PHYSICAL REVIEW E VOLUME 47, NUMBER 2 FEBRUARY 1993

Controlling chaos in highly dissipative systems: A simple recursive algorithm

R. W. Rollins, P. Parmananda, and P. SherardDepartment of Physics and Astronomy and Condensed Matter and Surface Sciences Program, Ohio University,

Athens, Ohio $5701-Bg79(Received 21 October 1992)

We present a recursive proportional-feedback (RPF) algorithm for controlling deterministic chaos.The algorithm is an adaptation of the method of Dressier and Nitsche [Phys. Rev. Lett. 68, 1 (1992)]to highly dissipative systems with a dynamics that shows a nearly one-dimensional return map ofa single variable X measured at each Poincare cycle. The result extends the usefulness of simpleproportional-feedback control algorithms. The change in control parameter prescribed for the nthPoincare cycle by the RPF algorithm is given by 6'p„= K(X„—XF) + Rbp„ i, where Xs is thefixed point of the target orbit, and K and R are proportionality constants. The recursive term isshown to arise fundamentally because, in general, the Poincare section of the attractor near X~ willchange position in phase space as small changes are made in the control parameter. We show howto obtain K and B from simple measurements of the return map without any prior knowledge ofthe system dynamics and report the successful application of the RPF algorithm to model systemsf'rom chemistry and biology where the recursive term is necessary to achieve control.

PACS number(s): 05.45.+b, 06.70.Td, 87.10.+e

I. INTRODUCTION

A control algorithm for stabilizing unstable saddle or-bits that densely populate a chaotic attractor by ap-plying small changes to a single control parameter wasfirst proposed by Ott, Grebogi, and Yorke [1] (OGY).The general OGY algorithm is dramatically simplified inhighly dissipative systems that are well characterized by aone-dimensional return map. Peng, Petrov, and Showal-ter [2], using model systems, and Hunt [3], by experi-mentally controlling the diode resonator circuit, demon-strated that control can be achieved by applying a bp„on the nth Poincare cycle that is simply proportional tothe deviation of X„from the fixed point X~ of the targetorbit on the return map. In this paper we refer to thistype of control algorithm as simple proportional feedback(SPF). Many nonlinear chemical and biological systemstend to be highly dissipative and SPF would seem to beapplicable.

However, Petrov, Peng, and Showalter [4] recentlydemonstrated that the SPF algorithm does not alwayswork for some choices of the control parameter even whenthe system is shown to have one-dimensional map be-havior. If the Poincare section of the attractor movesin phase space when the parameter p is changed and ifp„ i g p„, then X„is not on the attractor correspondingto p = p„and the usual static one-dimensional map isnot valid. Petrov, Peng, and Showalter suggest a clevermodification of SPF in such a case. They point out that,at least in some cases, a linear combination of control pa-rameters can be found that does not move the Poincaresection. The linear combination can then be used as thesingle control parameter with the SPF algorithm to af-fect control. But, in a real experimental situation, it maynot be possible to get experimental access to a particularcontrol parameter, and the required linear combination

of parameters needed to save the SPF method may beimpossible to realize.

Dressier and Nitsche [5] recently showed that a similarsituation exists when time-delay coordinates are used toreconstruct the phase space in which the OGY algorithmis applied. They show that the general OGY algorithmmust be modified such that the correction tIp„appliedduring the current cycle depends not only on the vectordeviation of the system from the fixed point but also onthe correction bp„~ made during the previous cycle.

In this paper we apply the Dressier and Nitsche [5]approach to the case where the system dynamics is re-constructed from measurements of a one-dimensional re-turn map. The result is a recursive proportional-feedback(RPF) algorithm that allows control with any choice ofa single control parameter. In the following sections wederive the RPF algorithm, show how the proportionalityconstants necessary for implementation of the algorithmare determined, and apply the algorithm to models ofboth chemical and biological systems where the recur-sive term is necessary.

