［Reports of the Faculty of Engineering at Yamanashi University， No．451994．12］
Original Paper
Observation of Quasi-Periodic Chaos in He Glow Discharge
in a Narrow Discharge Tube
TetsuyaAKITSU EijiOGAWA
Ab8tract Characteristics of chaotic behavior of ionization instability−the onset for the spectral broadening， the wave form non−linearities， the correlation dimension−are studied ia tem− perature stabilized， narrow discharge tube with internal diameter 3．0 mm in He． A chaotic state of ionization instability was observed when the primary oscillation at f、 was perturbed by the onset of secondary oscillations， a low frequency peak at f2 and side bands at f1−f2 and f1十f2 where， in a typical case， f1＝282 kHz， f2＝77．5±0．5kHz and f1／f2＝3．66． 1 The chaos and it’s application are attracting new lights of attentions in varieties of complex systems such as standing wave laser system 1）and the controlled thermonuclear fusion 2）． Ionization insta− bility is a dispersive unstable oscillations connected with creation and disappearance of the charged particles in weakly ionized plasmas． The spatial variations of the electron energy can be observed as the spatial modulation of the light emission。 Thus， ionization instability is one of good experimental resources for the studies on the irregularity in dispersive media． One of early researches on chaotic phenomena is related to the electrical discharges in aneon bulb by B．L， Van der Pol and J， Van der Mark3）in 1927． The characteristics of the forced relaxation oscillation in the Ne bulb circuit was studied in more detail by K． Masutani， K． Ha− shimoto and A． Yoshikawa4）in 1990． In the electri− cal discharges in small Ne bulbs， however， the ＊Dept． Electrical Engineering＆Computer Science， Yamanashi University． a）Present address：Hitachi Ltd． Kokubu Works， power System Control＆ Protection Dept，1・1 Kokubu−cho 1−chome， Hitachi city， Ibaragi Pref．316Japan
distance between electrodes is insufficient for the growth of ionization instabilities． P．Y． Cheung and A． Y． Wong5）showed a chaotic behavior， a period doubling excited with periodic emission of primary electrons in a plasma source． In1988，N．Ohno， M．Tanaka， A，Komori and Y．
Kawai6）showed a chaotic behavior of current carry− ing plasma in a multicusp plasma source． M． Kono， H．Nakashima and A． Komori7）showed the cascad− ed bifurcation to chaos in a current carrying ion sheath excited with external periodic oscillation． J． Qin， L． Wang， D． P． Yuan， P． Gao and B． Z． Zhang8） reported another example for the self・excited chaos in a plasma source excited with thermionic cathode． In their work， they pointed out the period−doubling qnd intermittent chaos． And J． Qin and L． Wang9） showed the fractal dimension as a quantitative measure of the chaotic state， Although similar experiments were carried out in numbers of beam driven plasma and voltage applied ion sheath， the malority of res’ults are not directly related to the ionization instability． In these works in almoSt co1− Iisionless plasmas， the chaotic phenomena are con− nected to the rnotion of charged particles in theP・tenti・l wel1・ 一
December 1994 Reports of the Faculty of Engineering， Yamanashi University No．45 In 1987， T． Braun， J． A． Lisboa and R E． Franckelo）reported the experimental observation of the cascaded period doubling in electrical dis− charges in He spectral tubes and proposed an expla− nation on the basis of the non−linearity of the ioniza− tion term， the current voltage characteristics of the cathode and unknown periodicity of the electrical discharge． Unfortunately， the quantitative measure of the chaos， such as the fractal dimension， was not described in their original report． In 1991， C． Wilke， R．