Controlling chaos in a fast diode resonator using extended time-delay autosynchronization:...

Controlling chaos in a fast diode resonator using extended time-delayautosynchronization: Experimental observations and theoretical analysis

David W. Sukow, Michael E. Bleich, Daniel J. Gauthier, and Joshua E. S. SocolarDepartment of Physics and Center for Nonlinear and Complex Systems, Duke University, P.O. Box 90305,Durham, North Carolina 27708

~Received 24 March 1997; accepted for publication 21 May 1997!

We stabilize unstable periodic orbits of a fast diode resonator driven at 10.1 MHz~corresponding toa drive period under 100 ns! using extended time-delay autosynchronization. Stabilization isachieved by feedback of an error signal that is proportional to the difference between the value ofa state variable and an infinite series of values of the state variable delayed in time by integralmultiples of the period of the orbit. The technique is easy to implement electronically and it has anall-optical counterpart that may be useful for stabilizing the dynamics of fast chaotic lasers. Weshow that increasing the weights given to temporally distant states enlarges the domain of controland reduces the sensitivity of the domain of control on the propagation delays in the feedback loop.We determine the average time to obtain control as a function of the feedback gain and identify themechanisms that destabilize the system at the boundaries of the domain of control. A theoreticalstability analysis of a model of the diode resonator in the presence of time-delay feedback is in goodagreement with the experimental results for the size and shape of the domain of control. ©1997American Institute of Physics.@S1054-1500~97!00104-3#

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In this article we investigate a control scheme that is ef-fective in suppressing deterministic chaos in fast dynami-cal systems. It is desirable to devise such schemes becauthe presence of deterministic chaos in devices generalldegrades their performance in many applications. Thesignatures of chaos include erratic, noise-like fluctuationsin the temporal evolution of the system variables, broad-band features in the power spectrum, and the long-termbehavior of the system is extreme sensitivity to applica-tions of small perturbations to the system variables. Re-cent research has demonstrated that feedback control algorithms that take advantage of the special properties ofchaotic systems are effective in controlling chaos. We review several specific control schemes and present oustudy of a continuous feedback scheme that is particu-larly well-suited for controlling chaotic systems that fluc-tuate on fast time scales. We present experimental resultof the control of a fast chaotic electronic circuit and findgood agreement with theoretical predictions.

I. INTRODUCTION

In many cases of practical importance, instabilities achaos can seriously limit the performance of nonlinearnamical systems. Obvious strategies for obtaining stablehavior are to redesign the entire system or integrate sevsub-systems that cover the entire range of operating cotions, but these options may not be possible or approprfor the application due to cost or space considerations,example. An alternate method for obtaining stable behais to use control techniques that suppress the instabiliRecently, researchers in the nonlinear dynamics commuhave demonstrated that such control can be achievedmaking only small adjustments to some accessible sys

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parameter. The strength and timing of the adjustmentsoften be determined from algorithms that do not requiretailed models of the dynamics.

The key idea underlying the new class of contrschemes is to take advantage of the presence of unssteady states~USSs! and unstable periodic orbits~UPOs! ofthe system~infinite in number!.1,2 Figure 1 shows an example of chaotic oscillations in which the presence of UPis clearly evident with the appearance of nearly periodiccillations during short intervals.~This figure illustrates thedynamical evolution of current flowing through an electrondiode resonator circuit studied in detail below.! The controlprotocols attempt to stabilize one such UPO3 by makingsmall adjustments to an accessible parameter when thetem is in a neighborhood of the state.4

Several research groups have demonstrated controunstable states in a variety of physical, chemical, and blogical systems.2 In most cases, as detailed in other articlin this volume, the control protocol involves feedback bason the difference between the present state of the systemsome externally generated reference signal representingdynamics of the ideal unstable state. We focus insteadsystems for which such a comparison is impractical, duethe speed of the oscillations or other difficulties in construing the correct reference state. One example of a hifrequency system in which matching to an external refereis difficult is the diode resonator circuit studied in this papA second example of significant technological importanthat clearly falls into this category are the semiconduclasers used in state-of-the-art optical devices.5–7

We recently reported a new technique for stabiliziUPOs in low-dimensional dynamical systems that allowscontrol over a large domain of parameters and is well su

560© 1997 American Institute of Physics

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561Sukow et al.: Controlling chaos in a fast diode resonator

for practical implementation in fast systems.8 We refer tothis scheme as ‘extended time-delay auto-synchronizati~ETDAS!: It uses a continuous feedback loop that attempto synchronize the system with its own past behavior raththan an external reference. It is based on a scheme first plished by Pyragas9 that involves the comparison of thepresent value of a state variable with its value one periodthe UPO in the past, but extends that approach to incluinformation from further in the past in a way that is easyimplement either electronically or optically. The efficacy oETDAS was demonstrated by stabilizing the dynamics ochaotic diode resonator driven at 10 MHz~a drive period of100 ns!.8 Since our report, Bleich and Socolar have deveoped a general mathematical technique for determiningstability of a nonlinear dynamical system in the presenceETDAS feedback.10

In this paper we present a detailed account of the expemental work discussed in our previous report and compthe experimental results with a theoretical analysis ofmodel of the controlled diode resonator. In the next sectiowe review previously developed techniques for controllinchaos and highlight the issues that arise when attemptto control fast dynamical systems. In Sec. III, we descriETDAS in detail and outline the general theoretical methfor determining the stability of a dynamical system in thpresence of time-delayed feedback. Section IV describvarious implementations of ETDAS including an all-opticascheme for controlling the dynamics of fast optical systemand the specific electronic implementation for controlling th

FIG. 1. Temporal evolution of a chaotic electronic circuit. The vertical axV(t) measures the voltage drop across a 50V resistor and is proportional tothe current in the circuit. The system ergodically visits the unstable perioorbits embedded with the chaotic attractor, three of which are indicated

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diode resonator. Section V summarizes our experimenfindings, and in Sec. VI we make concluding remarks anlook to the future.

II. CONTROLLING CHAOS USING SMALLPERTURBATIONS

A. Brief historical review

Techniques for stabilizing unstable states in nonlineadynamical systems using small perturbations fall into twgeneral categories: Feedback and non-feedback schemesnon-feedback~open-loop! schemes,11,12 an orbit similar tothe desired unstable state is entrained by adjusting an accsible system parameter about its nominal value by a weperiodic signal, usually in the form of a continuous sinusoidal modulation. This is somewhat simpler than feedbacschemes because it does not require real-time measuremof the state of the system and processing of a feedback snal. Unfortunately, period modulation fails in many cases tentrain the UPO~its success or failure is highly dependent onthe specific form of the dynamical system13!.

The possibility that chaos and instabilities can be controlled efficiently using feedback~closed loop! schemes tostabilize UPOs was suggested by Ott, Grebogi, and Yor~OGY! in 1990.1,14 The basic building blocks of a genericfeedback scheme consists of the chaotic system that is tocontrolled, a device to sense the dynamical state of the sytem, a processor to generate the feedback signal, and antuator that adjusts the accessible system parameter, as shschematically in Fig. 2.

In their original conceptualization of the control schemeOGY suggested the use of discrete proportional feedba

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FIG. 2. Building blocks of a feedback scheme for controlling chaos.

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562 Sukow et al.: Controlling chaos in a fast diode resonator

because of its simplicity and the fact that the control paraeters can be determined straightforwardly from experimeobservations. In this particular form of feedback control,state of the system is sensed and adjustments are madeaccessible system parameter as the system passes throsurface of section. The size of the adjustments is proptional to the difference between the current and desired stof the system. Specifically, consider a system whose dynics on a surface of section is governed by them-dimensionalmapzi 115F(zi ,pi), wherezi is its location on thei th pierc-ing of the surface andpi is the value of an externally accesible control parameter that can be adjusted about a nomvalue p . Feedback control of the desired UPO~characterizedby the locationz* ( p) of its piercing through the section! isachieved by adjusting the accessible parameter by an amdpi5pi2 p52gn•@zi2z* ( p)# on each piercing of thesection whenzi is in a small neighborhood ofz* ( p), whereg is the feedback gain andn is am-dimensional unit vectorthat is directed along the measurement direction.15 The loca-tion of the unstable fixed-pointz* ( p) must be known beforecontrol is initiated; it can be determined from experimenobservations ofzi in the absence of control. The feedbagain g and the measurement directionn necessary to obtaincontrol is determined from the local linear dynamics of tsystem aboutz* ( p) using the standard techniques of modecontrol engineering,1,16 and it is chosen so that the adjusmentsdpi force the system onto the local stable manifoldthe fixed point on the next piercing of the section. Successiterates then collapse toz* ( p). It is important to note thatdpi vanishes when the system is stabilized; the control ohas to counteract the destabilizing effects of noise.

