VOLUME 72, NUMBER 1 PH YSICAL REVIEW LETTERS 3 JANUARY 1994

Controlling Chaos in a Discharge Plasma

W. X. Ding, ' H. Q. She, W. Huang, and C. X. Yu'Chinese Center of Advanced Science and Technology (World Laboratory), P 0 .Bo.x 8730

Beijing l00080. People's Republic of ChinaDepartment of Modern Physics. Unit ersity of Science and Technology of ChinaH, efei, 230026, People's Republic of China

(Received 26 May 1993)

We have investigated experimentally controlling chaos in a discharge plasma. The power spectra, in-forrnation dimension, and especially the Lyapunov exponent spectra confirm that a small periodic modu-lation on the discharge voltage can be effectively used to control chaos in a discharge plasma system.

PACS numbers: 52.35.Ra, 05.45.+b, 52.80.-s

Nonlinear dynamics phenomena are abundant in na-ture and laboratory plasma. A rich variety of behaviorhas been observed in laboratory plasma, including self-oscillation, periodic doubling bifurcation, intermittency,quasiperiodicity, and chaos [1-4]. However, in manyreal physical systems, chaos is an undesirable phenom-enon which can have harmful consequences; in particular,chaos in magnetic confined plasma may evolve into fullydeveloped turbulence and lead to the anomalous energyand particle cross-field transport. Therefore, the abilityto control chaos is of much practical importance. Since anew method of controlling chaotic dynamical systems was

proposed by Ott, Grebogi, and York (OGY) [5], control-ling chaos has been a subject of intense research. It was

recently demonstrated in several experiments thatdynamical control of chaos can be achieved by the OGYmethod [6-9]. The method is based on the fact thatthere exist an infinite number of unstable periodic orbitsembedded in the chaotic attractor, and that only small,judiciously selected perturbation to an available systemparameter is needed to stabilize some of these. Ott, Gre-bogi, and York have given a prescription for finding theform of time-dependent parametric perturbation and am-plitude p* necessary to control chaos. While the methodhas been demonstrated to be efficient in the application tosome systems, one needs a fast responding feedback sys-tem that produces an external force in response to thesystem's dynamics, because all values needed to achievecontrol have to be calculated from an experimental signalwith the embedding technique [10]. In practice the formand amplitude of parametric perturbation can be deter-mined numerically and experimentally by directly observ-

ing the behavior of chaotic system response to the select-ed parametric modulation. This method has been suc-cessfully applied to the periodically driven pendulum[11,12] where chaos in a dynamics system can besuppressed by introducing small parametric perturbationto a system constraint, and to an experimental system ofmicrowave-pumped spin-wave instability [13]. Thenonfeedback aspects of this control add considerable flex-ibility to the present control options. In this Letter, we

would like to present the experiments on controllingchaos in an undriven plasma by a small periodic modula-tion of discharge voltage. The power spectrum, the infor-

mation dimension, and the Lyapunov exponent spectrumconfirm clearly that the chaos in the undriven plasma hasbeen suppressed when the modulation frequency and am-plitude are carefully chosen. Although the dynamicalcontrol of chaos has been demonstrated in other physicalsystems, to the best of our knowledge, this is the first timethat such control has been achieved in a plasma.

The experiment has been performed in an unmagnet-ized steady-state plasma device [4] which consists of anelectron-emitting cathode (six parallel tungsten fila-ments) at one end and a current-collecting anode (a rec-tangular stainless steel plate) at the other end, as shown

in Fig. 1. The cathode is connected to the chamber wall

and is electrically grounded. The anode is mounted onthe shaft of a probe so the distance between the cathodeand anode can be varied. The argon plasma is producedby a dc discharge. The discharge is controlled by argonpressure (P, ), filament current (If), discharge voltage(VD), and the distance between the cathode and theanode (d). The typical parameters of the steady-stateplasma measured by a Langmuir probe are electron den-sity n, =10 -10 cm, electron temperature T, =1-3 eV,and ion temperature T; ((T, . The electron saturationcurrent signal 1„(t)of the probe which is proportional tothe electron density and the discharge current ID isrecorded by digitizers (8 bit, N =8192 data points, sam-pling interval ht =16 ps). Power spectra, information di-mensions, and Lyapunov exponents calculated from the

Vg

PLASMA

Vo

FIG. 1. Schematic of experimental setup and dischargecurrent-voltage characteristic curve.

