A simple method for controlling chaosCathal Flynn and Niall Wilson Citation: American Journal of Physics 66, 730 (1998); doi: 10.1119/1.18940 View online: http://dx.doi.org/10.1119/1.18940 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/66/8?ver=pdfcov Published by the American Association of Physics Teachers Articles you may be interested in Effective suppressibility of chaos Chaos 23, 023107 (2013); 10.1063/1.4803521 Proportional feedback control of chaos in a simple electronic oscillator Am. J. Phys. 74, 200 (2006); 10.1119/1.2166367 An application of the least-squares method to system parameters extraction from experimental data Chaos 12, 27 (2002); 10.1063/1.1436501 The control of dynamical systems—recovering order from chaos— AIP Conf. Proc. 500, 551 (2000); 10.1063/1.1302686 New classification of chaos AIP Conf. Proc. 469, 299 (1999); 10.1063/1.58510

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A simple method for controlling chaosCathal Flynn and Niall WilsonDepartment of Physics, University College, Galway, Ireland

~Received 3 November 1997; accepted 27 January 1998!

A new method for the control of chaos is presented. The procedure is based on proportionalfeedback and is very simple to implement. The presentation is pedagogical, and in order to illustratethe method, we apply it to the He´non map and to the Lorenz equations. Finally, we look at thestability of the control algorithm in the presence of noise. ©1998 American Association of Physics

Teachers.

I. INTRODUCTION

In recent years, a considerable amount of interest has beenshown in attempts to control the chaotic behavior of systems.Since the presence of chaos in many systems is often unde-sirable~due to the irregular and unpredictable dynamics thatresult! a number of different methods have been developedwhich to a greater or lesser degree have succeeded in sup-pressing the chaotic behavior and in some cases renderingthe response of the system periodic. For low dimensionalsystems the two basic methods that have been developedinvolve ~a! the application of a small forcing term or modu-lation directly to the system parameters which results in theunderlying dynamics of the system being changed1,2 and~b!a type of proportional feedback such as that developed byOtt, Grebogi, and Yorke3 ~OGY!. In particular, the OGYmethod has proven to be very effective both in numericalsimulations4 and also in physical experiments.5,6 These ex-periments showed that the OGY method is capable of reduc-ing chaotic dynamics in a number of different systems toregular, periodic behavior.

The method of control presented here is conceptuallysimilar to, yet simpler than, the OGY method. Although themethod is less general than that of OGY, it is particularlywell suited to being used to demonstrate many important

concepts of nonlinear dynamics at an undergraduate levelwhere the more difficult mathematical details of the OGYprocedure may cause difficulty. Our method is based on thefact that unstable fixed points exist within a chaotic attractor.In the following sections we will first discuss the idea ofunstable fixed points along with methods of obtaining them.We outline our method and then apply it to both an iteratedmap and a system of differential equations. For the formerwe use the He´non map and for the latter the Lorenz systemof equations. In the final section the effects of noise will beinvestigated.

II. UNSTABLE PERIODIC ORBITS

Dynamical systems can be divided into flow~continuoustrajectory! systems and iterated maps~discrete systems!. Forthe former time varies continuously while for the latter timeadvances in discrete steps. Embedded within the chaotic at-tractors of flow systems are an infinite number of unstableperiodic orbits.7,8 Both the OGY and the method presentedhere use this fact to achieve control by continually perturbingthe system back toward one of these unstable periodic orbitsusing a proportional feedback technique.

