deliberate perturbations of a system constraint. The simplified method offers advantages for control of systems in which more complicated algorithms might not be feasible due to short time scales or limited computational resources. Applications to chemical and biological models are presented to demonstrate the utility and limitations of the method. Low-dimensional chaos can usually be stabilized through proportional feedback of one parameter; in some cases, however, a linear combination of several parameters must be utilized.

I. INTRODUCTION

The ability to deliberately select and stabilize a particu­lar unstable periodic state from the aperiodic oscillations of a chaotic system offers intriguing possibilities for practical ap­plications. Ott, Grebogi, and Yorke (OG Y) have developed a general method for stabilizing unstable periodic orbits em­bedded within a strange attractor. 1 Their algorithm is based on targeting the stable manifold of a particular unstable orbit by introducing a controlled perturbation of a system con­straint. The general utility of the method has been demon­strated by stabilizing unstable orbits in the Henon map, 1 and controlling chaos in an experimental system consisting of a buckled magnetoelastic ribbon has been reported by Ditto, Rauseo, and Spano.2

Controlling chaos in chemical systems is of special inter­est, both from a practical perspective, such as in tank reactor processes, and from the standpoint of implications for dy­namical behavior in biological systems, such as self-regula­tion. We recently proposed a simple geometric algorithm for controlling chaos in a chemical system based on targeting an unstable point in the return map corresponding to a particu­lar unstable orbit in the continuous system. 3 The map-based method offers significant advantages in applications where targeting procedures must be minimized, e.g., as in control­ling high frequency chaos.

Advantages and limitations of the map-based method for controlling chaos are further examined in this paper. The algorithm and its application to a simple three-variable chemical model4•5 are presented in Sec. II along with an analysis of the stability of the method to errors and perturba­tions. In Sec. III we generalize the algorithm to overcome limitations that may arise in applications of the simple one­parameter scheme. Application of the algorithm to a four­variable biological model recently studied by Lengyel and Epstein6 as an example of diffusion induced chaos is de­scribed in Sec. IV. A judicious choice of controlling param­eter, monitored variable, and Poincare section, or a linear

a) To whom correspondence should be addressed.

combination of two or more controlling parameters allows utilization of the map-based scheme for control. We con­clude in Sec. V with an assessment of the algorithm in poten­tial applications.

II. ALGORITHM AND APPLICATION TO A SIMPLE CHEMICAL SYSTEM

A. Proportional-feedback algorithm

Consider a ID map of the formxn + k (p) = F(xn,p) de­scribing the chaotic evolution of a system. The map can be linearized in the vicinity of any particular fixed point Xs; e.g., for the period-l unstable point,

xn+l(p)=f[xn-xs(P)]+xs(p), lfl>l, (1)

where p is a system constraint andfis the slope of the map at Xs' Suppose that varyingp by the small amount op moves the unstable point without significantly changing f, the linear­ized map becomes

xn+l (p+op) =f[xn -xs(P+op)] +xs(P+op), (2)

where

dxs (p) xs(P+op) =op +xs(p).

dp (3)

The transitivity of a chaotic system ensures that it will visit the vicinity of the fixed point at some point in time. Suppose Xn = Xs (p) + Ax, where Ax <x, (p). If, at this mo­ment, p is changed to p + op such that Xn + 1 (p + 8p) = Xs (p), the system state is directed to the original unstable point upon the next iteration. If p is then switched back to its original value, the system would remain at Xs indefinitely, were there no errors in targeting and no perturbations. The necessary variation of p can easily be determined according to

op = f A.x==.Ax/g. if - 1 )dxs/dp

(4)

In essence, the control scheme samples the difference between the system state and the particular unstable point to

7506 J. Chem. Phys. 96 (10), 15 May 1992 0021-9606/92/107506-08$06.00 @ 1992 American Institute of PhYSics This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.120.242.61 On: Tue, 25 Nov 2014 12:19:27