Ditto et al.17 demonstrated the practicality of this tecnique by using it to control the dynamics of a chaotic manetoelastic ribbon. They reconstructed the map using a tidelay embedding of a time series of a single variablefound that the discrete feedback control scheme is easimplement, robust to noise, and rather insensitive to impcise knowledge ofz* ( p) andg. Soon thereafter, Hunt18 andPeng, Petrov, and Showalter19 found that control can be obtained by occasionally delivering brief perturbatiodpi52g@z i2z* ( p)#Q(t i) to the system parameter, whez i5n•zi , andQ(t i) is a square pulse function of duratiot i . Carr and Schwartz20,21 have considered delays andt i ascontrol parameters. Often,g, n, z* , andt i are determinedempirically by adjusting their values to obtain controlled bhavior. Note again thatdpi vanishes when the UPO is controlled successfully. In the ensuing years, several discfeedback control schemes that refine the original OGY ccept have been devised and applied to experimental syswith natural frequencies ranging from 1022 to 105 Hz.2

B. Controlling chaos in fast dynamical systems

Scaling these schemes to significantly higher frequcies, such as that encountered in high-speed electronioptical systems, for example, is challenging for several r

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sons. One important issue in high-speed feedback controchaotic systems is the latency through the control loop, tis, the timet l between the sensing of the state of the systand the application of the control signal. The latency of tcontrol loop is affected by the propagation speed of the snals through the components of the loop and the procestime of the feedback signal. An additional important issuethat it is difficult to accurately sample the state of the systat discrete times in order to compare it with the referenvalue and to rapidly adjust the control parameter on a timscale comparable to the response time of the system.

Continuous feedback schemes avoid or reduce manthese problems and hence may be useful for controlling hspeed chaos. An obvious extension of the original OGY sgestion for controlling UPOs is to use continuous adjustmof the accessible system parameter by an amodp(t)52gn•@x(t)2x* (t)#, wherex(t) is the system tra-jectory,x* (t) is the trajectory of the UPO inm-dimensionalphase space, andg is a constant feedback gain.9,22 Thisscheme is not amenable for controlling the dynamics of hispeed systems, however, because it is difficult to accuradetermine, store, and regeneratex* (t).

Several researchers have suggested that unstable sstates can be stabilized in high-speed systems using a clacontinuous feedback techniques that do not require raswitching or sampling, nor do they require a reference sigcorresponding to the desired orbit. The USSs of a laser23 andan electronic circuit24 have been stabilized using continuoadjustment of the system parameter by an amodp(t)52gn•@dx(t)/dt#, often called derivative contro~these authors did not stress the high-speed capabilitiethis control scheme!. Note thatdp(t) vanishes when the system is stabilized to the USS and no explicit knowledge ofUSS is required to implement control. The feedback gg is determined from the local linear dynamics of the systabout the USS using the standard techniques of moderntrol engineering, or it can be determined empirically in eperiments.

As first suggested by Pyragas,9 the UPOs of a dynamicasystem can be controlled using continuous feedbacksigned to synchronize the current state of the system antime-delayed version of itself, with the time delay equalone period of the desired orbit. Specifically, UPOs of pert can be stabilized by continuous adjustment of the accsible parameter by an amount

dp~ t !52g@j~ t !2j~ t2t!#, ~1!

where g is the feedback gain,j(t)5n•x(t), and n is themeasurement direction. We refer to this method of contro‘time-delay autosynchronization’~TDAS!. Note thatdp(t)vanishes when the system is on the UPO sincej(t)5j(t2t)for all t. This control scheme has been demonstrated expmentally in electronic circuits25,26operating at frequencies o10 MHz, and in lower frequency systems such as an etronic circuit,27 a fiber laser,28 a glow discharge,29 amagneto-elastic ribbon,30 and a periodically driven yttriumiron garnet film.31 In addition, TDAS has been demonstrate

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theoretically to be effective for stabilizing the dynamics otunable semiconductor oscillator,32 neuronal networks,33

lasers,34,35 and pattern forming systems.36 The main draw-back to TDAS is that it is not effective at controlling highunstable orbits.25

III. EXTENDED TIME-DELAYAUTOSYNCHRONIZATION

A. The ETDAS feedback scheme

Recently, three of us8 introduced a generalization oTDAS that is capable of extending the domain of effectcontrol significantly37 and is easy to implement in high-speesystems. Stabilization of UPOs of periodt is achieved byfeedback of an error signal that is proportional to the diffence between the value of a state variable and an infiseries of values of the state variable delayed in time by ingral multiples oft. Specifically, ETDAS prescribes the continuous adjustment of the system parameter by

dp~ t !52gF j~ t !2~12R!(k51

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Rk21j~ t2kt!G , ~2!

where21<R,1 regulates the weight of information fromthe past.38 As we will motivate in Sec. III C and see in SeV C, highly unstable orbits can be stabilized asR→1. ThecaseR50 corresponds to TDAS, the scheme introducedPyragas.9 We emphasize that, for anyR, dp(t) vanisheswhen the UPO is stabilized sincej(t2kt)5j(t) for all tandk, so there is no power dissipated in the feedback lowhenever ETDAS is successful. Note that no property ofUPO must be known in advance except its period. In podically driven systems, where the period of the orbit is dtermined from the driving, no features of the UPO need ebe determined explicitly.

In some situations it may be impossible to producefeedback signal that faithfully reproduces the form of Eq.~2!due to the latency of the control loop. As a rough guidelithe latency becomes an issue when the time lagt l throughthe loop is comparable to or larger than the correlation tiof the chaotic system which is approximately equal tol21,wherel is the largest~local! positive Lyapunov exponent othe desired UPO in the absence of control. Under conditiwhen the latency is important, the actual adjustments ofsystem parameter is given bydpactual(t)5dp(t2t l ).

B. Linear stability analysis of ETDAS feedback control

The control parametersg andR can be determined empirically in an experiment or by performing a linear stabilianalysis of the system in the presence of ETDAS feedbcontrol for fixedn and p . Such an analysis can yield impotant information about the possible effects of control onsystem, such as identifying the orbits that may be controland the range~if any! of feedback gain needed to achiecontrol for a particularR. Briefly, the analysis proceeds bconsidering a system described by

x~ t !5f~x~ t !,t;p!, ~3!

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where the ETDAS feedback signaldp(t) @given by Eq.~2!#

is included as a perturbation to the parameterp . For simplic-ity, we consider the case where the latency of the conloop can be ignored (t l 50).

Following Ref. 10 we consider small perturbationy(t)5x(t)2x* (t) about at-periodic orbit x* (t). The dy-namics of the perturbations in a neighborhood of the oare governed by

y~ t !5J~ t !•y~ t !2gM ~ t !•Fy~ t !2~12R!

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trolled system andM (x* (t))[(]f/]pux*

(t), p) ^ n is anm3m dyadic that contains all information about how thcontrol is applied to the system and how small changes ipaffect it. The general solution to Eq.~4! can be decomposeinto a sum of periodic functions~modes! with exponentialenvelopes. The growth of an exponential envelope in ariod t is quantified by its Floquet multiplierm which can befound from the modified eigenvalue equation

Um21TFexpE0

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~5!

where the time-ordered product notationT@•••# representsthe operator which advancesy(t) forward in time by anamount equal tot, and1 is the identity matrix. In general, thetime-ordered product cannot be obtained analytically,though it can be found using numerical techniques. To splify our discussion, we denote the left-hand-side of Eq.~5!by g(m21). We note thatg has an infinite number of rootscorresponding to the infinite number of modes introducedthe time delay.

An UPO is linearly stable under ETDAS control forgiven set of parameters if and only if all the Floquet mulpliers lie inside the unit circle in the complex plane, that isumu,1 for all m satisfying Eq.~5!. Equivalently, the systemis stable if and only ifg has no roots on the unit disk sincg(m21) has no poles on the unit disk. The number of rooof g(m21) on the unit disk can be determined by countithe number of timesg(z) winds around the origin asztraverses the unit circle. A necessary and sufficient condifor linear stability of the UPO in the presence of ETDAfeedback control is that this winding number vanish.

In a typical situation, the choice of accessible systparameterp and the number of system variables that canmeasured is restricted, both by physical principles and ptical considerations. Thus, one wishes to map out the domof values ofg and R for which ETDAS is successful for agiven p and n over some range of values of a bifurcatioparameter. In principle,J(t) andM (t) can be estimated from

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experimental data. This may not be practical, however, escially in high-speed systems. It is therefore importantcompare theoretical predictions and experimental obsertions for simple systems in which accurate model equatioare available so as to reveal qualitative features that shobe kept in mind when trying to find a stable regime in a lewell characterized system. We note that an approximate sbility analysis of Eq.~3! has been undertaken recently.39

C. Frequency domain analysis of ETDAS feedback

While the time-domain stability analysis outlinedabove gives a complete picture of ETDAS feedback inneighborhood of the UPO, a frequency-domain analyshelps clarify the underlying reasons for its effectivenein stabilizing highly unstable orbits asR→1. Note that theETDAS feedback signal given by Eq.~2! linearly relates theinput signalj(t) with the output signaldp(t); hencedp( f )52gT( f )j( f ), wherej( f ) and dp( f ) are the Fourier am-plitudes of the input and output signals, respectively, and

T~ f !512exp~ i2p f t!