II. DEVELOPMENT OF RPF ALGORITHM

The one-dimensional return map is the result of highdissipation, which causes the reduction of the chaotic at-tractor to a very thin (nearly two-dimensional) structureembedded in a three-dimensional phase space (X,Y,Z)of the system. Neglecting the thickness of the attrac-tor, a Poincare section through the two-dimensional at-tractor at Z = Z~ gives a one-dimensional (1D) curve.(Note that Z could be dX/dt obtained from measure-ments of a time series of the single variable X so thatthe Poincare section, and one-dimensional map, can beexperimentally obtained from measurements of a singlevariable. ) For a particular value of the control param-

47 R780 1993 The American Physical Society

47 CONTROLLING CHAOS IN HIGHLY DISSIPATIVE. . . R781

X +i = f(X;p i, p~). (2)

Note that if Eq. (1) does not depend on p„ i (i.e. , thePoincare section of the attractor does not move in phasespace as p is changed), then the simple map of the previ-ous paragraph is recovered Figur. es 1(a) and l(b) showhow small changes in two different model parameters af-

eter p, the one-dimensional curve may be expressed asY = h(X; p), and the Poincare map equation for X is ofthe form X„+i —g(X„,Y„;p) = g(X„,h(X„;p); p), orX„+r = f(X„;p). This is the usual 1D return map thatdepends parametrically on the single parameter p.

Now we consider the case where p is changed by a smallamount bp about p = po on each Poincare cycle. If wechange from p„ i to p„at the beginning of the nth cycle,then the system will be on the attractor correspondingto p„q at the start of the cycle. We assume the highdissipation ensures that the system will settle onto theattractor corresponding to p„by the end of the Poincarecycle. Thus,

Y„' = h(X„;p„ i)

and at the end of the nth cycle X = X„+yg(X„,Y„;p„)= g(X„,h(X„;p„ i);p„). Thus, the gen-eral 1D return-map equation is

feet the Poincare section of a chaotic attractor for a modeldescribing metal passivation in an electrochemical celldiscussed in Sec. IV. If model parameter s is changedslightly, then Fig. 1(a) shows that the attractor movessuch that its Poincare section does not shift position.However, if model parameter p is changed, the Poincaresection moves as shown in Fig. 1(b). The SPF algorithmfails with p as the control parameter.

Now we consider the natural dynamics of the systemdescribed by Eq. (2) near the fixed point of the period-one orbit for p = pp' , Xp = f(XF,pp, pp). To first orderin 6X„=(X„—X~), bp„ i, and bp„, we have

bX„+g ——pbX„+ mbp„g + vs„,where p = Bf (Xy', pp, pp)/BX, iv = (Bg/BY)(B'h/Bp„ i), and v = Bf(X~,pp, pp)/Bp„Not. e that p isthe slope of the one-dimensional return map at the fixedpoint for p = pp and tv = 0 if h is independent of p.

We want to determine the bp„such that the system isbrought to the fixed point as quickly as possible. Severalstrategies could be applied. We use the strategy 6X„+2 =0 and b'p„+i = 0 and then the first and second iterate ofEq. (3) give the recursive control algorithm:

bp„= KbX„+Bbp„where

0.114

0.112—

0.110—Y

0.108—

0.106—

0 104—I

0.1170

0.114

0.112—

0.110—

Y0.108—

0.106—

(b)0.1 04—

I

0.1170

0.1095-

0.1090-

.1085-0.11715 0.11720 0.11725

I

0.1175X

I

0.1180

0.11725

l

0.1175X

I

0.1180

0.1095-le~ ~

0.1090-~ ~

0.1085-0.11715 0.11720

K=—,andR=-p PtU

(pv + tv)'

(pv + tv)'