PetroY, Peng, and Showalter: Controlling low-dimensional chaos 7507

be stabilized and proportionally varies a control parameter according to the difference. We require this difference (or the parameter variation) to be smaller than some limit defin­ing the control range; once within the range, the proportion­al-feedback algorithm is switched on. The magnitude of the control range is arbitrary and can be made very small; the smaller the range, however, the longer the average transient before it is visited. Once the system is stabilized, the param­eter range defined by the subsequent perturbations necessary to maintain stabilization is independent of the predefined control range and is typically extremely small. 3

The algorithm is applicable, in principle, to all the un­stable points corresponding to the unstable limit cycles of a strange attractor. The only requirement is that the attractor is described by effectively I D dynamics in the region around the unstable point of interest; whether or not the map is globally single valued is unimportant. The map need not be strictly single valued locally (as in a discrete system), but the extent of the local fractal structure should be small com­pared to the control range and the linear region of the fixed point.

The scheme can be further simplified by determining the proportionality factor g in Eq. (4) from the horizontal dis­tance of the shifted map to the original unstable point (see Fig. 3 of Ref. 3). In practice, the determination of g requires

1.4

1.0

0.6 . ' .. ": .. .... ". ". ". ". .. .. "" .....

" ............ ..

'. '.

.. '

measurements at two different parameter values to define two maps; the measured value of g is then used to calculate the necessary perturbations for control.

B. Application to a simple chemical system

The map-based algorithm has been applied to a three­variable model for isothermal chemical chaos. 3 The model is based on mass-action kinetics and can be studied as either an open4 or closed system.5 The rate equations for the open system are

da /3 2 -=fl(K+Y) -a-a , dr

(5)

a : = a + a/3 2 - /3,

dy O-=(3-y,

(6)

(7) dr

where a, (3, and yare dimensionless variables and fl, K, a, and 0 are dimensionless parameters. The general features of the attractor as well as the stabilized period-l limit cycle are shown in Fig. 1. The Poincare section defined by y = 15, r> 0 gives rise to a map that closely obeys ID dynamics. Because fl is directly related to the concentration of the reac­tant, it is chosen to be the controlling parameter. The vari-

., . ... ~.:"I"

FIG. 1. Strange attractor of chemical system defined by Eqs. (5)-(7). Param­eters are (jl,K,a,o) = (0.1540,65,5 X 10 - \2 X 10 - 2); Poincare section defined by r = IS, r> 0 shown as opaque sheet. Also shown is stabilized period· I limit cycle (solid trajectory). .' .:;, JO

:. : •. :r •.. :, ' .. " ............ ..

0.2

.... .. . ' . ...... .. .. .. . .. . . "~

" .. :, .

30 ,.4.0 10 ,.~O ,. ·

2.0 .... (J log

1.5

J. Chern. Phys., Vol. 96, No.1 0, 15 May 1992 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.120.242.61 On: Tue, 25 Nov 2014 12:19:27

7508 Petrev, Peng, and Showalter: Controlling low-dimensional chaos

ation of f.l is determined by monitoring the value of P on each return to the section, and period-I, period-2, and period-4 unstable limit cycles are readily stabilized (each with a dif­ferent value of g). The variable a may also be monitored for determining successive values of f.l for stabilization.

Intersections with the Poincare section before and dur­ing control yield values of Pn for construction of the map shown in Fig. 2(a). The nearly horizontal segment shows the mapping during control; it also defines the control range. Many different initial conditions were necessary to visualize the map in the control range, since once in the range the variation around Ps is extremely small. The enlargement in Fig. 2(b) shows that the map in the control range is not perfectly horizontal, corresponding to perfect control ac­cording to Eq. (10) (vide infra). The slope can be varied by

0 d 52

0 d co

~o + . cO

c:o... lD

(a)

0 d "'¢

0 d N

'?i

0

'4

o t<i

(\ i

i , I I

j

20.0

\

"'---40.0

\.