12Rexp~ i2p f t!, ~6!

is the transfer function for ETDAS feedback. The transffunction ‘filters’ the observed state of the dynamical systemcharacterized byj( f ), to produce the necessary feedbacsignal.

Figure 3 shows the frequency dependence ofuT( f )u forR50 ~TDAS! and R50.65 ~ETDAS! where it is seen thatthere are a series of notches that drop to zero at multiplesthe characteristic frequency of the orbitf * 5t21 and that thenotches become narrower for largerR. The existence of thenotches in the transfer function can be understood easilyconsidering thatdp(t), and hencedp( f ), must vanish whenthe UPO is stabilized. Recall that the spectrumj( f ) of thesystem when it is on the UPO consists, in general, of a serof d-functions at multiples of the characteristic frequencythe orbit f * 5t21; therefore, the filter must remove thes

FIG. 3. Transfer functionuT( f )u of ETDAS feedback. The dashed line is forR50 and the solid line is forR50.65. Note that notches atf / f * 5q(q50,1,2, . . . ) arenarrower forR50.65.

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frequencies~via the notches! so thatdp( f )50. The ETDASfeedback is more effective in stabilizing UPOs for largerRpartly because it is more sensitive to frequencies that copotentially destabilize the UPO. The narrower notches imthat more feedback is generated for signals with frequecomponents slightly different from the desired set. In adtion, uT( f )u is flatter and remains near one betweennotches for largerR, so the system is less likely to be destbilized by a large feedback response at these intermedfrequencies.

We note that other transfer functions that possnotches at multiples off * could stabilize the dynamics of thUPO; however, Eq.~6! is easy to implement experimentally

IV. IMPLEMENTATION OF ETDAS

A. General implementation

ETDAS feedback control of UPOs can be implementstraightforwardly using a variety of techniques, even on ftime-scales. One possible logical design of the feedback lis shown schematically in Fig. 4. We assume for the momthat the components impart negligible propagation delaysthe various signals~except, of course, for the intentional delay t necessary to form the ETDAS feedback signal!. Thedynamical state of the systemj(t) is converted to a voltageV(t) by a transducer. The power of this signal is split equabetween two different paths~point A!; half of the signal isdirected to one input port~point B! of a voltage subtractiondevice and is related to the first term on the right-hand-sof Eq. ~2!; the other half is directed to one input port~pointC! of a voltage addition device forming part of the groupcomponents~dashed box! generating the delay terms in Eq~2!. The signal in the latter path emerges from the additdevice, propagates through a variable delay line with

FIG. 4. One possible implementation of ETDAS feedback. The setup gerates the feedback signaldp(t) by combining the measured state variabof the systemj(t) with an infinite series of values of the state variabdelayed in time by integral multiples of the periodt of the desired UPO.The dashed box contains the components that generate the infinite deseries. This delayed series is subtracted~point F! from the current value ofthe system state variable~point B!, amplified, then converted to the feedbacsignaldp(t).

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delay time set equal to the periodt of the desired UPO~point D!, is attenuated by an amountR, and is injected intothe second port~point E! of the addition device where it iscombined with the original signal. A high-impedance buffsenses the voltage signal at the output of the delay line~pointD!. The signal emerging from the buffer is attenuated(12R), injected into the second port~point F! of the voltagesubtraction device, and represents the second term onright-hand-side of Eq.~2!. The signal emerging from thesubtraction device~point G! is proportional to the ETDASerror signal; it is amplified and injected into a transducer tadjusts the accessible control parameter bydp(t). We notethat all of the operations performed by the componentstween the input and output transducers could be accplished entirely with a digital computer equipped wianalog-to-digital and digital-to-analog converters, if the dnamics of interest were slow enough.

We emphasize that the unavoidable propagation dethrough components must be thoroughly characterized inreal implementation of ETDAS. As described in Sec. IVsmall additional time delays must be added to the feedbsystem and adjustments must be made to compensate fotime delays inherent in the components.

B. All-optical implementation

One useful feature of ETDAS feedback is the abilto generate the error signal using an all-optical tenique.8,34,35,40Specifically, the form of the ETDAS error signal given by Eq.~2! is identical to an equation that describthe reflection of light from a Fabry-Perot interferomete41

consisting of two equal-reflectivity mirrors, whereR5r 2

corresponds to the square of amplitude-reflection-coefficr of the mirrors, andt corresponds to the round-trip transtime of light in the cavity. In one possible scenario, the inptransducer shown in Fig. 4 generates a laser beam ofstrengthEinc(t) that is directed toward an optical attenuator amplifier~controllingg) and a Fabry-Perot interferometeas shown in Fig. 5. The fieldEre f(t) reflected by the inter-ferometer passes through the attenuator/amplifier and isverted to the ETDAS error signaldp(t) by the output trans-ducer. It may be possible to control fast dynamics of optisystems that generate directly a laser beam, such as semductor diode lasers, using the field generated by the lasethe measured system parameterj(t) and as the accessiblesystem parameterdp(t); no transducers are required.8,34,35,40

C. Specific implementation for controlling the fastdiode resonator

1. Electronic circuit layout

We control the dynamics of a fast chaotic electronic ccuit known as a diode resonator, described in the nexttion, using an analog-electronic implementation of ETDAas shown schematically in Fig. 6. The implementationtempts to mimic as closely as possible the generic dedescribed in Sec. IV A. It balances considerations of thenite propagation time of the signals through the compone~affecting the control-loop latency!, distortion of the signals

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~affecting the fidelity of the feedback signal!, and noise, withthe simplicity of the layout, ease of construction, and priNo transducers are required in this design because the asible system parameter, denoted byV(t), and control param-eter, denoted byVdp(t), are voltage signals.

The electrical signals propagate through the circuit infashion similar to that described in Sec. IV A for the geneimplementation of ETDAS; points~A-F! in Fig. 6 for thespecific implementation correspond to the same pointsFig. 4 for the generic implementation. One goal of the circlayout shown in Fig. 6 is to ensure that the differencepropagation time of signals from pointsA→G8 andA→D8→D→F→G8 is equal tot and the time fromD→E→D8→D is also equal tot in order to faithfully produce theETDAS error signal. This task is complicated by the nonzepropagation delays through the electronic components.

In our layout, the accessible system parameterV(t) issensed by a high-impedance buffer~B1! and the power associated with this signal is split equally between two differepaths~point A!. Half of the signal is directed to the inpu~point B! of an inverting, summing operational amplifie~A1! and the other half is directed to the input~point C! of asecond inverting, summing amplifier~A2! where it is filteredwith predistortion circuitry~F1!. The signal emerging fromA2 propagates through a long, variable delay line~D1! with adelay time set close to, but less than, the periodt of thedesired UPO, is inverted and amplified~A3! by a fixedamount to compensate for the average loss of the delaypasses through a short variable delay line~D2!, attenuated toset the control parameterR, and directed to the input~pointE! of A1 where it is filtered with predistortion circuitry~F2!.The signal emerging from the long delay line~point D! issensed with a noninverting amplifier~A4!, passes through ashort variable delay line~D3, point F!, and is directed to theinput of the summing amplifierA1 where it is attenuated bya factor equal to (12R) by adjustingP1. The signal emerg-

FIG. 5. All-optical implementation of ETDAS feedback using a Fabry-Peinterferometer consisting of two mirrors with amplitude reflection coecient r separated by a distance such that the round-trip-time of light incavity is equal to the periodt of the desired UPO. An optical attenuator oamplifier adjusted the feedback gaing.

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FIG. 6. Specific implementation of the ETDAS feedback circuitry for controlling the dynamics of a high-speed diode resonator. The signalV(t) is combinedwith an infinite series of previous values delayed by multiples of the orbital periodt. The dashed box marked ‘dc offset control’ compensates for small offsintroduced by the circuitry.A12A4 are operational amplifiers,B1 is a high-impedance buffer, andD12D3 are delay lines. The dashed boxesF1 andF2contain predistortion filters; their components depend on the particular choice of delay cableD1. Component values are: For stabilizing the period-1 UPL150.33mH, L250.33mH, R75150V, R85402V, R95100V , R105825V; for the period-4 UPO,L151.0mH, L251.0mH, R75150V, R85590V, R95100 V, R1051540 V. Other component values are:C150.1 mF, L353.3 mH, P155 kV, P251 kV, R1550 V, R25100 V, R35715 V,R45499 V, R55357 V, R65255 V , R115140 V, R125324 V, R13559 V, R14553.6V, R155215 V, andR1651 kV.

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ing from A1 ~point G! is proportional to the ETDAS errosignal; it is directed to additional amplifiers to set the feeback gaing and injected into the diode resonator.