W． Leven and H． Deutsch11）reported the experi− mental observation of the sequences of periodic， quasiperiodic and chaotic state of externally forced periodic oscillation in positive column of He dis− charges． In this work， the quasi−periodic chaos was excited with an external driving oscillation and Wilke explained his result on the basis of the non− linear field dependence of the electron drift velocity・ The experimental result can be classified into three types． The first one is the intermittent chaos， the second is the cascaded period doubling and the third is the quasi−periodic chaos． In the present paper we describe a novel discovery of a quasi− periodic chaos of ionization instabilities in He glow discharges：the spectrum of a coherent oscillation at f、 is perturbed by an onset of secondary oscilla・ tions at f2 and side bands of f1−f2 and f1十f2， in that almost infinite sequence is generated depending on f、／f，． Figure 1（a）shows the main part of the experimen− tal apparatus． The narrow part of the discharge tube was made of Pyrex，3mm in internal diameter and 93 mm in length， whereas the wall temperature was maintained at room temperature with water circulation． The narrow part of the discharge tube was designed to be identical with a He spectral tube that showed a chaotic oscillation． The discharge gas
was He with 99．9995％grade and the absolute
pressure was measured with Baratron−122A． The discharge electrodes were cut out from 99．99％Ni with a hollow shape，4．2 mm internal diameter and 15mm in length． The electrical discharge was energized with a high voltage power supply， maximum 2 kV，45 mA． The discharge current was l regulated with a series resistor，22．4kΩ． To measure the discharge current，TO
Figure l Experimental setup （a） Discharge tube，WATER
＼ OPTICAL ＼ ！！`
（b）Data processing system． H．V．ANODE
R Os〔：illoscopc HP5460UA 100MHz 2cliunnel Persnal Computer PC−286VG 8028G aresistor of 1．00×103Ωwas inserted into the dis・ charge circuit． Figure 1（b）shows a schematic diagram for the data acquisition and processing system． In the fol− lowing part， the AC signal of the discharge current was recorded at O．05 ＃s sampling rate with a digitaloscilloscope，．HEWLETT PACKARD， Model
HP54600A． The experimental data was collected with a personal computer in the laboratory site and the analyses of experimental data were carried out with remote work stations connected through the LAN（Local Area Network）． The spectral analysis was carried out using theMEM（the maximum entropy method）， the Burg
algorithm12）．The fractal dimension was determinedwith the method proposed by Grassberger and
Procaccia13）that furnishes the most practical
method for the approximation of the dimensionalityof the attractor measure：the dimension〃was
Observation of Quasi・Periodic Chaos in He Glow Discharge in a Narrow Discharge Tube Xn十△n
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v、 2000 0 500 1000 1500 2000 0 500 1000 1500 2000 Frequency［kHzl Frequency【kHz］ 26 28 30 32 Current［mA］ 34 36 Figure 2 Characteristics of the ionization instabilities in a sealed He tube． Phase space trajectory， spectrum， the correユation dimension and the amplitude of the perturbation of the discharge current for different discharge current． （a） Nearly sinusoidal oscillation， （b）Matured periodic oscillation， ． （c） Onset of the side band and the spectral broadening， （d）Matured chaotic state， Dc：Correlation dimension．December 1994 Reports of the Faculty of Engineering， Yamanashi University No． 45 determined from the asymptotic behavior of the integral correlation function， c（r2， ・（r）一融嘉、ΣH（r−llx・−x・ll）・ i，」＝1
wh・・e膚ll−max｛レ1−、、1・＜」＜∋
dis the embedding dimension． H is the Heavisidefunction：H（z）＝1for z≧Oand H（z）＝Ofor z＜
0． Although this algorithm has been proven to be useful in the calculation of the fractal dimension for known attractors（Henon， Lorentz）as well as in the experimental determination in hydrodynamic turbu− lence such as the RayleighBernard convectionl4）， N2 calculations are needed for N sampled points， resulting in large numbers of the total calculation． Thus， in the present work， L sample points were randomly chosen and the amount of the total calcu− lation was reduced to L×N， where N＝2000 and L＝100． Figure 2 shows variation of the correlation dimen− sion and the amplitude of the perturbed discharge current in He glow discharge as functions of the average discharge current． The discharge tube in Fig．1was replaced with a sealed He spectral tube． After the breakdown， the anode voltage was rapidly decreased to the lower limit of the stationary glow discharge， then gradually increased． In the insert of Fig，2，0ne can find typical examples for the two− dimensional phase space trajectory and the spec− trum corresponding to discharge states（a）一（d）．When the average discharge current became