A number of related techniques, including Poincare´ sec-tions, time series, and return maps, can be used to locate

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these unstable periodic orbits in dynamical systems. Closelyrelated to the idea of periodic orbits are fixed points—x* issaid to be a fixed point of the iterated mapf (x) if f (x* )5x* ~i.e., if the point is mapped onto itself!. To see howfixed points relate to dynamical systems we will consider thePoincare´ section. The Poincare´ section can be thought of as across section or slice of the attractor, thus if the attractor isndimensional the Poincare´ section will be (n21) dimen-sional. To construct the Poincare´ section of ~say! a three-dimensional system with state space variablesx, y, z we cansimply plot the point corresponding to a pair of these vari-ables ~e.g., x vs y! at equally spaced time intervals. Theappropriate choice for the length of this time interval is theone that shows the greatest amount of structure of the attrac-tor, but this is usually not critical. For a system with a peri-odic response, the time interval is this period and the Poin-care section is a single point. Hence we see that we canreduce the dynamics from a set of differential equations to aniterated map, i.e., a point on the Poincare´ section iterates to anew point. If the orbit is periodic then the point iterates ontoitself and we say that it is a fixed point. Unstable fixed pointsof iterated maps on the Poincare´ section of the flow corre-spond to unstable periodic orbits of the flow.

If, however, we are dealing with unstable periodic orbitsthe state space trajectory will only return to thevicinity of thefixed point a few times before resuming its previous irregularbehavior. Hence, in order to locate fixed points it is neces-sary to monitor successive points on the Poincare´ section. Ifthe separation between two of these successive points is suf-ficiently small then we may be reasonably confident thatthere is a fixed point nearby. A procedure which is equiva-lent to this involves monitoring the maxima of a time series.The time series is produced by numerically integrating theequations forward in time. After the transients have diedaway, the difference between the successive maxima of anysystem variable is calculated and again, a fixed point is in-ferred if this difference is sufficiently small. The average ofthe two maxima is taken as the approximate value for thefixed point. This was the technique we used to locate thefixed points of the Lorenz system. To see how the methodworks, Fig. 1 shows a time series of the Lorenz system withr 545.92. This particular time series was chosen because itshows that the system approaches an unstable periodic orbitat t'477. At this time two successive maxima differ by lessthan 0.01, i.e., 0.015%, and we can say that an unstable fixedpoint exists nearz564.95.

In order to see how the successive maxima evolve in timeand also as another way to locate fixed points, it is conve-nient to examine the return map of the system. It is con-structed by plotting then11th versus thenth maximum un-til enough points have been plotted to produce an almostcontinuous curve. Figure 2 shows a return map of the Lorenzsystem which can be used to follow the evolution of a pointon the curve. By following the arrows shown in the figure wecan see how the initial point iterates along the curve. Also,using this process we can see that the point at which the 45deg line intersects the curve gives the value of the fixed pointbecause at this pointzn is mapped onto itself. The slope ofthe curve close to the fixed point@i.e., f 8(x* )# gives us in-formation about the stability of the fixed point. Two separatecases exist.

~1! u f 8(x* )u,1.0. This indicates a stable fixed point sinceall nearby points on the curve iterate toward the fixedpoint.

~2! u f 8(x* )u.1.0. Here we have a state of unstable equilib-rium as any point that starts near the fixed point iteratesaway from it. This is the case in Fig. 2 so we can see thatthe fixed point here is unstable.

Before leaving the subject of return maps another pointthat should be noted is that for low values ofz the points lievery close to the 45 deg line. This can cause problems whencomparing successive maxima because if the tolerance used

Fig. 1. A time series of thez coordinate of the Lorenz system for parametervaluess516.0, b54.0, andr 545.92. The initial conditions werex510,y51, andz50. It is shown that the system approaches a periodic orbit ataboutt5477 but that because it is an unstable orbit, the system returns toirregular dynamics att5480. From this we can see that the fixed points canbe found by following the successive maxima of the time series.

Fig. 2. The return map of the Lorenz system forr 545.92. Both this figureand Fig. 1 show that the fixed point for thisr value corresponds toz'64.95.The arrows indicate how point that is initially near the fixed point movesaway.

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in the comparison is not small enough these points may re-sult in the incorrect identification of a fixed point. Because ofthis it is best to consider only maxima above a certain thresh-old value ~which can be estimated by looking at a samplereturn map!. For example, from the return map shown in Fig.2 it can be seen that only a maximum whose value is aboveabout 55 should be considered as a possible fixed point forrvalues around 45.92.