-------60.0

f3 n

80.0 100.0

~~----~------~----~~--~ 33.0 37.0 45.0 49.0

(b)

FIG. 2. (a) Map of /3" + I vs /3" from Poincare section in Fig. I before and during control. The control parameter Jl is varied around 0.1540 according to measured values of ~x = /3" - /3, in Eq. (4), where /3, = 40.568 and g = - 4.5 X 10". The horizontal section shows the control range as well as t~e ~!!lbilizing effect of control. (b) Blow-up of (a) around the control range, defined by 1/3" - /3, I..; 1.8.

varyingg, and provided that it is between - 1 and + 1, the orbit is stabilized. Thus the broken map describes the dy­namics of the overall system, Eqs. (5)-(7) and the propor­tional feedback provided by Eq. (4).

c. Effects of errors and perturbations

Errors in measurement as well as fluctuations from in­ternal and external sources are inevitable in any practical setting. In this section we examine how the algorithm be­haves under the influence of various sources of imperfection. We assume that the system is described by aID map and, except when indicated otherwise, assume that errors as well as noise are sufficiently small that the system remains in the linear region around the fixed point.

Consider a map ofxn + k vsxn near one of the k period-k fixed points. It is convenient to consider the system as having a new mapping equation during control,

x; =~ (xn -xs) +x" (8) /-1

xn+k =/xn - if-l)x;, (9)

where x; is the shifted fixed point. Substituting Eq. (8) into Eq. (9) yields

( 10)

Hence, in the absence of noise or errors in measuring proper­ties or setting parameters, the effects of which are discussed below, the system is stabilized at the original fixed point on the next return after falling within the predefined control range.

(a) Error in measuringxs. If the fixed point is measured as Xs + Es, x; and Xs in Eq. (8) are replaced by x; + Es and Xs + Es,

x; + Es = ~ (xn - Xs - Es) + Xs + E" /-1

(11)

and substituting Eq. (11) into Eq. (9) yields

xn+k =xs +/Es' (12)

Thus the system is stabilized at a position slightly away from the unstable point. The term/Es must be sufficiently small so that x n + k is in the linear region of the mapping around the original fixed point; fixed points with large slopes conse­quently require higher precision or wider linear regions for stabilization.

(b) Error in measuring X n • If the system state is mea­sured as Xn + En' the fixed point is shifted according to Xn + En instead of Xn; hence,

x; = ~ (xn + En - xs) + x s' /-1

and the new mapping becomes

with an effect analogous to case (a).

(13)

(14)

(c) Errorin measuringgor settingp. If there is a relative error Er in the shift of the fixed point due to errors in measur­ing g or setting p, Eq. (8) can be rewritten as

x; -Xs =~ (xn -xs)(l +Er ), /-1

(15)

J. Chern. Phys., Vol. 96, No. 10, 15 May 1992 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.120.242.61 On: Tue, 25 Nov 2014 12:19:27

Petrov, Peng, and Showalter: Controlling low-dimensional chaos 7509

and the mapping becomes

x,,+~ = -/€r(x" -x,) +xs •

The fixed point is stabilized when

I€rl < Ill/I·

(16)

(17)

The nonzero slope of the map in Fig. 2 reflects a slight error in the value of g. We note that higher precision is required to stabilize fixed points with large slopes.

(d) Nonlinearity in mapping or response. If the original mapping over the control range involves a second-order term,

x" + k = x; + /(x" - x;) + €I (x" - x; )2, (18)

the new mapping becomes parabolic with the extremum lo­cated at the fixed point,

(19)

If the response of the system to the controlling perturbation is slightly nonlinear,

x; -x, = /~ 1 (x" -xs) +€2(X" _X,)2, (20)

the new mapping is also parabolic,

X,,+k =Xs +€2(1-f)(X" _X,)2. (21)

In either case, the system is stabilized at the fixed point, only with a longer transient time.

