PHYSICAL REVIEW E 68, 066212 ~2003!

Controlling chaos in a Lorenz-like system using feedback

G. Kociuba* and N. R. HeckenbergDepartment of Physics, University of Queensland, St. Lucia, Queensland, Australia

~Received 10 December 2002; revised manuscript received 24 July 2003; published 31 December 2003!

We demonstrate that the dynamics of an autonomous chaotic laser can be controlled to a periodic or steadystate under self-synchronization. In general, past the chaos threshold the dependence of the laser output onfeedback applied to the pump is submerged in the Lorenz-like chaotic pulsation. However there exist specificfeedback delays that stabilize the chaos to periodic behavior or even steady state. The range of control dependscritically on the feedback delay time and amplitude. Our experimental results are compared with the complexLorenz equations which show good agreement.

DOI: 10.1103/PhysRevE.68.066212 PACS number~s!: 05.45.2a

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I. INTRODUCTION

To observe chaos in nonlinear systems with two degrof freedom one parameter must be modulated in ordecreate a third degree of freedom. Many laser chaos expments are performed this way@1#. If a system already hathree degrees of freedom, then it is possible for chaosemerge without modulation and thus without an externaimposed time frame. The Lorenz-like chaos in the ammolaser is an example of such autonomous chaos@2#.

Dynamics of such chaotic systems have been studiedtensively, both numerically@3#, and experimentally@4–6#.Recently, there has been much effort put into controlchaos. This can be separated into two categories: feedcontrol ~active control! and nonfeedback~passive! control. Anumber of feedback schemes can be used to control a chsystem, such as control by occasional proportional feedb@7#, control by synchronization@1#, and the well-known Ott-Grebogi-Yorke~OGY! method. All these methods vary icomplexity, and may be difficult to implement in a systewith limited bandwidth. This is because the computatitime to calculate detailed information about the systemsignificant. There are other feedback methods that dorequire any computation, but instead rely on extractingsystem variable and feeding that back into another variaor parameter. This control method is often called ‘‘feedbacontrol’’ and is understood in the literature to mean subtrtive feedback. That is, the control signal is expressed inform F(t)5G„x(t)2x(t2t)…, where t is the delay time,and G is some function with the condition thatG(0)50.One example is control of a chaotic CO2 laser by feedback oa variable which has been subtracted from its value atearlier time@8#. F(t) can be thought of as an error signwhich tends to zero as the system approaches control, comeaning a periodic state wherex(t)5x(t2t). The advan-tage of this type of scheme is that no knowledge of the stem other than the average pulsation period is required~con-trol is only achieved for certain values oft in the vicinity ofthe average pulsation period!. Experiments with subtractivefeedback on the Lorenz-like ammonia laser showed that ctrol to periodic and even steady state is possible@9#. The

*Electronic address: kociuba@physics.uq.edu.au

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feedback error signal was generated by analog subtracusing a coaxial delay line. When the laser was controlleda periodic state, the feedback signal was itself periodic,we can think of this as self-synchronization. It has also beshown that this type of laser operating above the chthreshold could be synchronized to another chaotic sysvia the pump@6#. Here we make a detailed theoretical anasis of the subtractive feedback system to elucidate the raof conditions under which control is possible. In an attemto further simplify the control of chaos we have examinthe possibility of avoiding the subtraction process in the geration of the feedback error signal. Now ifF(t)5G„x(t)…then F(t) is nonzero in general. This type of feedback hbeen investigated and is known to destabilize a systemgeneral@10#. The advantage would be that no subtractiondelayed signals would be required. This type of control wobserved in a loss modulated CO2 laser@11,12# where nega-tive feedback of subharmonic components extracted fromintensity signal was applied. This was achieved using a lorithmic amplifier followed by a ‘‘washout filter.’’ Here wenumerically investigate control without, and with, a low anhigh pass filter, which is simpler than the washout filtmethod.