greater than the lower limit for the growth of ion・ ization instabilities，26．2 mA， nearly sinusoidal mod− ulation appeared in the discharge current as well as in the optical emission from the plasma column． Thespectrum was monochromatic and 2000 sample
points converged into the trajectory of xn vs． x n＋△n plot［see Fig．2（a）］In the present result， one cycle of the trajectory was made up with 66 sample points and△n・一・ 4．With increasing the anode potential， the average discharge current increased and the oscil．lation at fl gradually increased in its ampli− tude． In the spectrum shown in Fig．2（b）， one can find higher harmonics at f＝nf1， where n is integer （2，3）．It should be noted that the correlation dimen・ sionレremained constant（1．0±0．5）during the peri− odic oscillation stage．When the average discharge current became
greater than a critical value， the secondary oscilla− tions appeared in the frequency spectrum． The tra− jectory of xn vS． x。＋△n plot became SCattered in the phase space as in Fig．2（c）． When the anode poten− tial was further increased， the wave form becamemoreとomplex．［See Fig．2（d）］The correlation
dimension〃increased to 1．7±0．5． The relation between the primary and secondary oscillations can be found in typical examples for the spectrum shown in Fig．3． The spectrum was mea−sured with a spectrum analyzer， ADVANTEST，
Model R3261A． In Fig．3（a）， one can observe a peak at f1＝297 kHz． The second and the third harmonics appeared at 590 kHz（1．98f1）and 888 kHz（2．98f1） respectively． In Fig．3（b）， side peaks appeared at Figure 3 Typical spectrum for periodic， quasiperiodic and chaotic states． （a）Appearance of higher harmonics， Frequency（rela− tive intensity）， P、：297 kHz（282．6μV）． P2：590 kHz （61．87μV），P3：888 kHz（16．13μV）． DC discharge current， Id＝29 mA． （b）Onset of side peaks， SP1：f1−f2＝241 kHz（89．49 μV），SP2 i f1十f2＝328 kHz（90．81μV）， where P、：f、＝ 284kHz（423μV）， f2＝43．5±0．5kHz（not observed） and f1／f2＝6．6． P2：562 kHz（118．8、μV）． at Id＝31．65 mA．怠
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’ 一 0 2△f 5△f7△f 9△f FRE（）UENCY（MHz） 1 （c）Matured side peaks near the onset of the chaotic state， SP1：f1−f2＝205 kHz（203μV）， SP2：f1ヰf， ＝＝ 360 kHz（179．8μV）， where Pl：f1＝282 kHz（316．6μV）， f2＝77．5±0．5kHz ＝80 kHz（158．2μV， the low fre− quency peak）and f1／f2＝3．66． at Id＝32 mA． （d）Chaotic state a．t Id＝33．89 mA （e），（f）Periodic windows at Id＝33．90 mA （W1）and Id＝35．26 mA（W2）， respectively． discharge current became more complex． And the spectrum became broader and fluctuated［Fig．3（d）］． Figure 3（e）一（f）shows typical examples for the spectrurn observed in periodic windows in the cha− otic state． The oscillation of the discharge currentbecame periodic and the correlation dimension
slightly decreased corresponding to each windows， as shown in Fig．2． The stable limit for the average discharge current varied for individual discharge tubes． Figure 4（a）shows a typical example for the chaotic oscillation and in Fig．4（b）one can find the Figure 4 Chaotic oscillation and the asymptotic behavior of the integral correlation function， Cω． （a）Chaotic wave form． （b） The integral correlation function， Cω， and corre− lation dimensionりvs embedding dimension． 0．8 0，G O．4 0．2 1ttt・N1 ｛〕 会〔｝．2 −o．1 ．O、G −0．8 −50 ±40 −30 −20 100000 ユoに ロ 0（り 1〔｝00 100a
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f1−f2 and f1十f2， where fl＝284 kHz， f2＝43．5±0．5 kHz and f1／f2＝6．6． In Fig．3（c）， the amplitude of the side band became greater and one can clearly firld the low frequency peak at 80 kHz（re f2）and side peaks at f1−f2 and f1十f2 where f1＝282 kHz， f2＝77．5±0．5kHz and f1／f2＝3．66． At slightly higher discharge current， the wave form of the asyrnptotic behavior of the integral correlation function and the correlation dimension． The correla− tion dimension was determined from the gradient of εωr。cγv／in the logarithmic plot at夕＝40 to 200 for the embedding dimensions unto 6．15）Although the saturation of the correlation dimension is incom一December 1994 Reports of the Faculty of Engineering， Yamanashi University No．45 plete for the chaotic state， the correlation dimension increased up to 1．7， as shown in the insert of Fig．4 （b）．