J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)256–259
000–000

Chaotic Oscillations via Quasi-Periodicity Caused by Applying
External Modulation in Ionization Waves
FUKUYAMA Takao, KOZAKOV Ruslan1,2, TESTRICH Holger1,2, WILKE Christian1,2
and KAWAI Yoshinobu
Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Fukuoka 816-8580, Japan
Institut für Physik, Ernst-Moritz-Arndt-Universität Greifswald, Domstrasse 10a, D-17489 Greifswald, Germany
2
Max-Planck-Institut für Plasmaphysik, Teilinstitut Greifswald, Wendelsteinstrasse 1, D-17491 Greifswald, Germany
1(Received: 9 December 2003 / Accepted: 23 February 2004)

Abstract
Dynamical behavior of nonlinear ionization waves excited in positive columns of glow discharge is investigatedwith external modulation. It is well known that competition of two frequencies leads the system into a state of chaotic
motion via quasi-periodicity as mentioned in Ruelle and Takens scenario [D. Ruelle and F. Takens, Commun. Math.
Phys. 20, 167 (1971)]. Here, quasi-periodic route to chaos is investigated experimentally in ionization waves system.
Quasi-periodicity with two frequencies is realized by competition between self-sustained wave and external periodicmodulation. According to the change of the rate of discharge current modulation, the system comes to show chaotic
oscillations via quasi-periodicity.
Keywords:
chaos, quasi-periodicity, ionization waves in glow discharge

1. Introduction

Hopf bifurcation into a two-frequency torus. Under further
changes of the parameter the torus becomes unstable and may
either develop into a three-torus via another Hopf bifurcation,or into a chaotic attractor with a dimension slightly larger
than two, which still can have some overall structure
resembling the torus. In particular, the two frequencies of thelimit cycle can dominate the chaotic motion.
The wave property can be mainly controlled by discharge
current and pressure, and it can be classified into three typesof coherent striations, which are Pomalý wave (P wave),
Souc̆ asný wave (S wave), and Rychlý wave (R wave)
according to “Novak constant” [8]. Novak constant N is
defined as follow: N [V] = E [V/cm] × λ [cm]. It shows 9[V],20[V] and 13[V] for P wave, S wave and R wave,
respectively. Periodic modulation is superimposed to thedischarge current, when the system is in R wave regime whichshows coherent striation. Quasi-periodicity with two
frequencies is realized by competition between self-sustained
R wave and external periodic modulation. According to thechange of the rate of discharge current modulation, the systemcomes to show chaotic oscillations via quasi-periodicity.

In dissipative physical systems, such as in plasmas,
fluids, chemicals etc., it is often observed that the systemsettles into a state of sustained chaotic motion [1-5]. Chaos
has been observed in various physical systems and several
routs to chaos have been identified. Ionization waves in a
glow discharge are of interest as a medium for testing theuniversal characteristics of chaos such as the low-dimensional
behavior and certain route to chaos [6]. It is well known thatcompetition of two frequencies leads the system into a stateof chaotic motion via quasi-periodicity as mentioned in Ruelle
and Takens scenario [7].
Here, quasi-periodic route to chaos is investigatedexperimentally in ionization waves system.

2. Chaos via quasi-periodicity
There will be a spectrum consisting of one frequency( f1), then two ( f1 and f2), and sometimes three frequencies( f1, f2 and f3). As soon as the third frequency arrives, thebroad-band noise characteristic of chaos should start toappear. This shows a system with small number of degrees of
freedom can engender a chaotic regime. Now it should beattempted to justify the necessity of having a quasi-periodic
regime with three frequencies before a strange attractor canappear by this route. Quasi-periodicity is quite a common
route to chaos. A stable fixed point turns into a limit cycle
via a Hopf bifurcation. The limit cycle goes through a second

3. Experimental setup
The experiments are performed using a glass tube withdimensions of 2 cm diameter and 75 cm length, as shown inFig. 1.

Corresponding author’s e-mail: fukuyama@aees.kyushu-u.ac.jp1256©2004 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchFukuyama T. et al., Chaotic Oscillations via Quasi-Periodicity Caused by Applying External Modulation in Ionization Waves

4. Results and discussions

High direct current voltage is applied to anode and theplasma is produced by a glow discharge between anode andcathode. Unstable ionization waves propagate from cathodeto anode, which is “traveling waves”. The experiments aredone under the condition of electrostatic without magneticeffects. The chamber is evacuated to vacuum, and then neon
gas is introduced into the chamber at a pressure of 0.83 Torr.
Typical electron temperature Te is about 10 eV. The current
source is used in order to adjust discharge current. Here,
discharge current is fixed at 5.4 mA and sinusoidal wave f1 issuperimposed to discharge current as an external force using
a function generator. Time series signals for analysis are
obtained from the photo diodes and sampled with a digital
oscilloscope.