2. Selection of components

One primary issue in designing the ETDAS circuitryminimizing the control-loop latencyt l . The latency must bemuch less than;100 ns to control the diode resonator dscribed in the next section based on our estimate of theues of the conditional Lyapunov exponents characterizthe UPOs. In this proof-of-principle investigation of ETDASwe select commercially available, inexpensive operatioamplifiers and buffers which add very little tot l yet are easyto obtain and incorporate into the design. For our choicecomponents,t l ;10 ns. Smaller latencies may be possibusing custom-built microwave circuitry; we do not explothis option.

The control-loop latency is governed by the time it takfor the signal to propagate from the input to the output ofETDAS following the pathA→B→G since the portion ofthe signal atG should be proportional ideally to the inpusignal ~with no time lag!. Propagation delays of the sign

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passing through the other components forming the infinseries of time-delay terms is not as important becausecan be compensated using a slightly shorter coaxial ca~D1!. We select the AD9620~Analog Devices,42 propagationdelay&1 ns, bandwidth 600 MHz! for B1 and the AD9618~Analog Devices, propagation delay.3 ns, bandwidth 160MHz! for A1 because they have short propagation delays,relatively easy to use, and are low cost. In the other partthe circuit where propagation delays are not as import~A2-A4!, we select the AD811~Analog Devices, propagationdelay .6 ns, bandwidth 140 MHz! because its stability isless sensitive to the precise layout of the components. Nthat the bandwidth of all components is significantly largthan the characteristic frequency of the chaotic system~10.1MHz!, thereby minimizing the distortion of the ETDASfeedback signal. Operational amplifiers and buffers wsimilar or better specifications should produce similarsults.

Due to the high bandwidth of these devices, we use sdard high-frequency analog electronic techniques. The cponents are laid out on a home-made, double-sided princircuit board in which the signal traces form a transmiss

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line ~nominal impedance 50V) and are kept under 2 cmproper ground planes are maintained under all componeand PC board mounted SMA jacks are used to transfersignals from the board to coaxial cables. The printed circboards are fabricated in a multi-step process: We creagraphical layout of the traces using a simple drawing pgram on a personal computer; the drawing is printed o‘transfer film’ with a laser printer;43 the traces are transferreto the clean, bare copper surface of the board; the plabacking of the transfer film is removed; the board is etchusing a hot ammonium persulfate solution; holes are drifor the leads of the components; and the componentssoldered to the board. The power supply leads to all optional amplifiers are bypassed with ceramic and tantalumpacitors as close as possible to the power supply pins,ferrite beads isolate the power supply lines between comnents. The dc offset voltage of the amplifiers is compensaat the last stage of the circuitry~point G8! by adding a dcvoltage to the signal from a filtered, adjustable power supas shown in Fig. 6. All signals routed between the ETDand other circuit boards propagate though short 50V coaxialcables~type RG-58/U!.

Another primary issue in the implementation of ETDAfeedback is minimizing the distortion~frequency-dependenattenuation and phase shift! of signals propagating througthe coaxial delay line that sets the time delayt. Previousimplementations of TDAS have used for the delay linevariable length transmission line,8 electro-optic delay line,28

or digital first-in-first-out delay device.29 Distortion is diffi-cult to avoid for the high-period UPOs because the delayis long, and for the case when the feedback parameterR→1since the signals circulates many times through the deline. To minimize the distortion, we use low-loss, semi-rig50 V coaxial cable~1/4 in. HELIAX type FSJ1-50A44!. Wefind that this delay line gives superior performance in coparison to standard coaxial cable~type RG-58/U!. The at-tenuation characteristics of the HELIAX cable is 0.175 d100 ft. at 1 MHz and 1.27 dB/100 ft. at 50 MHz whicshould be compared to 0.33 dB/100 ft. at 1 MHz and 3dB/100 ft. at 50 MHz for the RG-58/U cable.

3. Adjustment of circuit parameters

The final preparation step for controlling the dynamof the chaotic system using ETDAS feedback is to adjusttime delayt and the feedback parameterR. We note thatsettingt requires adjusting the propagation time of signfollowing two different paths. Referring to Fig. 6, the diffeence in time it takes for signals to propagate fromA→B→G8 and fromA→D→F→G8must be equal tot. In addi-tion, the propagation time around the delay loop~D→E→D8→D! must also equalt. Note that these times include thpropagation time through all components: The coaxial cathe amplifiers and attenuators, and the signal traces andnectors. The timing can be achieved straightforwardly usthree coaxial delay lines: A long delay lineD1 consisting ofa HELIAX cable whose propagation time is close to but lethant, a short coaxial cable~type RG-58/U! for fine tuning

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the propagation time ofD1, and a constant-impedance ajustable delay line~Hewlett-Packard model 874-LK10L45!for ultra-fine adjustment ofD1; a short variable delay lineD2 consisting of another adjustable delay line and coaxcables~type RG-58/U! whose length is just long enough tconnect this variable delay line to the printed circuit boaand a short, fixed delay lineD3.

Initial adjustment of the timing is achieved by injectingweak sinusoidal voltage whose period is equal tot at pointA. The first step is to adjust the circuitry for TDAS feedba(R50), that is, to set the path difference betweenA→B→G8 andA→D→F→G8 to t with the circulating loop tem-porarily disabled by disconnecting the attenuator~Kay Elem-etrics model 1/839 manual step attenuator46!. The lengthD1is set so that the sinusoidal signal at pointF is delayed byone period from the signal at pointB. The amplitude of thedelayed signal is adjusted so that the signal at the outputhe ETDAS circuitry~point G! is minimized using the vari-able resistorP1; iterating this procedure while monitorinthe signal atG results in precise adjustment ofD1. Next, thecirculating loop is reconnected with the attenuator set soR;0.1. The delayD2 and the variable resistorP1 are ad-justed to minimize the signal atG. Sensitive adjustment oD2 is achieved by decreasing the attenuation so thatR→1and iterating the setting ofD2 and P1. Typically, the 10.1MHz sinusoidal input signal is suppressed at pointG by ;50dB at 10.1 MHz.

Note that our procedure results in a ETDAS feedbasignal Vdp(t)5Vdp

ideal(t2t l ), where Vdpideal(t) is the ideal

feedback signal in the absence of control-loop latency,t l is the latency with contributions from the total time lagthe signal as it propagates from the diode resonator, throthe ETDAS circuitry and additional amplifiers setting thfeedback gaing, and the summing amplifier that injects thfeedback signal into the diode resonator.

Adjusting the feedback parameterR to any desired valueis straightforward once the timing of the circuit is achieveA new value ofR is selected by setting the attenuator aadjusting the variable resistorP1 which sets the value o(12R) in Eq. ~2!.

4. Compensating for nonideal circuit behavior

We characterize the quality of the ETDAS circuitry bmeasuring its frequency-domain transfer functionuT( f )u ~re-call the ideal shape ofuT( f )u shown in Fig. 3!. Figure 7showsuT( f )u for t5400 ns corresponding to a fundamenfrequencyf * 52.5 MHz ~for controlling the period-4 UPOof the diode resonator!, R50 ~dashed line!, and R50.65~solid line! where we adjust the circuit parameters so thatnotch at 10 MHz is as close to zero as possible. We optimthe circuit for operation at 10.1 MHz, rather than 2.5 MHbecause the spectrum of the period-4 orbit has a large pe10.1 MHz. It is seen that all other notches have a transmsion less than 0.047 forR50.65 and less than 0.017 foR50. The agreement between the observed and predi

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behavior is very good, especially when one considers thasensitivity of the shape of the notches on the distortion ofdelay line increases asR→1.

We find that these high-quality results are only possiwhen the signals are filtered with predistortion circuitry~F1andF2! that attempt to cancel the small distortion causedthe HELIAX cable. The filters, consisting of two resistoand an inductor, do not significantly effect the signals at lfrequencies and boost their amplitude a high frequenciesfashion opposite to the small frequency dependent attetion of the coaxial cable. For comparison, Fig. 8 shouT( f )u with the predistortion filters~F1 and F2! removed

FIG. 7. Experimentally determined transfer function of the ETDAS circuishown in Fig. 6 forR50 ~dashed line!, R50.65~solid line!, andt.400 ns( f * .2.5 MHz!.

FIG. 8. Transfer function of the ETDAS circuity shown in Fig. 6 with thpredistortion filters removed forR50 ~dashed line!, R50.65 ~solid line!,andt.400 ns (f * .2.5 MHz!. The fact that the notches do not drop to zeindicates the uncompensated distortion of the long delay line.

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where we make minor adjustments of the circuit parametto optimize the notch at 10.1 MHz. It is seen that the perfomance of the device is severely degraded, especiallyR50.65. We note that acceptable behavior of the ETDAfeedback circuitry witht5400 andR50.65 can not be ob-tained using standard coaxial cable~type RG-58/U! evenwith the predistortion filters.

V. CONTROLLING CHAOS IN A FAST DIODERESONATOR USING ETDAS

A. The fast diode resonator

In our experimental studies of controlling chaos on fatime-scales, we use a ‘diode resonator’ because it displaywide variety of nonlinear behaviors including period doubling bifurcations, hysteresis, intermittency, and crisis; iteasy to build and customize; and it is well characterizedslow time-scales.47–54 The low-speed diode resonator typically consists of an inductor and a diode connected in serdriven by a sinusoidal voltage.