III. CONTROL ALGORITHM

In order to demonstrate the method of control we willconsider two separate cases.

A. Case 1: Iterated map

As was done by OGY we shall use the He´non map as asimple example to illustrate our control method. The He´nonmap is:

xn115A2xn21Byn, yn115xn , ~1!

where we chooseB50.3. We will use the quantityA as thecontrol parameter and assume that it can be varied by a smallamount about some valueA0 . We will then show that thechaotic system can be forced to behave periodically bychanging the value of the quantityA via a feedback mecha-nism.

A feature of the He´non map which will be used to achievecontrol is the relationship between the control parameter,A, and the value of the fixed point,x* . For this mapping itis possible to analytically solve for the value of the fixedpoint for a particularA value. At the fixed pointxn115xn

5xn215x* and hence

x* 5A2x* 21Bx* . ~2!

The fixed points can then be found by solving for the posi-tive root of the following quadratic for a set ofA values:

x* 21~12B!x* 2A50. ~3!

Similarly we can see that if we known the value of a particu-lar unstable fixed point,x* , then we can solve for the corre-spondingA value,

A5x* 21~12B!x* . ~4!

The following steps comprise the algorithm which wasused to achieve control of the He´non map. Choose the initialconditions for the system~for the results shown in Fig. 3 weusedx5y50.5 andA5A051.29!. Because we only controlthe system near fixed points when it will be on the attractoranyway, it is not necessary to allow for the transients to dieaway.

~1! Iterate the map to get the next value forx andy.~2! If x5y then this point is a fixed point for some particular

A. Using Eq.~4! we can calculate the value thatA wouldhave if thisx value represented a true fixed point. Wewill call this valueA8. The reason thatx does not rep-resent a true fixed point is thaty in general is not equalto x.

~3! If the difference betweenA0 and A8 is less thansome predetermined small value,p* ~e.g., OGY havep* 50.2!, then letA5A8. If not then leaveA unchanged,unless the difference is greater than some value~0.3,say! in which case setA5A0 . This final step of setting

A5A0 if uA82A0u.0.3 is a precautionary one designedto stop the system from ‘‘blowing up’’ when noise isadded.

~4! Loop back to step~1!.

By using this simple procedure the system will lock onto afixed point after about 100 iterates and will remain in aperiodic state, as can be seen in Fig. 3 where the value ofxis plotted after each iteration. It should be noted thatp* ~i.e.,the maximum difference betweenA8 andA0 allowed! can bemade very small but at the expense of increasing the lengthof time it takes for the system to lock onto a fixed point. Forexample, if we makep* 50.01 it takes about 3400 iteratesbefore the system settles down on a fixed point.

An unstable periodic orbit becomes an unstable fixed pointon the Poincare´ section. This fixed point is a saddle point,i.e., there are two curves emanating from it: the stable andunstable manifolds. Points displaced from the saddle pointwhich are on the stable manifold move toward the saddlepoint while points which lie on the unstable manifold moveaway. An arbitrary point will on average move away fromthe saddle point. The position of the saddle point will changeas the control parameterp* changes. Our control mechanismis as follows: If the trajectory happens to be close to thesaddle point, we calculate the value ofp* which wouldmove the saddle point to the trajectory position. Since ourtrajectory will not be exactly at the new saddle point, due tonoise, etc., the trajectory will again move away from thesaddle point, but as it is closer to the saddle point than be-fore, it does not move away as quickly. By continuouslymoving the saddle point we can control the system so that atrajectory will remain at or very close to the unstable peri-odic orbit.

For the Henon map the situation described above can beseen by monitoring the successive values ofx, y, and A.What is seen is that for a whileA stays constant atA0 untilthe system happens to come close to the fixed point. At thistime A starts to change. The change becomes smaller witheach iteration asy converges onx until after a relativelyshort time the system finds a fixed point andA ceases to

Fig. 3. A plot of the value of thex term for the He´non map vs the iterationnumber. The control algorithm is switched on atn5400.