II. NUMERICS

The Lorenz model is used to describe the dynamics ofideal two level atom interacting with a traveling wave inring resonator. Although the atoms are pumped like a thlevel system, it can be shown that the three level systemreduce to a two level system to a good approximation@13#.Our autonomous system has been shown to have a wrange of dynamics for various parameter ranges. Nearlythis behavior has been reproduced using the Lorenz etions or the complex Lorenz equations@14#. The complexequations take into account the fact that in the case of lasystems the cavity frequency is detuned from the atoresonance in general. We use the complex Lorenz equatin our simulations of delayed feedback on a chaotic systThe complex Lorenz equations are

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where

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E, P, andD are the electric field, polarization, and inversiorespectively;l is the average pump level,f(t) is the modu-lation applied to the pump;d is the detuning of the cavityresonance relative to the atomic line center;k,g i , and g'

are the cavity, polarization, and inversion decay rates,spectively. In all our simulations the parameters ares51.5, b50.25, d50.2, andl546. For chaos to occur threlation between the decay rates must bek.g i1g' . This isknown as the bad cavity condition since a lossy cavityrequired.

Variations in the pump power directly affect the popution inversion so the feedback term appears asf (t). We in-vestigate two cases of feedback, the first being of the formsubtractive feedback@15# f (t)5A@ I (t)2I (t2t)# whereI (t)5E(t)E* (t) represents the laser intensity, andA is thefeedback amplitude. This type of feedback was experimtally implemented by driving an acousto-optic modula~AOM! with a signal generated using a coaxial cable deline to perform the subtraction@9#. To compare these resultwith our numerical results, we introduce an additional dein the feedback loop of our model to account for the progation delayT within the AOM used to convert the errosignal into a modulation of the pump power. For the secocase of nonsubtractive feedback, we dispense with thetraction step and set the feedback to bef (t)5AI(t2T) andinvestigate the amplitude-feedback delay parameter spacboth these feedback cases the system is operating well athe chaos thresholdl th so that the average pump levell ischosen such that inf(l1 f (t)).l th . In all our calculationstime is scaled@16# as kt which is dimensionless sincekrepresents the cavity decay rate, hence the feedback variT andt are also dimensionless.

A. Control by subtractive feedback

As explained above, in all our experiments there wasadditional delayT within the AOM, so we define the feedback term to be

f ~ t !5A@ I ~ t2T!2I „t2~T1t!…#. ~2!

The difference delayt is the time between the two measurments of output intensity, the difference between which cstitutes the error signal. The feedback delay is defined astime T for the feedback signal to enter back into the systeWe integrate the complex Lorenz equations using this feback term, and for different pairs of parameterst, T andamplitudeA, construct the time seriesI (t) associated witheach of the parameters. Periodic solutions were identifrom the time series and the periodicity ofI (t) was deter-mined ignoring the initial transient behavior before long tedynamics took over. The period of the time trace was ploton the difference delay–feedback delay parameter spaceperiods up to 6. Where no period was identified, the plot wleft blank. The first result was calculated using a very wefeedback amplitude 0.0004 and the plot is shown in Fig

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Since the intensity pulsations range between 80 andunits, this sets the maximum feedback amplitude to be ab0.04% of the average pump power. The average scaledsation period is 3.08. Only periods greater than 3 existvery weak feedback as is evident in Fig. 1. If the feedbaamplitude is increased to 0.001, equivalent to 10% ofaverage pump power, we find this increases the numbeperiodic states and leads to the existence of period 0—steady state. This is shown in Fig. 2, where the dashedindicates the average scaled pulsation period for the chasytem without feedback~3.08!. Many of the features in theweak feedback case in Fig. 1 are evident in Fig. 2, suchthe rings of period 4 in parameter space. In the stronfeedback case, these rings have been distorted and thetions shifted slightly. These rings contain regions of cont

FIG. 1. Control to various periods by subtractive feedbackmaximum amplitude of 0.04% of the pump. The difference delatand the feedback delayT are both dimensionless, see text for dtails.

FIG. 2. Control to various periods by subtractive feedbackmaximum amplitude 10% of the pump. The difference delay ist,and the feedback delay isT.

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CONTROLLING CHAOS IN A LORENZ-LIKE SYSTEM . . . PHYSICAL REVIEW E68, 066212 ~2003!