We conclude that the proportional-feedback algorithm is robust in the presence of small errors, random perturba­tions, and nonlinearities.

III. UTILITY AND LIMITATIONS OF THE MAP-BASED ALGORITHM

A. Single-parameter scheme

The above discussion is based solely on maps. A map constructed from a Poincare section in the state space of a continuous system reflects the behavior of the system on the strange attractor. The behavior off the attractor, on the other hand, is affected by the stable manifolds, and is not described by the map. Because the map is not applicable for points off the attractor, it cannot be utilized when the system state is significantly displaced from the attractor.

The proportional-feedback algorithm, as presented above, requires any movement of the unstable point to be along the original section of the attractor as the control pa­rameter is varied; otherwise, there is no direct correspon­dence between the map and the original system once the parameter is varied. Furthermore, unless the unstable point moves along the attractor, monitoring different variables may suggest different op values when projecting the section to a particular axis. This can be understood by considering an extreme case where the unstable point on a Poincare sec­tion is shifted nearly perpendicular to the axis for a particu­lar variable. Monitoring that variable would suggest a large value for op, while monitoring another variable would result in a smaller value.

The requirement that the unstable point move along the attractor may seem to unduly restrict application of the algo-

rithm. However, provided that the shifted fixed point re­mains close to the attractor, the dynamical behavior is gov­erned primarily by the unstable manifold, and the associated unstable periodic orbit is readily stabilized by the algorithm. Figure 3 (a) shows 12 Poincare sections of the attractor in Fig. 1 for two values of /1, one corresponding to the center and the other to the maximum of the control range. The two sets of sections overlap at the resolution of this figure. With the higher resolution shown in Fig. 3(b), we see that the attractor is shifted only slightly away from its original posi­tion on varying /1; however, the unstable limit cycle shifts significantly along the unstable manifold. This behavior seems to be typical for many systems exhibiting low-dimen­sional chaos. For example, in the Lorenz7 and Rossler8 sys­tems one can easily find Poincare sections and control pa­rameters such that the unstable limit cycles are moved along the attractors. We have applied the single-parameter scheme to each of these systems and readily stabilized the corre­sponding period-l unstable orbits.'l

If all the accessible parameters significantly shift the at­tractor away from the unstable manifold of the fixed point, it is often possible to retain the simplicity of the scheme by monitoring another variable or by adjusting the value of g around its measured value. Consider, for example, a two­dimensional Poincare section parallel to the x-y plane in a three-dimensional x-y-z state space. If the stable eigenvec­tor is perpendicular to the x axis, then monitoring x results in targeting the stable eigenvector (as in the OGY scheme), and the orbit is stabilized.

B. Multiple-parameter scheme

We now describe a more general approach which relies on a linear combination of parameters such that the unstable limit cycle is moved along the attractor. Consider a Poincare section in an n-dimensional state space and suppose that n-l parameters (PI ,P2 , ... ,p" _ I ) each shift the unstable point in different directions corresponding to the vectors JX,/Jpi. It is generally possible to find a linear combination of these vectors such that the sum vector lies along the attractor. Thus the system can be stabilized by varying the n-l param­eters according to this linear combination. Equation (4) is rewritten in the vector form

(OPI,OP2,.··,OP"_I) = (lIg'Y2/g, ... ,Yn_l/g)Llx, (22)

where 0Pi is the necessary variation in Pi and Yi is the linear combination coefficient for Pi. The g factor is determined from the shift of the map on varying the parameters accord­ing to the linear-combination coefficients. The effect of the combined-parameter perturbation is the same as that in the single-parameter scheme; therefore, the error analysis pre­sented in Sec. II C is applicable to the generalized scheme. A two-parameter application is demonstrated with a biological system in the next section.