The high-speed variation of the diode resonator, shoschematically in Fig. 9, has additional components neededimplement high-speed control. It consists of a rectifier dio~type 1N4007, hand-selected for low junction capacitance! inseries with a 25mH inductor~series DC resistance of 2.3V)and a resistorRs550 V driven by a leveled sinusoidal volt-age V0(t)5V0sin(vt) (v/2p>10.1 MHz! that passesthrough a high-speed signal conditioner~50 V output imped-ance!. The resistorRs converts the current flowing throughthe diode resonator~the measured dynamical variable! into avoltage V(t) which is sensed by a high-impedance buffe~Analog Devices AD9620, propagation delay&1 ns!. Thesignal conditioner consists of a high-speed inverting ampfier ~Analog Devices AD9618, propagation delay.3 ns!and serves two purposes: It isolates the sine-wave gener~Tektronix model 3325A55! from the diode resonator and itcombines the sinusoidal drive signalV0(t) with the ETDASfeedback signalVdp(t). The components are laid out onhome-made, double-sided printed circuit board in which t

FIG. 9. The high-speed diode resonator. The series-connected rectifieode, inductor, and resistor are driven by a leveled sinusoidal voltageV0(t).The signal conditioner/combiner isolates the driver from the circuit, ainjects the feedback voltageVdp(t) into the diode resonator. A high-impedance buffer measuresV(t), which is proportional to the current flow-ing in the resonator.

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traces are kept under 2 cm, proper ground planes are mtained under all components, and the signals enteringleaving the board pass from the traces to coaxial cathrough PC board mounted SMA jacks. A high-impedanbuffer ~Analog Devices AD9620! is also included to monitorthe ETDAS feedback signal. We find that it is crucialincorporate the current-to-voltage converter~the 50V resis-tor! into the architecture of the chaotic oscillator to minimithe latency of the loop; we believe that incorporating socomponents of the feedback loop into the chaotic sysmust be considered when attempting to control any hispeed dynamics.

We record a bifurcation diagram and first-return mapscharacterize the dynamics of the uncontrolled diode resotor using the sinusoidal drive amplitudeV0 as the bifurcationparameter. In the experiments, we capture multiple~typicallythree! 40-ms-long time-series data ofV(t) using a 100 MHz,8-bit digitizing oscilloscope~Tektronix model 2221A! anddetermine the maxima ofV(t), denoted byVn , using asecond-order numerical interpolation routine. The bifurtion diagram@Fig. 10~a!# is generated by plotting approxmately 300 values ofVn for each value ofV0 as it is variedfrom small to larger values~hysteresis in the system is evdent whenV0 is varied from the largest to smaller valuesV0 , but it is not shown in this diagram for clarity!. It dis-plays a typical period-doubling cascade to chaos wheretransition to chaos occurs atV0.1.5 V. Also evident are anarrow period-5 periodic window atV0.1.8 V and a largeperiod-3 window starting atV0.2.4 V. The first-return mapsare generated by plottingVn vs.Vn11 for fixed values ofV0 .The experimentally observed chaotic maps of the system

FIG. 10. Bifurcation diagrams of the diode resonator: Peak voltageVn as afunction of the drive amplitudeV0 . ~a! Experimental data.~b! Numericalsimulations.

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low (V052.2 V! and above (V053.6 V! the period-3 win-dow are shown in Figs. 11~a! and~b!, respectively. The general structure of these maps are similar to observed in lfrequency diode resonators.53

Considering the complexity associated with predictitheoretically the stability of the high-speed diode resonain the presence of ETDAS feedback based solely on expmental measurements, we use a simple model of the restor that captures the essence of its behavior. The model,scribed briefly in the Appendix, treats the real componentsthe resonator as collections of ideal components in a fashsimilar to that used in commercially available electronsimulation software packages. All parameters of the moare determined by independent measurements of the comnents. We find very good agreement between the experimtally observed and numerically generated bifurcation dgrams@Fig. 10~b!# and first-return maps@Figs. 11~a! and~b!#.It is seen that all features observed in the experimentpredicted by the model; however, there is a slight discrancy in the location of the periodic windows and in the sof Vn . As we will see in Sec. V F, this model is adequatepredict with good accuracy the stability of the controlleresonator.

B. Experimental observation of control

Our procedure for achieving control of the high-spediode resonator using ETDAS is to: set the delay-timet, thefeedback parameterR, and optimize the feedback circuitroff-line as described in Sec. IV C 3; connect the ETDA

FIG. 11. First return maps of the unperturbed diode resonator abovebelow the period-3 window. Experimental maps~a! V052.2 V and ~b!V053.6 V are in good agreement with numerically simulated maps~c!V052.2 V and~d! V053.6 V.

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circuitry to the diode resonator using short coaxial caband search for control while adjusting the feedback gaing.Control is successful whenV(t) is periodic with the desiredperiodt and the magnitude of the feedback signalVdp dropsbelow a level set byuT( f )u ~the transmission of the worsnotch! and the noise level in the system. In principle,Vdp

vanishes in an ideal situation. In practice, noise and nonidbehavior of the ETDAS circuitry~see Sec. IV C 4! lead to anonzero value ofVdp(t). Our criterion for control is thatuVdp(t)u,531023V0 , or that 20log(uVdp(t)u/V0),246 dB.We verify that the controlled orbits are indeed UPOs ofchaotic system to within the experimental uncertainty whthis criterion is fulfilled by comparing the return maps of tcontrolled and uncontrolled systems. We note that periostates that are not UPOs of the system can be obtained udifferent experimental conditions; in this case the feedbsignal is large~of the order ofV0).

Figures 12~a! and ~b! show the temporal evolution oV(t) when the ETDAS feedback circuitry is adjusted to sbilize the period-1 and period-4 UPOs, respectively, togetwith the associated feedback signal. The sinusoidal driveplitude is 2.4 V~the system resides in the chaotic regime jbelow the period-3 window in the absence of control! andthe feedback parameters areg56.2, R50.28 for the stabi-lized period-1 orbit andg54.2, R50.26 for the period-4orbit. It is seen that the feedback signal is a small fractionthe drive amplitude (,231023 for both cases!. The slightincrease inVdp(t) for the period-4 orbit is mainly due to

FIG. 12. Time series data and first return maps illustrating succesETDAS control. Temporal evolution~solid lines, scale on left! of the stabi-lized ~a! period-1 UPO (g56.2, R50.28) and~b! period-4 UPO (g54.2,R50.26) along with their associated ETDAS error signalsVdp(t) ~dashedlines, scale on right!, expressed as a fraction of the drive amplitude. Simdata in the form of a first return map for the controlled~c! period-1 (g54.4,R50) and ~d! period-4 (g53.1, R50.26) trajectories are highlighted bthe dark concentrations of points indicated by arrows. These maps arperimposed on maps of the uncontrolled chaotic resonator. For~a!-~d!,V052.4 V.

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imperfect reproduction of the form of the ETDAS feedbasignal by our circuitry. This effect is more prevalent for thperiod-4 setup because the delay line is longer and hecauses more distortion of the signals due to the disperand frequency-dependent loss of the coaxial cable.

Figures 12~c! and ~d! presents further evidence that thETDAS feedback indeed stabilizes the UPOs embedwithin the strange attractor of the diode resonator. We shthe return maps for the uncontrolled system~light dots! andthe controlled system~dark dots indicated by arrows! for thestabilized period-1 orbit@Fig. 12~c!, R50, andg54.4] andthe period-4 orbit@Fig. 12~d!, R50.26, andg53.1] forV052.4 V. It is clear that the stabilized orbits lie on thunperturbed map, indicating that they are periodic orbitsternal to the dynamics of the uncontrolled system.

C. Domain of control

We find that these UPOs can be stabilized using a wrange of feedback parameters which can be visualiquickly from a plot of the ‘domain of control.’ The domaiof control is mapped out by determining the range of valuof the feedback gaing that successfully stabilizes an UPOa function of the bifurcation parameterV0 for various valuesof the feedback parameterR. In general, increasing the feedback parameterR tends to enlarge the domain of control anshift it to slightly larger values ofg. We find that theperiod-1 orbit can be stabilized using ETDAS feedbackall V0 whenR50 andR50.68 as seen in Figs. 13~a! and~b!,respectively. It is clear that the range ofg that successfully

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FIG. 13. Experimental domains of control as a function of the drive amtudeV0 for period-1 and period-4 UPOs for two different values ofR. Theshaded regions illustrate period-1 domains of control forR50 ~a! andR50.68~b!, and period-4 domains of control forR50 ~c! andR50.65~d!.The dashed vertical lines mark the points at which the orbit becomes sin the absence of feedback.

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stabilizes the orbit for a given value ofV0 increases signifi-cantly for R50.68 in comparison toR50, especially forlarger values ofV0 .