96 0031-9007/93/72(1)/96(4) $06.001993 The American Physical Society

VOLUME 72, NUMBER 1 PHYSICAL REVIEW LETTERS 3 JANUARY 1994

time series signals are used to detect periodic and chaoticmotion.

We have previously studied this discharge plasma ex-tensively and found that the nonlinear dynamical phe-nomena are always associated with the occurrence of hys-teresis in the current-voltage (I-V) characteristic of thedischarge [41. Moreover, the chaotic behavior alwaysoccurs in the region near the turning point on either thelower or upper branch of the I-V curve, where the plasmapotential is negative with respect to the anode. This isthe regime where the plasma can be driven to chaos byvarying some of the discharge parameters (P„Iy, Vn, d).Because the chaotic state on the lower branch can bemaintained over a rather wide range of the discharge pa-rameters, in the experiments presented here we controlthe plasma state to approach the region near the turningpoint B on the lower branch by adjusting Vn, while keep-ing all other system parameters constant. For the givendischarge parameters P, 1.0X10 torr, Iy=26.0 A,and d 9.0 cm, when Vn is increased above a thresholdvalue of 12 V, the discharge develops a low-frequency (afew kHz) self-oscillation, which can be observed in bothsignals In(t) and 1„(t). A potential relaxation instabili-

ty, which has been observed experimentally in the posi-tively biased end plate Q machine [14], is suggested to beresponsible for the self-oscillation [15). The fundamentalfrequency of self-oscillation depends on the discharge pa-rameters. As a variation of pressure P„ this self-oscillation may transit to chaos through either the period-ic doubling, intermittency, or quasiperiodicity. As report-ed before, when the chaotic regime sets in not far abovethe threshold boundary, chaos is characterized by a low

dimension chaotic attractor (2 (Dz & 3) and one positiveLyapunov exponent, which represents that the plasma is

in a "weakly" chaotic regime. With the undriven plasmain chaotic operation, control was attempted by applying asmal/ modulation to one of the discharge parameterswithout creating new orbits with very difTerent propertiesfrom the existing ones. The most convenient parameterfor modulating was found to be the discharge voltage,which could be simply done by applying a sinusoidal sig-nal V sin(2' r) (V «Vn, f:0-5 kHz) (through 3:1transformer) in addition to VD. By adding a small modu-lation to VD with appropriate frequency and amplitudewe have been able to control the chaotic state. Figures2-4 show the results obtained with a sinusoidal voltagemodulation imposed after the plasma is operated in achaotic regime. Figure 2 shows that the power spectra ofID(t) evolve with increasing modulation amplitude V atthe frequency f~ 1.8 kHz matched with the characteris-tic frequency fn of the system. Before the V is appliedthe power spectrum as shown in Fig. 2(a) displays abroadband feature which is a characteristic of chaos.There is no significant variation of the power spectrumuntil V =60 mV From the power spectra shown in

Figs. 2(b)-2(d), it can be seen that as V increases abovethis value the level of the broadband noise decreases pro-

0.0I

2.0I

4.0

f(xHz)

I

60 8,0

fo (b)

0.0I

2.0I

4.0

f(KHz)

I

6.0 B.O

0.0I

2.0I

4.0

f(Kuz)

I

6.0 8.0

I I

4.0 6.0 8.0

f(KHx)

FIG. 2. Power spectra with the increase of the amplitude ofthe perturbation at a given frequency f 1.8 kHz for V 0,100, 200, and 300 mV, respectively. The other discharge pa-rameters are P, 1.0X10 torr, II 26 A, VD 12 V, andd 9.0 cm. All the plots are on a log-linear scale with the samevertical axis calibration. The full dynamic range of all spectrais 80 db.