IV. CONTROLLING CHAOS IN A DIFFUSIVELY COUPLED BIOLOGICAL SYSTEM

Lengyel and Epstein6 recently reported a study of diffu­sion-induced oscillations and chaos in a two-cell system con­sisting of coupled two-variable oscillators. Each cell is de-

J. Chern. Phys., Vol. 96, No. 10, 15 May 1992 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.120.242.61 On: Tue, 25 Nov 2014 12:19:27

7510

30.0

20.0

10.0

0.0

{al

0 . 0 0 ..--

0 0 00

0

c:a.. O lO

o . o ...q-

o o

\ \

Petrov, Peng, and Showalter: Controlling low-dimensional chaos

30.0 ~ 60.0

40.0

90.0

•• ~!J • .. .

• . "'

rP

°eo o

--.----~-------

~~------~------~--------~----~ 0.00 0.01 0.02 0.03 0.04

(bl

J. Chem. Phys., Vol. 96, No.1 0, 15 May 1992

FIG. 3. (a) Twelve different Poincare sections of two attractors, one corre­sponding to Fig. I with.u = 0.1540 and the other slightly shifted with .u = 0.1544. Period-I unstable limit cy­c1eon each section for.u = 0.1540,.; at­tractor shown by solid circles. Period-I unstable limit cycle on each section for .u = 0.1544, 0; attractor shown by smaller solid circles. The Poincare sec­tions are defined at 30· intervals with re­spect to the a-f3 plane around an axis at f3 = 17, r = IS. (b) Poincare sections at r= IS, r>O of attractors with .u = 0.1540 and .u = 0.1544. Period-I unstable limit cycle on section for .u = 0.1540; .; attractor, D. Period-I unstable limit cycle on section for .u = 0.1544, .; attractor, O. The inset shows the unstable point moving pri­marily along direction of the unstable manifold while shifting only slightly away from the original section.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.120.242.61 On: Tue, 25 Nov 2014 12:19:27

Petrov, Peng, and Showalter: Controlling low-dimensional chaos 7511

scribed by the Degn-Harrison model lO for the respiratory behavior of a Klebsiella Aerogenes bacterial culture, each with identical parameter values. The diffusively coupled sys­tem can be written in the dimensionless form

XI = bl - XI - xlyl/(l + qlxn + D"c(x2 - XI)'

(23)

YI = al - XIYI I( 1+ qlxi) + Dy C(Y2 - YI), (24)

x2 = b2 - Xl - x 2Y zl( I + q2X~) - D"c(xz - XI ),

(25)

Y2 = a2 - X 2Y2/(l + q2X~) - Dyc(yz - YI), (26)

where X and yare variables, a, b, and q are parameters of the uncoupled model, Dx and Dy are diffusion coefficients for X

and y, respectively, C is the coupling strength related to the physical measures of the cells (assumed to be of the same volume) and the permeable membrane, and subscripts 1 and 2 denote each of the two cells. The coupled system exhibits very complex behavior including multistabiIity among oscil­latory states and inhomogeneous chaos, even though the un­coupled cells exhibit only stationary behavior for the same parameter val ues. (,

We choose a coupling strength corresponding to a point within the chaos region and carry out integrations with the same parameter values in each cell. In order to simulate an experimental study, the attractor is embedded in a three­dimensional time-delay state space using XI (t), XI (t - r), and XI (t - 2r), as shown in Fig. 4. A Poincare section de­fined by XI (t - 2r) = 1.6, XI (t - 2r) <0 yields the map shown in Fig. 5. Monitoring XI (t - r) and varying a l ac­cording to the single-parameter scheme results in stabiliza­tion; however, it is necessary to adjust the value of g from its measured value. Monitoring Xl (t) fails to provide correct predictions for control, even when g is adjusted. This indi-

2.41 ___ L_

1.9

1.4

0.9

FIG. 4. Strange attractor of diffusively coupled Degn-Harrison model for bacterial respiration in time-delay state space of X, (t), X, (t - r), and x, (t - 2r) with r = 3. Parameters: (a , = a2,b , = b2,q, = q2,Dx>Dy,c)

= (8.9,11,0.5,1 X 10-\1 X 1O-',4X 10-'). Initial conditions: (XI,X2,Y"Yz) = (\,1,1,0.5).