III. DETERMINATION OF PARAMETERS

The parameters necessary to implement the RPFmethod (X~, p, to, and v) can be easily determinedfrom two types of 1D return-map measurements. For aconstant parameter value near po, a plot of X„+~ ver-sus X„defines the 1D mapping function f,(X;bp)f (X;pp + bp pp + bp). The value of X~ = f,(X+, 0) andthe slope p of the function f, at X = X~ are obtainedfrom X'„data taken with 6p = 0. The values of v and tp

are obtained from the sequence X„'s measured while re-peatedly alternating the control parameter up to go+ bpfor one Poincare cycle and then back to po for the next cy-cle. Alternate pairs (X„,X„+r) taken from this measuredsequence lie on two maps f„(X;6p) and fb(X; 6p) wheref„(X„;6' ) —= f(X„;po,po + 6p) = f,(X„;0)+ 6p dfb(X„;6p)—:f(X„;pp + 6p, pp) = f,(X„;0) + to6p. Anexample of the three return maps is shown in Fig. 2. Itis easiest to measure the fixed points of these two maps,XF = f„(XF;bp) and X+ ——fb(X+, bp). We define g„and gg such that X+ ——X~+g„bp and X& ——X~+gybe.Then, as shown in Fig. 2, tv and v can be expressed interms of p, g„, and gb by tv = (1—p,)g„, and v = (1—p, )gb.Now, Eq. (5) becomes

FIG. 1. Poincare sections for the chemical model dis-cussed in Sec. IV. (a) Circles, for model parameter s = s0 =9.7 x 10;and squares, for s = so+ bs, where bs = 7 x 10(b) Circles, for model parameter p = p0 = 2 x 10; andsquares, for p = go+ bp, where bp = 7 x 10 . The enlargedsymbol is the position of the fixed point for the period-oneorbit in each case.

2P & Pgb

(p —1)(pg~+ gb)'

(pg~+ gb)

Following OGY, we define the shift in the fixed pointfor constant shift bp in control parameter as gbp. Then,from the above definitions, we have the following sumrule:

R782 R. W. ROLLINS, P. PARMANANDA, AND P. SHERARD 47

0 1 1 725 s ~ I ~ I ~ ~ 1 ~ ~ I ~ ~ ~f' '.~ g /

~0

rrr

/

g QP, r'~ I g ——~i„.

/X„+, '. r

'II.,'gb6 P

r I ~ 4

/I

I/

r I ~ ~ ~

r It g~ a

~, h0.11710.' ~

0.11710 0.11715 0.11720

0.11715—

0.11725X„

FIG. 2. Diagram showing the three (nearly 1D) mappingfunctions defined in the text f,(x; 0), f„(x;6p), and fb(z; 6p)in the neighborhood of the fixed point X~. The data are takenfrom the numerical solution of the chemical model discussedin Sec. IV. The fractal nature of the return map is clear atthis scale and we choose corresponding fixed points on eachmap. The geometrical relationships between the parametersn and v and the parameters related to the shift in the fixedpoints g„and gg are shown.

IV. APPLICATION TO MODELS

We now apply the RPF control algorithm to stabi-lizing periodic orbits within a chaotic attractor in twomodel systems where SPF fails. The first system is akinetic rate-equation model for metal passivation [6] andthe second system is a diffusively coupled two-oscillatormodel [4, 10, 11] for the respiratory behavior of a bacte-rial culture. Both cases lead to nearly 1D return mapsbut SPF fails for several control parameters. A stan-dard fourth-order Runge-Kutta algorithm was used tointegrate the model equations.

In preparation for performing experimental control ofchaotic behavior in a chemical system, we attempted con-trol by computer simulation using a mathematical modelof a similar system. We found that the SPF algorithmfailed in controlling chaos in the model and, out of ne-cessity, the RPF algorithm described in the previous sec-tions was developed and successfully applied. The chem-ical system under consideration is the passivation of thereactive surface of a metal electrode in an electrochemi-cal cell. The chemical kinetics of the passivation modelincludes the formation of two surface films, MOB andMO, where M represents the metal atom. It combineselements from surface reaction models by Talbot and Ori-ani [7] for MOH and by Sato [8] for MO formation. Theehemieal kinetics lead to the dimensionless equations [9]

m+vg = = g~+gb.(1-~)

Also, note that if m = 0 then gb = 0, g„=g, and R = 0,so that the RPF algorithm reduces to the SPF algorithmbp„= [p/(p —l)g]6X„ofPeng, Petrov, and Showalter [2).

