Plasma and Fusion Research: Regular Articles

Volume 14, 3401070 (2019)

Spatiotemporal Structure of Ionization Waves
in a Glow Discharge Plasma∗)Takao FUKUYAMA, Kazutaka ISHIDA and Hiroki KANZAKI
Faculty of Education, Nagasaki University, 1-14 Bunkyo-machi, Nagasaki 852-8521, Japan(Received 26 December 2018 / Accepted 5 March 2019)
Spatiotemporal structures formed in ionization waves are experimentally investigated in this study. A system
involving ionization waves in a discharge tube has a few degrees of freedom in time and space. In our experiment,neon plasma is produced in a glass tube by a glow discharge between electrodes after the tube is evacuated to forma high vacuum. Spatiotemporal signals for the analysis are sampled as fluctuations in the light intensity by using
a line-scan camera and photodiodes. The largest Lyapunov exponents are calculated from the time-series data
sampled from the line-scan camera and photodiodes. Reconnection in the spatiotemporal structure is observed,
which is caused by Eckhaus instability. Topological defects in the spatiotemporal structure are observed, which
result in the appearance of spatiotemporal chaos; this leads to an order structure when an electric pulse is applied
to the system as an external force or when coupled oscillators are synchronized.
c 2019 The Japan Society of Plasma Science and Nuclear Fusion ResearchKeywords: glow discharge, ionization wave, chaos and periodicity, spatiotemporal structure, Eckhaus instability,synchronization, chimera state
DOI: 10.1585/pfr.14.3401070

1. Introduction
In recent years, studies on nonlinear dynamics, suchas chaos, fractals, and pattern formation, in dissipative systems have gained attention. Such studies are also important in plasma science because plasma is a typical medium,
which includes nonlinearity [1–5]. In this paper, we focus
on the dynamic behaviors of ionization waves [6,7] in glow
discharge plasma. Ionization waves show various kinds ofnonlinear behaviors, including chaos, as a function of thedischarge current. Furthermore, spatiotemporal structures
can be observed using a simple detector such as a photodiode and line-scan camera. Thus, ionization waves can beconsidered a suitable medium to study the universal properties of nonlinearity. The dynamics of ionization waves
has been thoroughly investigated experimentally and theoretically [8–13]. The mechanism of spatiotemporal structures in ionization waves has been discussed from theviewpoint of Eckhaus instability as well [14, 15].
In this paper, in Sec. 2, three types of experimental
configurations are described. In Sec. 3, the results obtained
from the three experiments on spatiotemporal structures inionization waves are discussed, and the obversations of thestudy are summarized in Sec. 4.

2. Experimental Setup
The configurations of the experimental setups areshown in Fig. 1. In this study, three types of experiments,

as shown in Figs. 1 (a) - (c) are performed using a glass tube
with a diameter and length of 0.02 m and 0.75 m, respectively. Neon gas is introduced into the tube at a pressure ofabout 478 Pa after evacuating the tube to a high vacuum.
In all our experiments, neon plasma is produced by glow
discharge between the electrodes when dc electric field isapplied to the electrodes. Typical electron and ion temperatures are about 10 eV and 0.025 eV, respectively. The discharge current is a typical control parameter that governs
the system.
Figure 1 (a) shows the experimental configuration using one discharge tube. Figure 1 (b) shows an experimental configuration using one discharge probe by applying
an electric pulse as an external force. Figure 1 (c) shows
an experimental configuration to study dynamical behaviors of two coupled nonlinear oscillators by using two discharge tubes. We sampled 52 data points in Figs. 1 (a) and(b) and 16 data points in Fig. 1 (c), all sampled at every1 cm interval. Light fluctuations along the central axis areobserved using a line-scan camera. The positive column inionization waves shows various kinds of dynamic behavior, including chaos and solitons [16], when varying thedischarge current as the control parameter of the system.
Herein, in most of the discharge current situations, groupvelocity propagates from the cathode to the anode as a traveling wave; in contrast, phase velocity propagates from anode to cathode, i.e., the ionization waves show features ofthe backward wave.

author’s e-mail: fukuyama-takao@nagasaki-u.ac.jp∗) This article is based on the presentation at the 27th International Toki
Conference (ITC27) & the 13th Asia Pacific Plasma Theory Conference(APPTC2018).

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c 2019 The Japan Society of Plasma
Science and Nuclear Fusion Research

Plasma and Fusion Research: Regular Articles

Volume 14, 3401070 (2019)

Fig. 2 Spatiotemporal structures: (a) typical order structure(25.8 mA) and (b) typical chaotic structure (24.8 mA) areshown.