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change. It is important to note that this final value ofA willnot, by itself, ensure a periodic response. In general, only thefinal A value coupled with the particularx andy values thatthe system ends up on after the control algorithm is appliedwill result in periodic behavior. It is also possible to stabilizehigher order orbits using the procedure outlined above andwe have specifically done so in the case of a period-2 orbit.

B. Case 2: Coupled differential equations

To demonstrate the control method for this case we willuse the Lorenz system of equations:

x5s~y2x!,

y5x~r 2z!2y, ~5!

z5xy2bz.

For the following analysis we will set the parameter valuesas follows;s516.0,b54.0, andr will be the control param-eter which will initially be set to 45.0 in order to make theresponse of the system chaotic.

Because this system is described by coupled differentialequations and not a mapping, the algorithm we use to controlthe system will be slightly different. Specifically, there is noanalytic relationship between the fixed points and the controlparameter,r . To overcome this we can find an empiricalrelationship between the fixed points,z* and r in the rangeof r values that we will use to obtain control of the system.To obtain this relationship we use a technique outlined inSec. II and monitor the time series of thez variable. Whenthe difference between two adjacent maxima is less thansome tolerance we take the average of the two maxima to bea fixed point. If we are monitoring the maxima ofz then weknow from the return map~Fig. 2! that the fixed points willoccur in the range of about 55–70 for ther values we willuse. A tolerance of 0.01 ensures that the adjacent maxima arewithin approximately 0.015% of each other. Thus we can bereasonably confident that the average of these two maximacan be taken as a fixed point. Using this procedure we cal-culated the fixed points for a range ofr values as is shown inFig. 4. The fixed points are clearly scattered closely about astraight line and using a standard least-squares fit routine wecalculate the slope and intercept of this line. Then, for anyparticular fixed point we can calculate the correspondingrvalue by using the slope and intercept. The relationship be-tween r and z* is approximately linear as can be seen inFig. 4. However, this approximation is accurate enough forthe control algorithm to work. We have found that themethod will work even if we calculate the slope and inter-cept using only three well spaced points.

The algorithm for controlling the Lorenz attractor is out-lined below and it is closely related to the one for the He´nonmap. In particular we will seek to control the system byvarying r by small amounts about some valuer 0 . A fourth-order Runge–Kutta integration technique~with a time step of0.01! is used to solve the differential equations and so followthe dynamics of the system.

~1! Integrate the equations forward until a local maximumfor z ~or other variable! is found.

~2! This z maximum,zmax, is near a fixed point for someparticular r value. We can calculate this newr value~call it r 8! as follows:r 85(zmax2intercept)/slope.

~3! If ur 82r 0u is less thandr –max, setr 5r 8. The param-

eter dr –max is the maximum distance we are allowed

to mover away fromr 0 ~1.0 in this case!.~4! Loop back to step~1!.

Figure 5 shows the results of applying this algorithm to theLorenz system. For the time series shown, we setdr –max to

1.0 and switch on the control algorithm att5400. The wayin which control is achieved is essentially the same as in thecase of the He´non map, with ther value being changed so asto cause the fixed point to move toward the present maxi-mum.

Fig. 4. A graph showing the relationship between the fixed points,z* , andthe control parameter,r , for the Lorenz system. A least-squares fit algorithmwas used to get the slope~51.263! and the intercept~56.918!.

Fig. 5. A plot of the successive maxima of thez coordinate of the Lorenzsystem as a function of time. The control procedure was initiated att5400.The values of the other relevant parameters aredr –max51.0 andr 0545.0while the initial conditions arex5y5z51.0.

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The procedure described above can also be used to stabi-lize period-2 and higher order orbits. In order to control asystem about a period-2 orbit we must find a relationshipbetween these period-2 fixed points and the control param-eter. Thus instead of comparing thenth and then11thmaxima to see if they indicate a fixed point, we must nowmonitor thenth and then12th maxima to see if their dif-ference is less than the predetermined tolerance. While ex-perimenting with these higher order orbits we found that itwas quite easy to stabilize a period-2 orbit but that forperiod-3 and higher it was more difficult.