Y = p(l —8QH —8Q) —qY;

8QH = Y(1 8QH 8Q) [exp ( P8QH) + I ]8QH

+ 2s8Q(1 —8QH —8Q)

8Q = r8QH —s8Q(l —8QH —8Q),

(9)(10)

Xl 61 Xl 2 lyl/(1 + ql2'l) + D C(X2 2 1)

y'r = ar —xryr/(1 + qlx, ) + D„c(y2 —yl),

where Y is the concentration of metal ions in the elec-trolyte, 8QH and 8Q are the respective fraction of themetal surface covered by each film, p, q, r, and s are pa-rameters related to chemical rate constants, and P repre-sents the non-Langmuir nature of MOH film formationin the Talbot-Oriani model. The system has been studiedrecently in some detail [9] and is chaotic for parametervalues (p, q, r, s, P) = (2.0x 10 4, 1.0x10 s, 2.0x109.7 x 10 s, 5.0). The discussion below concerns the be-havior of the system in the neighborhood of this point inparameter space.

In an experimental cell, the parameter p is related tothe anodic potential, which can be changed by the ex-perimenter. The parameter s, on the other hand, is re-lated to rate constants that are not easily accessible tothe experimenter. Figures 1(a) and 1(b) show that thePoincare section moves when the experimentally accessi-ble parameter p is changed (the variable X in this case isthe metal-oxide film coverage 8Q and the Y' coordinate isproportional to the concentration of metal ions in solu-tion). Attempts to control using the SPF algorithm weresuccessful when using s but failed when using the experi-mentally accessible parameter p. The RPF algorithm wasapplied to stabilize the period-one unstable saddle orbit.We used bp = 5 x 10 when alternating p to obtainf„(X';6p) and fb(X; 6p). Figure 2 shows the three returnmaps f,(), f„(), and fb() for X —= 8Q taken when 8QH

passes the Poincare plane 8QH = 0.3125 with 8QH ( 0.The values X~ = 0.1171692,p = —2.44, g„=4420, and

gb = —3230 were obtained from data as shown in Fig. 2.Equation (6) then gives K = 1.24 x 10 4 and R = 0.562for the RPF algorithm. While the system was on thechaotic attractor the control was turned on and the RPFcorrection was applied whenever IX„—XF

I( 4 x 10

The results of turning the control on and off are indicatedby the time series of Ho at the Poincare section shown inFig. 3. We never observed the system to fall off the stabi-lized period-one orbit so long as the RPF algorithm wascontinually applied. We also used the RPF algorithm tostabilize a period-two orbit by using the second iterateof the Poincare return map. The properties of the sec-ond iterate 1D maps gave X~ = 0.117497 88, p, = —4.75,g„= 3870, and gb = —3259. This gave K = 1.18 x 10 4

and R = 0.715. The results of turning the control on arealso shown in Fig. 3.

Finally, we report the successful application of theRPF algorithm to a system for which the SPF was al-ready shown to fail (if a single control parameter wasused) by Petrov, Peng, and Showalter [4]. The respira-tory behavior of bacterial culture has been modeled bythe dimensionless equations [4, 11]

47 CONTROLLING CHAOS IN HIGHLY DISSIPATIVE. . . R783

0.1180 2.3 ' '~ i.r I

0.1 1 76 —.

0-2.1—

I ~

0.1172— 1.9—

0.1 1 68 '-'-: "-:-.'-"-.—.'--:-'-.-'-"-"'-=~ ~ '

500 1000 1500 2000

Iteration NumberFIG. 3. Sequence of 8o values taken at the Poincare sec-

tion (described for the chemical system in Sec. IV) as the RPFcontrol for period-one orbit is turned on and o8' and then RPFfor control of period-two orbit is turned on.