FIG. 5. Map of x;' + I (t - r) vs x;' (t - r) constructed from Poincare sec­tion at x, (t - 2r) = 1.6, x, (t - 2r) <0 of attract or in Fig. 4. The period-I unstable limit cycle is located in the section at x; (t), x; (t - r) = 1.196,

2.018.

cates that the stable eigenvector lies almost perpendicular to the XI (t - r) axis in the time-delay space.

We were unable to find a parameter that moves the peri­od-I fixed point along the attractor in the time-delay state space. Shown in Fig. 6(a) (top) is a segment of the section and the period-I unstable point. Also shown are sections and associated unstable points resulting from changing the value of bl (middle) and a l (bottom). Neither of these param­eters results in a shift of the unstable point along the unstable manifold; rather, the entire attractor is shifted away from its original position.

The multiple-parameter scheme described in Sec. III B allows control when the attractor is shifted as in Fig. 6(a). The procedure is a simple extension of the single-parameter scheme. Monitoring XI (t - r) on the Poincare section and varying the parameters a l and b l according to an appropri­ate linear combination allows the period-l unstable limit cy­cle to be stabilized. From Fig. 6(a), we find the linearcombi­nation

ax~ ax~ ---1.2--, Ja l Jb l

(27)

where x; = [x~ (t - r) ,x~ (t) ] , results in a sum vectorlying along the attractor, as shown in Fig. 6(b). From the map in Fig. 5 and the map shifted according to the linear combina­tion, the g factor associated with a I is found to be - 1.3. Therefore, the variations of parameters a I and b l are

t5a l = - Llxl (t - r)/1.3, (28)

t5b l = £lx l (t - r)/1.08, (29)

corresponding to g = - 1.3 and Yz = - 1.2 in Eq. (22). We apply the control algorithm to only one of the cells

by monitoring £lx l (t - r) and varying parameters a l and b l ; both cells, however, are brought under control. Figure 7 shows time traces for Xl and Xz before and after stabilization

J. Chern. Phys., Vol. 96, No.1 0, 15 May 1992 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.120.242.61 On: Tue, 25 Nov 2014 12:19:27

7512

"i"' ~ ....-

X

(al

(bl

Petrov, Peng, and Showalter: Controlling low-dimensional chaos

....­

~-?----------------------------------~ ....-

0 ~ ....-

m ....-• ....-

CO ....-. ....-

....-~ ....-

o N . ....-

m ....­. ....-

CO ....-

4, ... .--r-..... -~ .

."fPl' ,,..

,..,J- ......... - . ..... "';~ . .,....

_~ •• JII".c ,

2.00

" • :rr'" .,co. ...

.... 'f" ... .c-­..•.

.~ . .: ~ ...

2.03

~~--------~----------r-------~ 2.00 2.03

J. Chem. Phys., Vol. 96, No. 10, 15 May 1992

FIG. 6. (a) Poincare section of attractors for (a"b,) = (8.9,11), top; (8.9,10.99), middle; and (8.89,11), bottom. Shift of the period-l unstable limit cycle is indicat­ed by arrows. The two shifted attractors were obtained by varying the correspond­ing parameters during integration. (b) Ef­fect of varying linear combination of a, and b, determined from (a) according to Eqs. (27)-(29). The unstable limit cycle is moved along the attractor with obI/Oat = r2 = - 1.2.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.120.242.61 On: Tue, 25 Nov 2014 12:19:27

Petrov, Peng, and Showalter: Controlling low-dimensional chaos 7513

a ...;,.----------------,

a N

IX)

d (q

r:--. ~ .;t::.. X

a 0+------..,--------1

3600.0 3800.0

t 4000.0

FIG. 7. Time traces for XI and X2 shortly before and after onset of control showing chaos and subsequent stabilization of period-I limit cycle in cell I (top) and cell 2 (bottom). Control in cell 1 was switched on at t = 0 and control range was visited at t = 3834; control range defined by Ix~ (t - r) - x: (t - r) I <0.03. (Average time to visit control range in ten calculations was about 50 oscillations or t z. 1000.)