The largeness and shape of the domain of control isportant for several practical reasons. Since the domailarge, the system is rather insensitive to drift in the feedbparameters or the state of the chaotic system, and controbe obtained even with imprecise,a priori knowledge of theproper feedback parameters. Also, the ETDAS feedbackautomatically track changes inV0 over its entire range with-out adjustingg because the shape of the domain is a wihorizontal band.

Increasing the size of the feedback parameterR givesrise to more dramatic effects on the domain of control whstabilizing the period-4 orbit, as shown in Figs. 13~c! and~d!.It is seen that TDAS feedback (R50) fails for moderate andlarger values of the bifurcation parameter. The domaincontrol can be extended to include the entire rage ofV0 usingR50.65, illustrating the superiority of ETDAS for controling highly unstable orbits. Also, the size and shape ofdomain of control for this case has all of the advantamentioned in the previous paragraph for the period-1 domof control.

D. Transient behavior

We find that the period-1 and period-4 UPOs shownFig. 12 can be stabilized by initiating control at an arbitratime; we do not have to wait for the system to naturaapproach the neighborhood of the UPO nor target the systo the UPO before control is initiated. Such control is posible for several reasons: The system is particularly simplthat there is only one period-1 and one period-4 UPO;basin of attraction of these UPOs in the presence of ETDfeedback is large; and we apply large perturbations@Vdp(t)is comparable toVd(t)] to the system during the transienphase.

The dynamics during the initial transient can be qurich; the nature of the convergence of the system to the Uis a function of the precise state of the system when conis initiated and the value of the control parameters. Whencontrol parameters are adjust such that the system is neawithin, the domain of control, the system often displays brintervals of periodic behavior other than the desired behaduring the transient phase. Figures 14~a! and ~b! are twoexamples of the approach to the period-1 UPO forV051.9V, R50.68, andg55.2 ~a! or g57.0 ~b!, where we plot thepeak valueVn of V(t) as a function of the peak numbern~control is initiated nearn50). In one case@Fig. 14~a!#, aninitial chaotic transient gives way to a period-2 behavior tdecays exponentially to the desired UPO. In the [email protected]~b!#, an initial chaotic transient abruptly switches to cotrolled behavior.

Richer transient dynamics is observed when stabilizthe period-4 UPO, as shown in Figs. 14~c! and ~d! forV051.9 V, R50.26, andg56.1 ~c! or g52.6 ~d!. In onecase@Fig. 14~c!#, we observe a brief interval of period-behavior that eventually destabilizes and decays in an o

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latory fashion to the desired UPO. In the other [email protected]~d!#, the system passes through intervals of period-2period-16 behavior before converging to the period-4 UPIt is not surprising that the system resides near a period owith a period longer than the desired UPO since the ETDfeedback cannot distinguish between the two differenthaviors. However, the low-period behavior is not stablelong times since the ETDAS feedback is not sensitive to adoes not correct for noise and fluctuations that destabithis undesired low-period orbit.

E. Average time to attain control

While the transient dynamics are quite rich near tboundary of the domain of control, ETDAS feedback stalizes rapidly the desired UPO over most of the domain. Fure 15 shows the average time^t r& to obtain control of theperiod-1 UPO from an arbitrary starting time as a functionthe feedback gaing for V051.9 V andR50 @Fig. 15~a!# andR50.68 @Fig. 15~b!#. The hashed region indicates the ranof g over which stable period-1 behavior is not observeThe data points represent the average of 20 trials anderror bars indicate the standard deviation of the times totain control. It is seen that^t r& is less than 3ms and typicallyless than 1ms ~10 orbital periods!, except near the edge othe domain of control wheret r& grows rapidly. In addition,it is seen that the range over which rapid control is obtainincreases significantly by increasingR to 0.68. This high-lights one of the advantages of ETDAS feedback conover TDAS feedback. Figures 16~c! and ~d! show similardata for the period-4 UPO. We note that^t r& should increaserapidly near the edges of the domain of control for this UP~as it does in the period-1 case!. In our experiment, however

FIG. 14. Transient behavior of the diode resonator following initiationETDAS control. The dots represent the peak valuesVn of V(t). The dashedvertical line marks the point at which control is initiated. Convergence tcontrolled period-1 UPO forV051.9 V,R50.68,~a! g55.2 and~b! g57.0.In addition, we show two approaches to the period-4 UPO forV051.9 V,R50.26, ~c! g56.1 and ~d! g52.6. Note that the transients have briesections of decaying periodic behavior.

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F. Theoretical analysis

We find that the experimentally measured domainscontrol can be predicted theoretically with good accurausing the techniques described in Sec. III B. Briefly, the pcedure to determine the domain is to:10

• determine the trajectory of the desired UPO for ovalue of the bifurcation parameterV0 by analyzing thecoupled nonlinear differential equations describing the dnamics of the diode resonator@see Eq.~A4!#;

• choose the feedback parametersn andR ~note thatn isset throughout the experiment by our choice of accesssystem variable and control parameter!;

• find the boundaries of the domain of control@thepoints at which the winding number jumps from zero topositive integer, see Eq.~5!# for one value of the bifurcationparameterV0 using standard numerical root-finding algorithms;

• follow the boundary by repeating this procedure fthe other values ofV0 .

Note that this analysis does not include the effectscontrol-loop latency.

We compare the observed@Fig. 16~a!# and predicted@Fig. 16~b!# domain of control for the period-1 UPO wheR50.68. It is seen that the theoretical analysis correctly pdicts the general horizontal banded shape of the domainthat the low-gain boundary of the domain is in good quantative agreement with the experimental observations. Hoever, in contrast with the observations, the theoretical ana

FIG. 15. Average transient times^t r& as functions of feedback gaing for theperiod-1 orbit with V051.9 V and R50 ~a! and R50.68 ~b!, and theperiod-4 transient times forV051.9 V andR50 ~c! andR50.65 ~d!. Theerror bars indicate the standard deviation of^t r& and the hashed regionsindicate the range ofg for which control is not obtained.

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sis predicts that values of the feedback parameterg largerthan 1000 will give rise to successful control. Somewhcloser agreement between the observed and predictedmain of control is obtained for the period-4 orbit (R50.65)as shown in Figs. 16~c! and~d!, respectively. Similar resultsare obtained for other values of the control parameterR.

We find that the discrepancy between the theoreticapredicted and experimentally observed domains of controdue primarily to the control-loop latency (t l 510 ns for ourimplementation of ETDAS!. To investigate the effects oft l ,we turn to direct numerical integration of the time-delay dferential Eqs.~2! and ~A4!. The equations are integrated uing a fourth-order Adams-Bashforth-Moulton predictocorrector scheme with a step size of 0.1 ns. Succescontrol is indicated when the ETDAS error signal falls to tmachine precision in a time equal to 20,000 orbital perioWe note that this procedure may underestimate the sizthe domain of control when the convergence is slow. Tothe initial conditions for the time-delay differential equtions, we integrate the equations without the time-determs ~equivalent to the diode resonator in the absencecontrol!, initially using a fourth-order Runge-Kutta algorithm. We store variable values, derivatives, and calculatime-delay in arrays as we integrate; these values are nefor the predictor-corrector routine and to determineVdp(t).We switch to the predictor-corrector integrator after tcycles of Runge-Kutta and add the control terms. We nthat despite the infinite series in Eq.~2!, it is not required toretain all past variable values to calculateVdp(t); ETDAScan be expressed recursively, requiring only storage ofinformation from one past orbital cycle.8

FIG. 16. Comparison of experimental and theoretical domains of conWe compare period-1 domains~a! and~b! at R50.68, and period-4 domains~c! and~d! at R50.65. The vertical dashed line marks the point at which torbit becomes unstable in the absence of feedback. Note that the thecally predicted domains assumet l 50.

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Figure 17 shows the experimental@period-1, Fig. 17~a!;period-4, Fig. 17~c!# and numerically predicted@period-1,Fig. 17~b!; period-4, Fig. 17~d!# domains of control for theUPOs with t l 510 ns andR50.68 ~period-1! and R50.65~period-4!. It is seen that the agreement is improved, higlighting the fact that small control-loop latencies~only 10%of the orbital period for the period-1 UPO! can shift thedomain of control.

G. Effects of control-loop latency

To illustrate the deleterious effects of large control-lolatency, we add an additional delay to the feedback signainserting various lengths of coaxial cable~type RG-58/U!between the output of the ETDAS error generating circuand the signal conditioning device at the input to the dioresonator. In this case,Vdp(t)5Vdp

ideal(t2t l ), where thetime-lag t l includes the latency of the ETDAS circuitry~10ns! as well as that incurred by the coaxial cable. By intrducing the additional time-lag, we mimic the situationwhich slower electronic components are used in the ETDcircuitry.