x2 —b2 —x2 —xzy2/(I + q2x2) —D,c(x2 —xi), (13)g2 = &2 x2y2/(I + q2x2) Dyc(y2 yl) (14)where x and y are system variables; a, b, and q are param-eters of the uncoupled oscillators; D~, D„are the x andy diffusion coeflicients; and c is the coupling strength.Petrov, Peng, and Showalter [4] found. that SPF con-trol using a single control parameter generally was notpossible. They were unable to find a parameter thatdid not move the Poincare section of the attractor inphase space. They were successful if they chose a spe-cial linear combination of aq and bq such that the Bxedpoint moved along the attractor that remained station-ary in the phase space. We chose the single parameteray to affect the control using the RPF algorithm. Con-trol was applied to a period-one orbit embedded in thechaotic attractor at the same point in parameter spaceas used by Petrov, Peng, and Showalter [4] (ai = as,bi = b2, qi = q2, D» D» c)= (8.9, ll, 0.5, 1 x 101 x 10, 4000). We used bai = 0.03 when alternatingai to obtain f„(x;bai) and fb(x; bai). The return mapswere taken from a series of xq„ taken when yq passes theplane yz = 13.45 with y'z ) 0. The values X~ = 2.033,p, = —2.34, ga = —0.2302, and gb = —0.1665 weretaken from data like that shown in Fig. 2. Equation(6) then gives K = —4.403 and R = —1.046, which wereused to implement the RPF algorithm. While the sys-tem was on the chaotic attractor the control was turnedon and the RPF correction was then applied whenever]xi„—X~] ( 0.01. The results of turning the control

.7, . . . ' '". I ~ ~ ~ . i ~ . ~' ~ ~ ~ I

500 1000 1500 2000

Iteration NumberFIG. 4. Sequence of x~ values taken at the Poincare sec-

tion (described for the biological system in Sec. IV) as RPFcontrol is turned on and ofF several times.

on and off are indicated by the time series of xi at thePoincare section shown in Fig. 4. As with the RPF con-trol of the chemical system, we never observed the systemto fall oif the stabilized periodic orbit so long as the RPFalgorithm was continually applied.

V. CONCLUSION

A RPF algorithm was developed for controlling chaosin systems described by one-dimensional return maps.Previous proportional-feedback algorithms, at least inthe small-perturbation regime, are generally limited tocontrol parameters that do not afFect the position of thePoincare section of the attractor in phase space. We showthat the recursive algorithm removes this limitation andyet is easy to implement in real experimental systems.We believe the simple RPF algorithm is widely applica-ble to systems found in physics, chemistry, and biology.Our preliminary results on applying RPF to control areal chemical system are encouraging.

ACKNOWLEDGMENTS

We have benefited greatly from discussions with AlanMarkworth and Kevin McCoy of the Battelle MemorialInstitute in Columbus and with Jerry Orosz, Don Week-ley, Gregg Johnson, Earle Hunt, and Howard Dewald atOhio University. This work was supported in part by aBattelle subcontract of the Electric Power Research In-stitute (EPRI) research project Contract No. RP2426-25 and by Ohio University Research Challenge GrantNo. RC89-107.

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[3] E. R. Hunt, Phys. Rev. Lett. 67, 1953 (1991).[4] V. Petrov, B. Peng, and K. Showalter, J. Chem. Phys.

96, 7506 (1992).[5] U. Dressier and G. Nitsche, Phys. Rev. Lett. 68, 1 (1992).[6] Alan J. Markworth, J. Kevin McCoy, Roger W. Rollins,

and Punit Parmananda, Apphed Chaos, edited by JongHyun Kim and John Stringer (Wiley, New Pork, 1992),Chap. 12.

[7] J. B. Talbot and R. A. Oriani, Electrochim. Acta 30,1277 (1985).

[8] N. Sato, in Passivity of Metals, edited by R. P. Franken-thal and J. Kruger (The Electrochem. Society, Penning-ton, NJ, 1978).

[9] A. J. Markworth, J. K. McCoy, R. W. Rollins, and P. Par-mananda, EPRI Report No. RP2426-25 (1991);J. KevinMcCoy, Punit Parmananda, Roger W. Rollins, and AlanJ. Markworth (unpublished).

[10] H. Degn and D. E. F. Harrison, J. Theor. Biol. 22, 238(1969).

[11] I. Lengyel and I. R. Epstein, Chaos 1, 69 (1991).