I

2.00.0

gressively and some of the sharp lines characteristic of aperiodic signal become more and more visible. The fun-damental frequency fo and its harmonic frequencyemerge from the noise background. This observation is inqualitative agreement with the numerical results obtainedin studying the effect of a small parametric perturbationon the Du5ng-Holmes equation [I lj. This is a good in-dication that "the degree of chaoticity" has been reducedand the chaotic motion has been converted to periodicmotion. In addition it should be noted that the maximum

VOLUME 72, NUMBER l PHYSICAL REVIEW LETTERS 3 JAXUARV 1994

perturbation amplitude applied to the discharge voltage is

only a few percent of VD and when the discharge voltageis increased or decreased by an increment that is thesame as the maximum V, the original chaotic attractorremains almost unchanged. On the other hand we have

applied the small perturbation to a periodic attractor andfound that the perturbation cannot alter the basic topo-logical structure of the orbits on the attractor. This factconfirms that chaos is controlled by a small periodic per-turbation of an available system parameter.

The control of chaos is further demonstrated by thevariation of information dimension [16] of the chaotic at-tractor with increasing V . The information dimension is

calculated with the embedding technique from experi-mental time series signals [10], and saturates for theembedding dimension larger than ten. Figure 3 showsthe dependence of the information dimension on themodulation amplitude V~ for a given frequency. ForV (60 mV the information dimension shows virtuallyno change (Di =2.75), characterizing a chaotic attractor.When the modulation amplitude increases above the criti-cal value V 60 mV, the information dimension de-creases toward D i 1 with increasing V, characteristicof a periodic trajectory. This result further confirms thatthe chaotic motion in a discharge plasma can be convert-ed to periodic motion by a small periodic modulation ofthe discharge voltage. It is worth mentioning that in theexperiments we have examined the effects of the modula-tion phase on controlling the system. It has been foundthat all the observed phenomena are independent of theinitial phase of the parameter modulation, which agreeswith the numerical results obtained in studying controlchaotic dynamics with weak periodic perturbation [121.This is not surprising since the behavior of long-term evo-

lution of the chaotic plasma system is determined by amotion on the attractor, where the unstable periodic or-bits are dense in a typical chaotic attractor. Thereforewhen the modulation frequency and amplitude are select-ed appropriately, the unstable orbit becomes phase lockedto the external parameter modulation after a time delayrelated to the applied modulation. It is suggested that the

2.0I

1.5

1,0

0.5

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-0.50.0 1.0 2.0

1.5

10

0.5

0.0

-0.50.0 1.0

1

2.0

(b)I

30

system has the ability to find an appropriate periodic or-bit for a given parameter perturbation.

To attain further insight into how the system respondsto the periodic perturbation, we performed a series of ex-periments where the measurements were carried out by

scanning the modulation frequency f at constant V

and recording the Iluctuation signals of ID(t) by a dataacquisition system. The Lyapunov exponent spectrum is

calculated from the experimental time series signal ID(r)by a practical algorithm [4,17-19] in order to character-ize the behavior quantitatively. Once the exponents aredetermined, the Kolmogorov entropy can be obtainedeasily by summing all of the positive Lyapunov ex-ponents. For the low dimension chaotic attractor report-ed here, the Kolmogorov entropy is the same as the lead-

ing Lyapunov exponent because of only one positive ex-ponent that exists in the chaotic attractor. The resultsare depicted in Fig. 4 where we give the leading

Lyapunov exponents ki as a function of P (P f /fo) forseveral values of V . For V~ 100 mV, which is not farabove the critical value V~, the leading exponent li is re-duced significantly in some narrow ranges of P near therational values (e.g., P z, 1, and 2), and less sig-

1.5

1.0

0.5

0.0

—0.50.0

I

1.0 20

{c}

I I I I I I I 1 I i I I I I I I I10 100 200 300 400

V (mV)

FIG. 3. The information dimension D~ vs voltage modulation

amplitude for f 1.8 kHz. The other discharge parametersare the same as in Fig. 2.

pFIG. 4. The leading Lyapunov exponents as the function of P

(P f /fp) for V -IOO, 200, and 300 mV, respectively. Theother discharge parameters are the same as in Fig. 2. X,] 1.SO

for V~ =0. The Lyapunov exponents presented are normalized

by the fundamental frequency fo The points repres. ent experi-mental results and they are connected by lines to guide the eye.