Figure 18 shows the range ofg that gives rise to successful control of the period-1 UPO as a function oft l forV051.9 V, andR50 @Fig. 18~a!# andR50.68 @Fig. 18~b!#.For both cases, it is seen that increasingt l causes the rangof g to decrease until control is unattainable when the timlag is approximately one-half oft ~100 ns!. Stabilization ofthe orbit is possible for larger values oft l as a new region ofcontrol appears, although with the opposite sign of the fe

FIG. 17. Comparison of experimental and numerically simulated domof control. The numerical simulations include a control-loop latencyt l 510ns. As in Fig. 16, we compare period-1 domains~a! and~b! at R50.68, andperiod-4 domains~c! and ~d! at R50.65. The vertical lines again mark thpoint at which the orbits become unstable in the absence of feedback.that inclusion of control-loop latency improves the quantitative agreementhe results.

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back parameterg. It is also seen that the range ofg thatgives rise to control whenR50 is much less than that wheR50.68, indicating that ETDAS is more effective thaTDAS in the presence of a time-lag. Control is not possifor t l .120 ns. For comparison, the correlation~memory!time of the system is approximatelyl21.140 ns, wherel isthe largest conditional Lyapunov exponent characterizingperiod-1 UPO in the absence of control. Hence, the corrtion time appears to be the relevant parameter with regardcontrol-loop latency.

Similar results are obtained when stabilizing tperiod-4 orbit as shown in Fig. 19, although control fails fmuch smallert l . It is seen in Fig. 19~a! that control is pos-sible only tot l .26 ns forR50 andt l .70 ns forR50.65in Fig. 19~b!. The correlation time for the period-4 UPOl21.250 ns. Also, the range ofg that gives rise to successful control is much larger forR50.65. The fact that ETDASfeedback extends significantly the ranget l in comparison tothat achievable by TDAS suggests that ETDAS may befective in controlling the dynamics of systems that fluctuaon the nanosecond or even sub-nanosecond scale for wrelatively smallt l may be difficult to achieve.

In contrast to the behavior observed for the period-1bit, we find no islands of stability forg,0 at the longertime-lags. We do find, however, a small region whereorbit is stabilized intermittently aroundt l .1072117 nswhenR50.65. The system alternates irregularly betweentervals of ‘noisy’ period-4 behavior and intervals of chaobehavior. The indicated region of Fig. 19~b! is the where thesystem resides near the period-4 orbit for at least 10% of

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FIG. 18. Effects of control-loop latencyt l on the domain of control for theperiod-1 UPO. Controlled regions are shown forV051.9 V, ~a! R50 and~b! R50.68. The dashed vertical line att l 510 ns indicates the minimumlatency attainable with our implementation of ETDAS.

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time on average. Under some conditions, the systemsmains near the UPO for'16 ms, corresponding to 40,00orbital periods. Figure 20 shows examples of the dynamevolution of the peaksVn of V(t) during this intermittentbehavior. It is seen that the orbit can lose stability abrup@Fig. 20~a!#, or through a transition to other periodic behaiors @Fig. 20~b!#. We emphasize that these transitions doresult from a change in the feedback level, but occur sptaneously in the system.

H. Dynamics of the system outside the domain ofcontrol

We find it instructive to investigate the dynamics of tdiode resonator in the presence of ETDAS feedback out

FIG. 19. Effects of control-loop latencyt l on the domain of control for theperiod-4 UPO. Controlled regions are shown forV051.9 V, ~a! R50 and~b! R50.65. Note the region of intermittency forg,0 at larget l in ~b!.The dashed vertical line indicates the minimum latency of our circuitry.

FIG. 20. Evolution of peak valuesVn of V(t) showing intermittent chaoticbursts away from period-4 UPO atV051.9 V andR50.65.~a! A short burstwith g57.4 and t l 5116 ns. ~b! Transition to a burst through periodibehavior withg56.8 andt l 5112 ns.

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the domain of control to serve as a guide when attemptingcontrol the dynamics of new systems. We explore the raof different behaviors by adjusting the feedback gaing whilefixing the other control parameters atVd51.9 V, t5100 ns~period-1 UPO control!, R50, andt l 510 ns~its minimumvalue!. For these parameters, the range of successful conof the period-1 orbit is approximately 3.14,g,16.9. Simi-lar behaviors can be obtained by fixingg and adjusting adifferent parameter or when attempting to control longeperiod orbits.

As g increases from zero to the lower boundary of tdomain of control (g;3.14), the system goes from the chotic state to the desired period-1 UPO through an inveperiod-doubling cascade. Figure 21~a! compares the returnmap of the system without ETDAS feedback~light collectionof points! and with feedback~dark collection of points indi-cated by arrows! for g52.5. It is seen that the system witfeedback displays period-2 behavior that is not part oforiginal chaotic attractor since the points do not fall on tchaotic return map. The error signalVdp(t) is large in thiscase.

We find that the system displays quasi-periodic behavwhen the feedback gain is larger than the higher boundarthe domain of control (g ;16.9). Figure 21~b! shows thereturn maps of the system with and without feedbackg517.5 where it is seen that the points fall on an oval cetered on the location of the period-1 fixed point. The tempral evolution ofV(t) displays oscillations around 10.1 MHwhose amplitude is modulated by low-frequency (;2 MHz!oscillations. For large values ofg, the low-frequency oscil-lations become more pronounced and the oval-shaped remap increases in size.

A more sensitive technique for investigating the dynaics close to the boundary of the domain of control isperform a spectral analysis ofV(t). Figure 22~a! shows thespectrum of the signalV( f ) for g53.1, just outside thelower boundary of the domain of control. Clearly evident aspectral features at multiples of 5 MHz~the subharmonic of

FIG. 21. First return maps illustrating the dynamics of the system outsidthe domain of control.~a! The system displays period-2 behavior~shown bythe two dark sets of points indicated by arrows! when the feedback gain istoo weak (g52.5). These dots do not fall on the attractor of the unperturbsystem~lighter dots!, indicating the stabilized orbit is not a true UPO of thoriginal system.~b! The system undergoes quasi-periodic dynamics~whichappear as the dark oval centered on the period-1 fixed point! wheng is toolarge (g517.5). System parameters areV051.9 V andR50.

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the fundamental frequency! indicating period-2 behavior.Note that subharmonic peaks are.30 dB below the peaks ofthe fundamental frequency. Figure 22~b! shows the spectrumfor g517.2, just above the upper boundary of the domaincontrol. Sidebands about the fundamental spectral featudisplaced by;2 MHz appear in the spectrum, indicating thlow-frequency modulation of the fundamental frequencNote that these frequencies appear where the ETDAS feback is less sensitive~recall the form of the transmissionfunction of the ETDAS circuitry shown in Fig. 3!. Hence,increasingR tends to suppress the growth of these frequenand enlarge the domain of control.

VI. CONCLUSIONS

Continuous feedback is a technique well-suited to cotrolling fast systems. Rather than generating a feedbackattempts to synchronize the system to an ideal referestate, we use in this experiment the ETDAS technique tcontinuously synchronizes the system to many cycles ofown past behavior. We stabilize period-1 and period-4 orbof a diode resonator driven at 10.1 MHz, and demonstrthat the feedback signal becomes on the order of the nolevel in the system when control is successful.

We show that control of the system is possible in preously uncontrollable areas of parameter space by increathe parameterR, which corresponds to using more information from further in the past. IncreasingR also broadens therange of feedback gains that effect control for a given dristrength. We also consider the time-lagt l on the effective-ness of the control, and demonstrate that control is possat large time lags, particularly for larger values ofR.

Given the optical analog of ETDAS, the next step in thwork may be control in fast optical systems. A Fabry-Pecavity may be able to stabilize the GHz frequency instabties in semiconductor lasers subjected to weak streflections.8,34,35,37Also, a version of ETDAS may be usefufor controlling unstable steady states in dynamical system

FIG. 22. Power spectra ofV(t) outside the domain of control for theperiod-1 UPO.~a! The spectrum has features at multiples of 5 MHz, indcating period-2 dynamics when the feedback gain is too weak (g53.1). ~b!The spectrum displays sidebands displaced by 2 MHz, indicating lofrequency modulation of the fundamental frequency when the feedbackis too strong (g517.2). System parameters areV051.9 V andR50.

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ACKNOWLEDGMENTS

DWS and DJG gratefully acknowledge financial suppfrom the U.S. Air Force Phillips Laboratory under ContraNo. F29601-95-K-0058, the U.S. Army Research Office uder Grant No. DAAH04-95-1-0529, and the National Sence Foundation under Grant No. PHY-9357234. MEB aJEES gratefully acknowledge financial support from the Ntional Science Foundation under Grant No. DMR-94-124

APPENDIX A: A MODEL OF THE DIODE RESONATOR

To compare our experimental findings with theory, wdescribe the dynamics of the diode resonator by a secoupled nonlinear differential equations delineated by Neell et al.57 This model is well suited for rectifier diodes, sucas the type 1N4007 in our resonator. Other types of dioare more accurately described by different models.50–53

The model treats the real circuit as a collection of idecomponents as shown schematically in Fig. 23. The resiRs and the inductorL accounts for the total reactance aninductance of the circuit, respectively. The rectifier dioconsists of the parallel combination of: An ideal diode whovoltage-to-current characteristics is described by the Sholey formula

I d5I s@exp~eVd /akBT!21#, ~A1!

where Vd is the voltage drop across the diode,I s is thereverse-bias current,e/kBT is the thermal voltage, andaaccounts for carrier recombination in the depletion zonevoltage-dependent capacitorCt accounting for the junctioncapacitance whose form is given by

-in

FIG. 23. Real and idealized diode resonator. The rectifier diode is conered to be made up of an ideal diode~described by the Shockley formula! inparallel with two voltage-dependent capacitances.