to less than period 4 where period 0 dominates. As mighexpected, islands of control exist at multiples of the averpulsation period along the feedback delay axis. The saapplies for the difference delay axis. If we leave the feedbamplitude at 0.001 but change the sign of the feedbacnegative, the result is shown in Fig. 3. There are simstructures here as in the previous figure except that thelands are displaced half an average pulsation period upwaThis can be understood by considering the change of sigthe feedback term to be equivalent to a phase shift of haperiod. These well defined islands of stability are destroif the feedback amplitude is too large. We increased the feback amplitude to 0.006 which would correspond to anerage of 60% of the average pump power. The resulshown in Fig. 4. Since the feedback amplitude modulatioso large it is no longer perturbative and the system islonger Lorenz like. Figure 4 shows there are now islandscontrol which are not at multiples of the average pulsatperiod of the unperturbed system in either the differencelay axis or the feedback delay axis. The size of the perioislands are significantly smaller in the strong feedback ccompared to the moderate feedback case as in Fig. 2.would expect the islands to be smaller since if the systemnot very close to the periodic state, then the large feedbamplitude drives the system quickly away from the periostate. Previous experiments on the ammonia laser@9# showedthat the laser was controlled to period 0 when the feedbamplitude was 3% and 7%. The difference delayt used wasabout one laser pulse period. Control to period 1, 2, 4, 6observed at 5% modulation depth. Numerically, we foucontrol to period 0, 1, 3, 4, 5, 6 with a modulation depth10% for the second multiple of the average pulsation peras well as the first multiple. We did not find any period 2this amplitude. This could be due to the fact that numericawe calculated the period of the intensity traces which ctained about 1000 pulsations. The experiments have the ltation that only about 70 pulses during control could be

FIG. 3. Control to various periods by subtractive feedbackmaximum amplitude 10% of the pump. The difference delay ist,and the feedback delay isT. The error signal has been inverted wirespect to Fig. 2.

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corded. Thus it is possible that what appeared to be periowas actually a long transient which approaches periodThis is always a problem with any finite time series since ocannot be sure whether the dynamics in a finite time seriepermanent or transient.

The islands of stability have a definite preferred oriention, in particular the period 0 islands all have a slope2 1

2 . This can be understood the following way. Periodoccurs after the oscillation is completely damped so justfore it is extinguished it is a sinusoid to a very good appromation. So we can write the signal asf (t)5asin(vt) and thedelayed signal will bef (t1f)5asin„v(t1f)…, where fis the phase difference between the two signals duethe subtraction timet so f5tv. Since f (t)2 f (t1f)522asin(vf/2)cos„v(t1f/2)…, the generated difference signal has an effective delay equal to half the delayt within thecontrol island which can be compensated by an equalopposite change in the feedback delayT.

We now look at the stability of the fixed points of thnonlinear system to gain some insight into the mechanismcontrol. The eigenvalues of the feedback system cannoobtained analytically since the determinant of the Jacobiaa transcendental function, so this can only be solved numcally. There are an infinite number of complex solutionsthis type of equation in general, and the system will be staif all the eigenvalues are negative. We search the samerameter space as in Fig. 2 and setA50.001 and plot~Fig. 5!whenever all the eigenvalues of the determinant of the Jabian are less than or equal to zero. Without the feedb@ f (t)50 in Eq.~1!# the eigenvalues of these fixed points apositive. It is clear that the position of the islands of contto period 0 and period 1 are contained within the islandsFig. 5. This shows that the eigenvalues of the fixed pointsthe system are all negative during control to period 0 aperiod 1. Control to a periodic state by stabilizing an existiunstable periodic orbit of a chaotic attractor has previou

f FIG. 4. Control to various periods by subtractive feedbackmaximum amplitude of 60% of the pump. The difference delay ist,and the feedback delay isT.

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G. KOCIUBA AND N. R. HECKENBERG PHYSICAL REVIEW E68, 066212 ~2003!

been demonstrated experimentally and theoretically ascussed in the Introduction. These methods of control relythe state of the system having a reasonable probabilityvisiting the desired unstable periodic orbit where a conmechanism can take full effect. This is not the case forriod 0 since the state never visits the unstable fixed p@17#. We overcome this by applying the feedback. Thchanges the stability of the unstable fixed points, allowinginitially inaccessible region of phase space to be visitedcertain values of feedback parameters as shown in Fig.

In previous feedback experiments@9# and in our experi-mental results following this section, the feedback signathe laser system is ac coupled. This will only give a nonzfinal error signal if the unfiltered error signal is varying witime, and a constant unfiltered signal will appear as zero fierror signal. To check what effect this might have, weclude this effect in our model by applying a high pass anlow pass filter tof (t) and using this modifiedf (t) as thefeedback signal in the differential equations. The high pfilter models the ac coupling while we also include a lopass filter to model the finite bandwidth of the AOM. Wapply this procedure to the subtractive feedback case fordifferent values of low pass cutoff angular frequencies 61, 0.85, 0.75, and 0.5 where in each case the high pass cwas set to 0.01. The result for the high cut off angular fquency of 1 is shown in Fig. 6. The characteristic scapulsation period is 3.08~hence the scaled angular frequenis '2), where time has been scaled to the parameterk. Thecutoff frequency 6 is' three times the average pulsatiofrequency and we found that this results in only a sligdecrease of the amplitude of the fundamental frequency cponent allowing control to proceed. As the cutoff frequendecreases, there is a greater attenuation of the fundamfrequency which gets fed back into the equations. Thissults in an effective lower modulation amplitude at the fudamental frequency and therefore the range of controlcomes narrower. This is evident by comparing Fig. 6 to F