VOLUME 72, NUMBER 1 PH YSICAL REVIEW LETTERS 3 JANUARY 1994

nificantly as P is beyond these ranges, as shown in Fig.4(a). Here the curve of leading Lyapunov exponents as

the function of P has the appearance of "tongues" similarto those observed in phase-locked phenomena [13].When V is increased further to 200 mV, it can be seen

from Fig. 4(b) that the tongues have become wider, and

the values of Xf have vanished in the vicinity of the p= 2,1, and 2. Moreover, at these levels of control signal mag-

nitude, the values of A, 2 have become negative aroundthese P values. Furthermore, a comparison between Figs.4(b) and 4(c) immediately shows that the tongues be-

come wider, and the range of frequency over which chaoscan be controlled is extended more significantly with in-

creasing V . In fact, when the phase locking is achieved

chaos can be suppressed more easily with increasingmodulation amplitude V . This implies that when an un-

stable chaotic motion becomes phase locked to an exter-nal drive force the transition from chaos to periodicitycan be realized.

The experimental results demonstrate how a relativelyweak periodic perturbation can control the chaos in the

discharge plasma. A possible mechanism explaining this

phenomenon is as follows. By applying an externalmodulation, one of the system parameters p can be modu-

lated periodically about its mean value, with periodT~=nTO/m (T~ ~f~ ', To fo '), where n and m areintegers. By adjusting the modulation amplitude ap-

propriately, the system parameter p can be periodicallyshifted such that the dynamics is forced to fall on thestable manifold of the original unstable periodic orbits,thus resulting in a controlled trajectory. Therefore, the

method used for controlling chaos in our experiments is

essentially similar to the OGY method. The differencebetween these two methods is in the procedure to find the

amplitude p* of the parameter variation necessary toachieve control of chaos. In contrast to the OGYmethod, in our experiments p is determined experimen-

tally by observing the behavior of the chaotic system's

response to the increasing modulation magnitude and

scanning the modulation frequency. In the undriven

chaotic plasma system, it may be more convenient totame the autonomously chaotic dynamics with weak

periodic perturbations. When a natural periodicitycharacteristic of the system is found and the perturbationmagnitude is chosen appropriately, one may stabilize one

of the unstable orbits in the chaotic attractor so as tosuppress chaos. Furthermore, this simple control methodhas been applied to the hyperchaotic attractor indicated

by two positive Lyapunov exponents; we have never no-

ticed tongues as mentioned above unless modulation am-

plitude is much increased so the system has been greatlychanged. It is demonstrated that the control methodpresented here can be applied to "weakly" chaotic attrac-tor.

In conclusion, we have demonstrated experimentallythat a small periodic perturbation on an accessible pa-rameter of the system can be effectively used to suppress

the undriven chaos in a dc discharge plasma systemwithout any a priori knowledge of the models or the

equations governing the dynamics. The power spectra,information dimension, and Lyapunov exponent spectraquantitatively confirm our experimental results. The re-sults reported here indicate that this control techniquemay be relevant to the controlling of plasma turbulence.

The authors would like to acknowledge discussionswith Professor Y. Gu, Professor B. H. Wang, and Profes-sor X. W. Hu. We also thank Professor E. Ott for send-

ing us the reprints. This work is supported by NationalBasic Research Project "Nonlinear Science, " Huo Ying-dong Foundation from the National Education Commit-tee, and Youth Science Foundation of the University ofScience and Technology of China.

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