of the period-l limit cycle. Note that during control the peri­odic oscillations in the two cells differ significantly from each other in amplitude, reflecting the asymmetry of the sys­tem. As in the three-variable chemical system discussed in Sec. II B, the response to the control perturbation occurs rapidly, and after stabilization subsequent perturbations are very small (typically around 10 - 4 for both al and hi ). The biological system requires a much longer time to visit the control range, however, due to the relatively small range. A smaller control range was necessary in this system to avoid overlapping multiple attractors at these parameter values.

V. CONCLUSION

Chaotic systems offer great flexibility for adjusting from one type of periodic behavior to another through deliberate perturbations of a system constraint. I

,3 For systems giving rise to effectively 10 maps in the vicinity of an unstable fixed point corresponding to an unstable limit cycle, control can be achieved through a simple map-based algorithm. The in-

completeness of maps in describing the transient dynamics of perturbed continuous systems may occasionally prevent successful application of the single-parameter algorithm; however, it is usually possible to retain the simplicity by ap­propriately selecting the Poincare section, the control pa­rameter, and the monitored variable, or by adjusting the pro­portionality factor. The more general method involving use of a linear combination of two or more parameters with con­stant coefficients provides greater assurance of control. In experimental settings, the multiple-parameter scheme may be realized in a straightforward manner analogous to the application of the simpler algorithm. A practical considera­tion for experimental applications is that the algorithm can be employed using embedded time-delay state space, allow­ing control when only one system variable is accessible.

Note added in proof. A recent experimental study of con­trolling chaos with proportional feedback in a multimode laser containing an intracavity crystal has been reported by Roy et alY This method has also been used by Huntl 2 to stabilize high-period orbits in a chaotic diode resonator. These studies used intermittent rather than continuous ap­plication of the algorithm; we have also found this modifica­tion to be advantageous in certain cases.

ACKNOWLEDGMENTS

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

I E. Ott, C. Grebogi, and J. A. Yorke, Phys. Rev. Lett. 64, 1196 (1990). 2 W. L. Ditto, S. N. Rauseo, and M. L. Spano, Phys. Rev. Lett. 65, 3211

(1990). 3B. Peng, V. Petrov, and K. Showalter, J. Phys. Chern. 95, 4957 (1991). 4B. Peng, S. K. Scott, and K. Showalter, J. Phys. Chern. 94, 5243 (1990). 5S. K. Scott, B. Peng, A. S. Tomlin, and K. Showalter, J. Chern. Phys. 94,

1134 (1991). 6 I. Lengyel and I. R. Epstein, Chaos 1, 69 (1991). 7E. N. Lorenz, J. Atmos. Sci. 20,130 (1963). 80. E. Rossler, Phys. Lett. A 57,397 (1976). 9Period-1 was stabilized in the Lorenz system (Ref. 7) (parameters:

a = 10, R = 28, b = 2.6666) by monitoring Zmax and varying R and in the Rossler system (Ref. 8) (parameters: a = 0.2, b = 0.2, c = 5.7; Poincare section: X = 2.0, x> 0) by monitoring y and varying c.

IOH. Degn and D. E: F. Harrison, J. Theor. BioI. 22, 238 (1969). II R. Roy, T. W. Murphy, T. D. Maier, Z. Gills, and E. R. Hunt, Phys. Rev.

Lett. 68, 1259 (1992). 12E. R. Hunt, Phys. Rev. Lett. 67,1953 (1991).

J. Chem. Phys., Vol. 96, No. 10, 15 May 1992 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

129.120.242.61 On: Tue, 25 Nov 2014 12:19:27