Fig. 1 Experimental setup of three experimental configurations.
The pressure in the tubes is fixed at about 478 Pa, andthe discharge current is varied as a control parameter.
The time-series signals are obtained from fluctuations in
the light intensity by using photodiodes and a line-scancamera. (a) Experimental configuration using single discharge tubes, (b) experimental configuration using a single discharge tube by applying an electric pulse as an external force, and (c) experimental configuration to study
dynamical behaviors of two coupled nonlinear oscillators
by using two discharge tubes.

3. Results and Discussion
A system involving ionization waves in a discharge
tube has degree of freedom in time and space, and time-series signals are obtained from fluctuations in the lightintensity by using photodiodes and a line-scan camera. In
our previous works [13, 17], only time-series data for theanalysis were sampled using a photodiode. Moreover, in
this study, data corresponding to the spatial position issampled simultaneously using a line-scan camera, wheretime-series signals are obtained every 35.0 µs and the spatial data are obtained as the fluctuations in the light intensity.
First, the results corresponding to the experimentshown in Fig. 1 (a) are described. When the discharge current is varied as a control parameter, the system showsa variety of behaviors, as mentioned before. Figure 2
shows spatiotemporal structures: (a) typical order structure(25.8 mA) and (b) typical chaotic structure (24.8 mA).
We now discuss the mechanism through which spatiotemporal structure is formed in ionization waves. In a
spatiotemporal structure, reconnections are caused by slip
between two wave numbers: in the cathode side (right sideof the figure) and anode side (left side of the figure). Theseare chimera states of two waves, and it can be explainedthat these states are caused by the Eckhaus instability [15],which causes a bifurcation of the wave number.
If the wave number bifurcates in orderliness, the spatiotemporal structure shows a regular pattern and not a
chaotic one. In contrast, in the case that the wave number bifurcates in disorderliness, the spatiotemporal structure shows a chaotic pattern. In Fig. 2, the spatiotemporal

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Plasma and Fusion Research: Regular Articles

Volume 14, 3401070 (2019)

Fig. 3 Spatiotemporal structures (a) before and (b) under electric pulse application of a chaotic system, are shown. The
electric pulse has a frequency of 1.8 kHz and the discharge current is 29.0 mA. (a) Chaotic and (b) order structure is shown, respectively.

structure shows chaotic dynamics, with several topological defects [18, 19] observed in the structure (Fig. 2 (b)).
Discontinuous reconnection of two wave numbers in thepositive column of ionization waves caused topological defects.
The results corresponding to the laboratory experiment shown in Fig. 1 (b). The eﬀect of electric pulse application on the nonlinear dynamic systems is investigatedusing one discharge tube.
For chaotic analysis, Lyapunov exponents are calculated to examine the eﬀect of the electric pulse application. The Lyapunov exponents are calculated from thetime-series signals of waves sampled from the line-scancamera on a tube, based on the algorithm proposed inRef. 20 and the time-series data obtained in our experiments. The largest Lyapunov exponent shows a positive
value for chaotic oscillations; this value is higher for amore chaotic state and approaches zero for a system withperiodic oscillations.
Figure 3 shows spatiotemporal structures before(Fig. 3 (a)) and during electric pulse application (Fig. 3(b)), when the original system shows spatiotemporal chaos,i.e., under applied electric pulse to a chaotic system. The
electric pulse has a frequency of 1.8 kHz, which is almost half the fundamental frequency of the original system. Here, the discharge current as a control parameter ofthe system is 29.0 mA. The chaotic system changes to aperiodic one under this applied electric pulse.
The transformer incorporated in the circuit ofFig. 4 Spatiotemporal structures (a) before and (b) under electric pulse application of a periodic system, are shown.
The electric pulse has a frequency of 1.6 kHz and thedischarge current is 26.0 mA. (a) Order and (b) chaotic
structure are shown, respectively.

Table 1 Values of the largest Lyapunov exponents are shown for(a) pulse application to a periodic system and (b) pulse
applied to a chaotic system.