FIG. 5. Islands represent the nonpositive Lyapunov spectrumEq. ~1! for the feedback parameterA50.001. The difference delayis t, and the feedback delay isT.

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2 ~which is unfiltered, but is found to look similar to a cut ofrequency of 6! where the islands of control retain their orentation but decrease in size. All the period 0 islands wfound to have been extinguished for a cutoff frequency0.5, and the lowest period was 4. These results show thaeffects of ac coupling and the limited bandwidth of the AOcan reasonably be neglected in the theoretical modelThere is only a significant difference when the bandwidththe feedback signal is equal to or less than the characterfrequency of the system, and this just causes the controlands to shrink and get destroyed if the bandwidth is muless than the characteristic frequency.

B. Control by nonsubtractive feedback

We now simplify the feedback term so that there issubtraction and simply takef (t)5AI(t2T). As before weintegrate the complex Lorenz equations using this feedbterm, and for different pairs of parametersT and amplitudeA, construct the time seriesI (t) associated with each of thparameters. The periodicity ofI (t) was calculated the samway and a map of the results as a function of delay afeedback amplitude was constructed as shown in Fig. 7save time, relatively larger steps in the feedback amplitudAwere used. Again the scaled average pulsation period ofunperturbed system is 3.08 as indicated by the dashedPeriod 0 dominates the regions near multiples of the scaaverage pulsation period, 3.08 and 6.16, which first appeaa feedback amplitude of 1.131023 corresponding to abou11% modulation depth. As this amplitude increases moreferent period numbers emerge. The numerical results in7 show that control to period 1 occurs only on the right haside of the large period 0 block. This segment of periodbegins at slightly more than the average pulsation periodthe unperturbed system. This is the only region we canplore experimentally since there are delays in the acouoptic modulator which cannot be removed.

of FIG. 6. Control to various periods by subtractive feedback fohigh and low pass cutoff scaled frequenciesv are 1 and 0.01, re-spectively. The scaled~dimensionless! average pulsation frequencv0 is 2.

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CONTROLLING CHAOS IN A LORENZ-LIKE SYSTEM . . . PHYSICAL REVIEW E68, 066212 ~2003!

The eigenvalues of the Jacobian of Eq.~1! are calculatedand regions where all eigenvalues are non-positive are deated in Fig. 8. The period 0 and period 1 islands from F7 appear in a similar position to the islands in Fig. 8. Tshows that the stabilization of the fixed points during contto period 0 and period 1 allows an initially inaccessiblegion of phase space to be reached.

Three measures are introduced to extract informaabout the dynamics during and near a control window. Tperiodicity is calculated as before, and the number of trsient pulses before control was achieved is calculatedreferred to as ‘‘transient pulses.’’ The difference betweennumber of cycles of output intensity and the feedback sigwas calculated~which is related to the average frequendifference! then this difference is determined over the dution of the feedback. This measure is referred to as the ‘‘erage phase difference.’’ Finally, ‘‘phase slips’’ are calcula

FIG. 7. Control to various periods by nonsubtractive feedbas a function of the difference delayt and feedback amplitudeA.

FIG. 8. Islands represent an all nonpositive Lyapunov spectof Eq. ~1!. The feedback delay isT and amplitude isA.

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by comparing each successive intensity peak time and feback peak time keeping track of their relative orientatioand counting the number of times the sign of the differenbetween these peak times changes. This measure issince there are situations where the average frequencietwo quantities are the same yet there may be an equal nber of positive and negative phase slips which cannotextracted from the average phase difference measure al

The feedback delay has been set toT53 which corre-sponds to a vertical line atT53 in Fig. 7, and the measureare calculated for the same range of amplitude points as7, except the number of amplitude points was increasedfold for higher resolution. The number of transient pulsbefore control emerged is shown in Fig. 9~b! indicating thatthe fastest rate of convergence to the periodic state is' inthe center of the amplitude control window shown in F9~a!. There is a period 6 orbit atA50.0004 which has acorresponding low number of transient pulses.