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Ct5H Cb~12Vd /VJ!2b Vd,VJ/2

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whereCb is the zero-bias junction capacitance,VJ50.6 V isthe junction potential,b quantifies the variation of the doping concentration across the junction, andb15 (12b)/2 andb2522(b11) ensures thatCt is continuous; and a voltagedependent capacitorCd accounting for the diffusion capactance, arising from the finite response time of carriers tchanging field, whose form is given by

Cd5C0exp~eVd /akBT!, ~A3!

whereC0 is the zero-bias diffusion capacitance.Using standard circuit analysis, the differential equatio

describing the diode resonator are given by

dVd

dt5~ I 2I d!/~Cd1Ct!,

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dt5@2Vd2IRs1V0~ t !#/L,

where I is the current flowing through the diode resonatV0(t)5V0sinvt with v52p(10.13107) 1/s under normaloperating conditions, andL525 mH is the measured lowfrequency inductance of the inductor.

We determine the values of the parameters by measuseveral of the circuit characteristics. The parameters ofideal diode are found by measuring the current flowthrough the real diode~we remove it from the diode resonator for this measurement! as a function of an applied dvoltage. Reasonable agreement between the measuredpredicted behavior is found withI s58.8 nA anda51.79.The ac characteristics of the diode resonator are determby measuring its response to a weak sinusoidal modulawith a dc offset. Specifically, we measureI (t) when theresonator is driven by a voltageV0(t)5Vdc1dVsinvt as afunction ofv for 24 V<Vdc<0.5 V anddV.890mV. Thefrequency response of the resonator displays a resonancto the combined action of the inductor, capacitors, and retors. The remaining circuit parameters are determined byting the predicted form of the resonance curve@Eq. ~A4!# tothe observed behavior. We find that the parameters chasomewhat over the range ofVdc which may be a result osaturation of the inductor as the higher currents, an effectaccounted for in our model. The best overall fit yielRs5245 V, b50.33,C050.2 pF, andCb517 pF.

1F. J. Romeiras, C. Grebogi, E. Ott, and W. P. Dayawansa, Physica D58,165 ~1992!.

2E. Ott and M. Spano, Phys. Today48, 34 ~1995!.3In the nomenclature of modern control engineering, a control protocolstabilizes a system about a state that is inherently part of the systecalled a regulator control scheme.

4A system ergodically explores the neighborhoods of the infinite numbeunstable states when it is chaotic. Targeting techniques can be usdirect the system to the desired state.

5K. Petermann, IEEE J. Sel. Top. Quantum Electron.1, 480 ~1995!, andreferences therein.

6G. H. M. van Tartwijk, A. M. Levine, and D. Lenstra, IEEE J. Sel. ToQuantum Electron.1, 466 ~1995!, and references therein.

Chaos, Vol. 7,

a

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,

nge

and

edn

dues-t-

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7Y. Liu, N. Kikuchi, and J. Ohtsubo, Phys. Rev. E51, 2697 ~1995!, andreferences therein.

8J. E. S. Socolar, D. W. Sukow, and D. J. Gauthier, Phys. Rev. E50, 3245~1994!.

9K. Pyragas, Phys. Lett. A170, 421 ~1992!.10M. E. Bleich and J. E. S. Socolar, Phys. Lett. A210, 87 ~1996!.11M. Ciofini, R. Meucci, and F. T. Arecchi, Phys. Rev. E52, 94 ~1995!, and

references therein.12R. Mettin and T. Kurz, Phys. Lett. A206, 331 ~1995!, and references

therein.13R. Mettin, A. Hubler, and A. Scheeline, Phys. Rev. E51, 4065~1995!.14E. Ott, C. Grebogi, and J. A. Yorke, Phys. Rev. Lett.64, 1196~1190!.15Typically, Modern Control Engineers use a different, but equivale

method for expressing the perturbations. For example, it is commofind the perturbations expressed asdpi5pi2 p52KT@zi2z* ( p)# whereKT is a 13m feedback matrix~Ref. 16!.

16K. Ogatga,Modern Control Engineering, 2nd Ed.~Prentice–Hall, Engle-wood Cliffs, NJ, 1990!.

17W. L. Ditto, S. N. Rauseo, and M. L. Spano, Phys. Rev. Lett.65, 3211~1990!.

18E. R. Hunt, Phys. Rev. Lett.67, 1953~1991!.19B. Peng, V. Petrov, and K. Showalter, J. Phys. C95, 4957~1991!.20T. W. Carr and I. B. Schwartz, Phys. Rev. E50, 3410~1995!.21T. W. Carr and I. B. Schwartz, Phys. Rev. E51, 5109~1995!.22Z. Qu, G. Hu, and B. Ma, Phys. Lett. A178, 265 ~1993!.23S. Bielawski , M. Bouazaoui, D. Derozier, and P. Glorieux, Phys. Rev

47, 3276~1993!.24G. A. Johnson and E. R. Hunt, IEEE Trans. Circuits Syst.40, 833~1993!.25K. Pyragas and T. Tamas˘evicius, Phys. Lett. A180, 99 ~1993!.26D. J. Gauthier, D. W. Sukow, H. M. Concannon, and J. E. S. Soco

Phys. Rev. E50, 2343~1994!.27A. Namajunas, K Pyragas, A. Tamas˘evicius, Phys. Lett. A204, 255

~1995!.28S. Bielawski, D. Derozier, and P. Glorieux, Phys. Rev. E49, R971~1994!.29Th. Pierre, G. Bonhomme, and A. Atipo, Phys. Rev. Lett.76, 2290~1996!.30T. Hikihara and T. Kawagoshi, Phys. Lett. A211, 29 ~1996!.31M. Ye, D. W. Peterman, and P. E. Wigen, Phys. Lett. A203, 23 ~1995!.32E. Scholl and K. Pyragas, Europhys. Lett.24, 159 ~1993!.33C. Lourenco and A. Babloyantz, Neural Comput.6, 1141~1994!.34W. Lu and R. G. Harrison, Opt. Commun.109, 457 ~1994!.35C. Simmendinger and O. Hess, Phys. Lett. A216, 97 ~1996!.36R. Martin, A. J. Scroggie, G.-L. Oppo, and W. J. Firth, Phys. Rev. Le

77, 4007~1996!.37K. Pyragas, Phys. Lett. A206, 323 ~1995!.38A map-based version of ETDAS was proposed independently by E

Abed, H. O. Wang, and R. C. Chen, Physica D70, 154 ~1994!.39W. Just, T. Bernard, M. Ostheimer, E. Reibold, and H. Benner, Phys. R

Lett. 78, 203 ~1997!.40M. E. Bleich, D. Hochheiser, J. V. Moloney, and J. E. S. Socolar, Ph

Rev. E55, 2119~1997!.41M. Born and E. Wolf,Principles of Optics, 6th ed.~Pergamon, New York,

1980!, Sec. 7.6.1.42Analog Devices, One Technology Way, P.O. Box 9106, Norwood, M

sachusetts 02062-9106.43Press-n-Peel film #100PNPR, Techniks, Inc., P.O. Box 463, Ringoes,

Jersey 08551.44Andrew Corporation, 10500 W. 153rd Street, Orland Park, Illinois 60445Hewlett-Packard, 3000 Hanover Street, Palo Alto, California 94304-1146Kay Elemetrics Corp., 12 Maple Avenue, P.O. Box 2025, Pine Bro

New Jersey 07058-2025.47J. Perez and C. Jeffries, Phys. Lett. A92, 82 ~1982!.48C. Jeffries and C. Perez, Phys. Rev. A26, 2117~1982!.49R. Hilborn, Phys. Rev. A31, 378 ~1985!.50P. S. Linsay, Phys. Rev. Lett.47, 1349~1981!.51J. Testa, J. Perez, and C. Jeffries, Phys. Rev. Lett.48, 714 ~1982!.52R. W. Rollins and E. R. Hunt, Phys. Rev. Lett.49, 1295~1982!.53E. R. Hunt and R. W. Rollins, Phys. Rev. A29, 1000~1984!.54Z. Su, R. W. Rollins, and E. R. Hunt, Phys. Rev. A40, 2698~1989!.55Tektronix, Inc., 26600 SW Parkway, P.O. Box 1000, Wilsonville, Oreg

97070-1000.56A. Chang, G. M. Hall, J. R. Gardner, and D. J. Gauthier~preprint!.57T. C. Newell, P. M. Alsing, A. Gavrielides, and V. Kovanis, Phys. Rev.

51, 2963~1995!.

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