Fig. 1 (b) has a resistance that it has an eﬀect on the dynamic behaviors of discharge current values, which madeit diﬀer from one in Fig. 1 (a).
Figures 4 (a) and 4 (b) show spatiotemporal structures
before and during the electric pulse application, to a periodic system. The electric pulse has a frequency of 1.6 kHz,
which is almost half the fundamental frequency of the original system. Here, the discharge current as a control parameter of the system is 26.0 mA. In contrast to results
shown in Fig. 3, the electric pulse application, changes theperiodic system to a chaotic system.
Table 1 lists the values of the largest Lyapunov exponents for: (a) pulse applied to a periodic system, and (b)3401070-3

Plasma and Fusion Research: Regular Articles

Volume 14, 3401070 (2019)

Fig. 5 Spatiotemporal structures (a) before and (b) under coupling, when two original oscillators show spatiotemporalchaos, are shown. The discharge currents of wave 1 and2 are 15.80 mA and 24.20 mA, respectively. (a) Chaoticand (b) order structure is shown, respectively. Horizontaland vertical ranges indicate 10 cm and 0.25 ms, respectively.

pulse applied to a chaotic system. Error bars are calculated using standard deviating several measurements. As
mentioned before, the anode side (left side of the tube)
and cathode side (right side of the tube) have diﬀerent
wave numbers through reconnection; therefore, the largest
Lyapunov exponents for both sides are calculated independently.
From these results, the structures on the cathode side
do not change significantly with the pulse application;in contrast, the structures on the anode side are aﬀectedby pulse application. Pulse application to a chaotic system result in chaos suppression because the value of thelargest Lyapunov exponents becomes smaller; in contrast,
the pulse applied to a periodic system causes chaos excitation because the value of the largest Lyapunov exponents
becomes larger.
Finally, the results corresponding to the experimentshown in Fig. 1 (c) is described. The spatiotemporal dynamic behaviors of two coupled nonlinear oscillators arestudied here using two discharge tubes.
Figures 5 (a) and 5 (b) show the spatiotemporal structures before and under coupling when the two original oscillators show spatiotemporal chaos. The discharge currents of wave 1 and 2 are 15.80 mA and 24.20 mA, respectively. It means that the spatiotemporal structure changes
from chaos to order because of the coupling interaction.
Figure 6 shows the spatiotemporal structures (a) before and (b) during coupling when the two original oscil-Fig. 6 Spatiotemporal structures (a) before and (b) under coupling, when two original oscillators show spatiotemporal periodic states, are shown. The discharge currents ofwave 1 and 2 are 20.30 mA and 13.95 mA, respectively.
(a) Order and (b) chaotic structure are shown, respectively. Horizontal and vertical ranges indicate 10 cm and0.25 ms, respectively.

Table 2 Total values of the largest Lyapunov exponents of two
oscillators are shown for (a) coupling interaction of two
chaotic systems and (b) coupling interaction of two periodic systems.

lators show spatiotemporal periodic states. The discharge
currents of wave 1 and 2 are 20.30 mA and 13.95 mA. It
suggests the spatiotemporal structure changes from orderto chaos are caused by the coupling interaction.
Table 2 and Fig. 7 list the total values of the largest
Lyapunov exponents of the two oscillators (a) before coupling and (b) in coupling. The Error bars are calculatedusing standard deviating many measurements. The largest
Lyapunov exponents for both sides (right and left sides) arecalculated independently.
From these results, the structures on the cathode side
do not transform by coupling; in contrast, the structures on
the anode side are aﬀected by coupling interaction. Coupling interaction of two chaotic systems results in chaos
suppression because the value of the largest Lyapunov ex-3401070-4

Plasma and Fusion Research: Regular Articles

Volume 14, 3401070 (2019)• Spatiotemporal structures are aﬀected by pulse application. Pulse application to a chaotic system causes
chaos suppression; in contrast, pulse applied to a periodic system result in chaotic excitation.
• Spatiotemporal structures are aﬀected by coupling interaction. Coupling interaction of two chaotic systems causes chaos suppression; in contrast, couplinginteraction of two periodic systems results in chaoticexcitation.

Fig. 7 The largest Lyapunov exponents of two oscillators corresponding to Table 2 (a) and (b) are shown.

ponents becomes smaller; in contrast, coupling interaction of two periodic systems causes chaos excitation because the value of the largest Lyapunov exponents becomeslarger.
The value of discharge current as the control parameter of systems is selected in such a way that dynamic behaviors of two oscillators change by coupling, therefore,
a detail study on the relationship between the value of discharge current and the change of dynamic behavior by coupling is considered as future work.

4. Conclusions
Experimental study on spatiotemporal structures
formed in ionization waves is summarized as follows.
• Reconnections and topological defects in the spatiotemporal structure in ionization waves are observed, and spatiotemporal chaos appears because oftopological defect.

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