The average phase difference, which is related to theerage frequency, is displayed in Fig. 9~d!, and the differenceis less than 4p rad where the number of cycles is'1000.This shows that the weakest form of generalized synchrozation occurs for the whole amplitude range calculatedthe fixed feedback delay ofT53. To determine which re-gions of the generalized synchronization states are phlocked, the number of phase slips was calculated andshown in Fig. 9~c!. There are generally few phase slips duing the controlled state, indicating phase locking. Phase swere not calculated for the amplitude range correspondinperiod 0 as synchronization has no meaning at steady s

There are some phase slips forA,0.0011 but thisamounts to only 18% of the data points in this region givia high synchronization ratio of 82%.

The measures used in this analysis show that generasynchronization occurs for a wide range of feedback paraeters, and perfect synchronization occurs before conemerges for 0.0011,A,0.0013. The phase slip measurcan show when a dynamical state is approaching the edg

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FIG. 9. The period number, number of transient pulses befcontrol emerges, phase slips, and average phase difference,function of feedback amplitude are shown in~a!, ~b!, ~c!, and ~d!,respectively.

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G. KOCIUBA AND N. R. HECKENBERG PHYSICAL REVIEW E68, 066212 ~2003!

an Arnold tongue which is not observed by comparing avage frequencies, and quantifies the quality of the synchnized state.

We now modify the nonsubtractive feedback systemincluding filtering and the result for the cutoff angular frquency of 1 is shown in Fig. 10. Again as the low pass cuis decreased there is less modulation at the fundamentaquency so a greater feedback amplitude is required to cpensate. This is evident in the figure as the islands of conmove towards a higher amplitude of feedback as the cutodecreased. These results show that limiting the bandwidtthe error signal has the effect of raising the thresholdcontrol and for a sufficiently large bandwidth the resultsessentially the same as the unfiltered case.

III. EXPERIMENTAL

Our experimental system consists of a13CO2 laser whichoptically pumps a15NH3 ring laser though a vibration transition at 10.78mm. The lasing occurs through a rotationtransition at a wavelength 0.153 mm. We use a semi-confring cavity as shown in Fig. 11 to achieve unidirectionlasing, where the backward traveling wave is chosen in perence to the forward wave because the ac Stark effect sthe gain line in the forward direction@18#. This allows us touse the Lorenz equations to describe the dynamics oflaser system@13#. The intensity of the backward wave in thring laser is measured with a Schottky barrier diodeB. Thissignal is monitored by a spectrum analyzer and recordeda digital storage oscilloscope. In order to implement the nsubtractive feedback scheme a signal proportional to theser output has to be fed back to modulate the pump poThe signalI (t) is therefore fed into a buffering amplifieamplified, then applied to the acousto-optic modulator. Tfinite acoustic velocity in the AOM creates a delay that cobe varied by' 20% of the fundamental pulsation period badjusting the AOM’s position transverse to the CO2 pump

FIG. 10. Control to various periods by nonsubtractive feedbfor a scaled cutoff angular frequency of 1. The low pass cutof0.01.

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IV. EXPERIMENTAL RESULTS

The delay timeT was adjusted so that this correspondto the average pulsation period of the laser. Control to per1 was observed as shown in Fig. 12. Here the average fback amplitude was 5%. Before the feedback control wturned on ~at t51.2 ms) the laser produced Lorenz-likchaos. Initially the control signal caused the Lorenz-like psations to break into transient pulses before the syssettled to period 1 pulsations, and the feedback signalalso periodic. The phase difference between the feedbsignal and the intensity output was locked only during tcontrolled state as expected. The smallest feedback dwhich could be achieved in the experiment was about twthe average pulsation period of the unperturbed systemthat we could not exploret,6.0 in Fig. 7.

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FIG. 11. Experimental schematic. Gr is a blazed grating atpump wavelength which doubles as a mirror for the lasing walength. wm is a wire mesh used as an output coupler. Theinfrared~FIR! intensity output is measured, delayed, and appliedthe AOM

FIG. 12. Control to period 1 using nonsubtractive feedbackthe FIR laser withA'0.05. The average pulsation period befoand during control is 1.179ms and 0.9747ms, respectively. Theintensity is in arbitrary units.