The accuracy of the present estimation is lim− ited by the length of sample points． In Malraison’s work14）15，000 sample points were used for the estimation of the correlation dimension in chaotic states． Thus， probably longer sequential sampling with higher resolution should be necessary for fur・ ther improvements of the estimation of the fractal dimension． Figure 5 shows the pressure dependence of the discharge current limits and the classification of the discharge characteristics． At pressures in the hat− ched region， unstable self−excited oscillations were observed that had lost the periodicity and fluctuated in very irregular’manner． The open circles near the horizontal axis indicate the lower current limit for the stationary glow discharge． The smaller open circles indicate the current limit for the onset of the sinusoidal oscillation． The discharge current limit varied depending on the conditioning of the dis− charge tube and the broken line indicates the lowest limit． Between the hatched region and the stable region， self−excited ionization instabilities were observed with a periodic motion along the whole narrow part of the discharge tube． The quasi− periodic chaos was observed at the boundary of the hatched indicated with triangles． The phase velocity was measured in experi− mental conditions indicated with filled circles in Fig．5（a）． The relative phase shift of the optical emission was measured with two optical fibers displaced along the plasma column． Figure 5（b）、 shows a typical example for the phase shift as a function of the axial distance． The・frequency of the ionization instability was 267．4 kHz． The wave length was 8 mm． The phase velocity of the ioniza− tion instability was 2．3×105 cm／s directed towards the cathode． The voltage drop at the narrow part was determined from the voltage vs． current charac− teristics of discharge tubes． The reduced strength of the electric field per unit Torr， per unit length was estimated as 12．0±0．25 V／ cm Torr． The potential difference per one wave length was 24．5±0．5V／λ． In He discharges， this result is slightly higher than the first excited poten一 Figure 5 Current limits and the classification of dis− charge states． （a）He pressure dependence of discharge current limits． （b）Relative phase shift of the optical emission． 1【mA］ ψ ［rαd］ 35 30 25 20 15 10 5 0 2π 巨 3 史 3 π ≧ 3 1 3 0 He，φ＝3inm，．乙＝93mm 1
a
1．5 2 2，5 Ptessure［rTorr1 ．2a°s 3 3．5 0b
2 4 Z【mm］ 6 8 10 tial and comparable with the ionization potential （24．5eV）． From the typical frequency， phase velocity and the strength of the electric field per wave length， our result can be classified as a type of ionization instability， so℃alled r−wave．16∼22）The characteristics of r−wave have been studied， for example in Amemiya’s work in He discharges22）， and reasonable agreements have層been pointed out between the theoretical model and observed charac− teristics． Nonetheless， no description of similar quasi−periodic chaos have been mentioned． The reproducibility of the present experiment was very sensitive to the stability of gas temperature． When the discharge medium was heated up by discharge current， the slower wave（10wer frequency）was suppressed by faster waves（higher frequency）andthe chaotic state was overridden by another
sinusoidal state． The stability of the positive column was also sensitive to the selection of the materials and shape of electrodes． When electrode material was replaced with 99．99％aluminum， we could notObservation of Quasi−Periodic Chaos in He Glow Discharge in a Narrow Discharge Tube establish the experimental conditions for the quasi −periodic state． This result indicates that the secon− dary electron emission coefficient of the cathode surface is another important factor for the re− producibility of the experirnental result． In conclusion， chaotic state of ionization instabil− ity has been studied in He discharge in temperature stabilized small diameter tubes． A primary mode of the ionization instability， r・wave， was perturbed with another wave． then chaotic state was generat− ed through the quasi−periodic state． The experimen− tal conditions have been established for the repro− duction of the chaotic state． Finally， ionization in− stabilities show varieties of discharge phenomena that are sometimes difficult to explain in simplified models． On the other hand， this discharge phenom− ena can be one of the most interesting research materials to study on non−linear phenomena in dispersi寸e media．