IV. EFFECTS OF NOISE

To study the effect of noise on the stability of the controlprocedure we add Gaussian noise.9 A term ed is added to theequations whered is a random variable with a Gaussian dis-tribution ~^dx&5^dy&50 and^dx

2&5^dy2&51! and e is theamplitude of the noise. For the He´non map our equationsbecome:

xn115A2xn21Byn1edx ,

~6!yn115xn1edy .

Figure 6 shows a plot ofxn vs n for 1500 iterates of the mapwith noise added. The noise amplitude for the results shownis 0.01 andp* 50.2. As can be seen from the figure, themethod is effective in keeping the orbit near the fixed point.As the noise amplitude is increased, however, there are oc-casional bursts of chaos which become more frequent forlargee values. For example, whene is increased to 0.03 thecontrol method is no longer capable of eliminating the cha-otic response of the system. The numbers used here as wellas the plots can be compared with those for the OGY methodin Ref. 3.

For the Lorenz system we add a Gaussian noise term tothe equations as shown below. As for the He´non map^d i&50 and^d i

2&51 wherei 5x,y,z,

x5s~y2x!1edx ,

y5x~r 2z!2y1edy , ~7!

z5xy2bz1edz .

Again we find that the method is stable for small noise am-plitudes but that as the noise level is increased, the bursts ofchaos that interrupt the periodic behavior become more fre-quent and of longer duration. Figure 7~a! shows the Lorenzsystem being controlled with noise amplitudee55.0. If e isincreased to 10.0 some element of control is maintained butthe approximately periodic behavior is interspersed withbursts of chaos as can be seen in Fig. 7~b!.

V. CONCLUSIONS

In this paper we have presented a very simple controlmethod which eliminates the chaotic response of a systemand is capable of controlling the response about an unstableperiodic orbit. This control method was demonstrated forboth an iterated map and a system of differential equations. It

Fig. 6. It is shown how the control of the He´non map is affected by theaddition of noise. Again the control procedure is initiated atn5400 and theamplitude of the noise,e, is 0.01.

Fig. 7. A graph showing how the control of the Lorenz system is affected bythe addition of noise. Control is initiated att5400 and the amplitude of thenoise,e, is ~a! 5.0 and~b! 10.0.

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was also shown that the method is capable of controllingchaos even in the presence of moderate levels of noise. Es-sentially the procedure works by exploiting a known rela-tionship between the fixed points and a system parameter.This information then allows the control parameter to bechanged in such a manner as to move the fixed point towardthe system’s present location in state space. This is differentthan the OGY method in which the state space trajectory iscontinually perturbed toward a certain fixed point.

Also, in order to establish the empirical relationshipbetween the fixed points and the control parameter of adynamical system we have seen that it is a simple matterof observing the system’s behavior for a range of valuesof the control parameter and plotting the values of the fixedpoints over this range. The method of control presentedis also capable of stabilizing period-2 and higher periodorbits. Our procedure has been shown to be capable of con-trolling chaos even in the presence of moderate levels ofnoise, although noise reduced the effectiveness of the controlprocedure. Because of the way in which control is achievedin this method, we would expect to see the system drift alittle when noise is added, although the behavior of the sys-tem is still much more predictable than if there were nocontrol algorithm applied. For the He´non map the maximumamplitude of the noise,e, that we can use and still maintaincontrol of the system is slightly less than for the OGYmethod.

Another system of equations was used to further test theeffectiveness of the procedure. This system is the set ofequations developed by Rikitake10,11 to describe a model forthe irregular reversals of the earth’s magnetic field,

x5yz2mx,

y5~z2A!x2my, ~8!

z512xy.

This system is quite similar to the Lorenz and could be madeto behave periodically by applying the control algorithm asbefore. We also tried some preliminary experiments withmodels of a forced, damped pendulum4 but with less success.More work is needed to establish whether or not the controlmethod presented in this paper is applicable to such systems.

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