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CONTROLLING CHAOS IN A LORENZ-LIKE SYSTEM . . . PHYSICAL REVIEW E68, 066212 ~2003!

We also find that when the sign of the feedback isversed, control can still be achieved by adjusting the delathe feedback~not shown since it looks the same as nonversed feedback!.

We also found that outside the range where control wachieved, the initially chaotic intensity can be synchronizso that the output and feedback is fully phase locked, andtime series is shown in Fig. 13. It appears that a differtype of chaos is produced during feedback. This resemLorenz-like chaos operating closer to the chaos thresh~from above! compared to the unperturbed system due tolengthening of the spirals. To check for phase slips a hisgram is calculated for the time difference between the Fintensity peaks and the feedback signal, and the resushown in Fig. 14. During synchronization there is a signcant lag between the FIR intensity, and the feedback sigand all time differences are within the average pulsationriod T51.068.

The histogram contains two peaks where the larger pcorresponds to the start of the spiral, and the smaller toend of the spiral which is preceded by' ten cycles with asignificantly larger period than the average. The averagesation period decreases with increasing pump strength wno feedback is applied, so the feedback has increasedperiod during the last few cycles of the spiral despite the fthat the average energy of the pump during that regionslightly higher than at the start of the FIR spiral.

These results show that control to period 1 canachieved by choosing an appropriate delay time. Synchrzation can also be achieved, which can be used to creamodified chaos which has a higher bandwidth than theperturbed system which is phase locked to the delayednal.

V. CONCLUSION

We numerically investigated control of Lorenz-like chato various periodic states including period 0 using two fe

FIG. 13. Synchronization by nonsubtractive feedback. Averpulsation period before feedback and during feedback1.1933ms and 1.068ms, respectively. There are no phase slwhen the non subtractive feedback is turned on. The intensityarbitrary units.

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back control methods. The first case was subtractive feback of intensity including loop delay. We found that forsmall amplitude control to periods greater than 3 existedmoderate feedback amplitude, control to period 0, 1, 3, 4and 6 emerged where large islands of period 0 dominatedifference delay–feedback delay parameter space. Theslands are separated by the average pulsation period ofsystem. They can be shifted half a period by invertingfeedback signal. We showed that islands of period 0 aperiod 1 correspond to a nonpositive set of eigenvaluesthe chaotic system with feedback emphasizing that a nonturbative picture is necessary to understand the full rangopportunities for control, which are not limited to preexistinunstable periodic orbits. We then examined the effect ofplying an ac filter to the feedback signal and varied the bawidth before feeding it into the chaotic equations. Controstill possible even if the high pass cutoff frequency wslightly less than the characteristic frequency of the systThe results are in good agreement with previously publisexperimental results.

The second feedback case consisted of a simsubtraction-free feedback with only a loop delay. We foucontrol to the same periodic cycle numbers as in the subttive case at the same feedback amplitude, and with the ation of period 2. The period 0 and period 1 islands dominthe feedback delay–amplitude parameter space. Theslands corresponds to the fixed points of the feedback syscontaining no positive eigenvalues. Phase slips were calated indicating not only perfect synchronization just befo~and during! control, but also for very low feedback ampltudes. Modifying the feedback signal by applying the saa.c. filtering and finite bandwidth as in the subtractive feeback case, we find that this has the effect of raisingthreshold amplitude for control as the bandwidth approacthe characteristic frequency of the system. Experimentwe were able to control a chaotic Lorenz-like laser to per

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FIG. 14. This shows the frequency of occurrence for the tidifference between the FIR intensity peaks, and the associpump fluctuation peaks, corresponding to Fig. 13. The averagesation period is 1.068ms. Peak detection error ranges from 0.050.1 ms.

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G. KOCIUBA AND N. R. HECKENBERG PHYSICAL REVIEW E68, 066212 ~2003!

1 by this nonsubtractive method and found control to per1 but were prevented from demonstrating period 0 by tidelays in the AOM. Phase synchronization was experimtally observed for a relatively large frequency mismatch11.7% between the initial and final average pulsation fquencies.

Overall, the concordance of experimental and theoret

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results confirms that control of a strongly chaotic systembe achieved by controlling a single parameter using an esignal based on a single variable, without any computatioFurther, the system can be controlled not only to periostates but also to the technically more useful steady seven though this region of phase space is inaccessible inoriginal system.

ys.

an,

ff,

um

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