At firstly, dynamical behavior is studied whenmodulation frequency f1 is changed. Experimental conditions
are as follows: f1 is a control parameter, the modulation rateof discharge current Iˆ = 0.41 %, fundamental frequency ofthe system f0 = 20.75 kHz, wave length λ = 3.60 cm, electricfield E = 3.49 V/cm, Novak constant N = E × λ = 12.6 V
which is corresponding to “R wave” region.

Fig. 2
Fig. 1 Schematic diagram of the experimental apparatus.

Dynamical behavior when modulation frequency ƒ1 ischanged.

Fig. 3 Dynamical behavior when the modulation rate of discharge current Iˆ is changed.

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Fukuyama T. et al., Chaotic Oscillations via Quasi-Periodicity Caused by Applying External Modulation in Ionization WavesFig. 4

Details of the frequency in Fig. 3(b) and (f). Every peak frequency in a quasi-periodic state can be calculated as a combinationof three frequencies ƒ1, ƒ2 and ƒ3

The results are shown in Fig. 2. Peak frequency f0 , f1 ,and f0 – f1 are observed. For f1 > 15 V, many peak frequencies
are observed and the system shows quasi-periodic or chaoticstate.
On the other hand, dynamical behavior is studied whenthe modulation rate of discharge current Iˆ is changed.
Experimental conditions are as follows: f1 is 18 kHz (here, f1 /f0 = 0.867…), the modulation rate of discharge current Iˆ isa control parameter, fundamental frequency of the system f0= 20.75 kHz, wave length λ = 3.60 cm, electric field E =3.49 V/cm, Novak constant N = E × λ = 12.6 V which iscorresponding to “R wave” region. The results are shown inFig. 3. For 0 % ≤ Iˆ ≤ 2.91 %, the system shows a periodic
oscillation at self-sustained frequency 20.75 kHz, as shownin Fig. 3(a). For 2.91 % ≤ Iˆ ≤ 3.96 %, the system shows aquasi-periodic oscillation with an interaction between f0 andf1, as shown in Fig. 3(b) and (c). For 3.96 % ≤ Iˆ ≤ 5.33 %,the system shows a chaotic oscillation with an interaction
between f0 and f1, as shown in Fig. 3(d) and (e). Here, chaotic
analysis is performed using each time series for Fig. 3(d) and(e) in order to quantitatively evaluate whether two systems
are chaotic or not; calculations of the correlation dimension
are performed. In the calculation, the method of Grassberger
and Procaccia [9] is used. As a result, it is confirmed thatsystems are chaotic since the value of the correlation
dimension is not an integer, which is the characteristic of thechaotic state. For 5.33 % ≤ Iˆ ≤ 8.09 %, the system shows amode-locked or chaotic oscillation, as shown in Fig. 3(f) and(g). For Iˆ ≥ 8.09 %, the system shifts to the external force f1= 18 kHz, namely quenching occurs, as shown in Fig. 3(h).
Furthermore, every peak frequency in a quasi-periodic state
can be calculated as a combination of three frequencies f1, f2
and f3 as shown in Fig. 4. Thus, quasi-periodic route to chaos
has been experimentally observed in ionization waves system.
In low-pressure region (less than 0.75 Torr), the ion
acoustic waves have been observed. Therefore, detail studies
from the viewpoint of the transition to the ion acoustic waves
are considered as future work.

5. Conclusion
Dynamical behavior is studied when modulation
frequency f1 is changed. Peak frequency f0 , f1 , and f0 – f1
are observed. For f1 > 15 V, many peak frequencies areobserved and the system shows quasi-periodic or chaotic state.
Moreover, dynamical behavior is also studied when themodulation rate of discharge current Iˆ is changed. For 0 % ≤Iˆ ≤ 2.91 %, the system shows a periodic oscillation at selfsustained frequency 20.75 kHz. For 2.91 % ≤ Iˆ ≤ 3.96 %, thesystem shows a quasi-periodic oscillation with an interaction
between f0 and f1. For 3.96 % ≤ Iˆ ≤ 5.33 %, the system showsa chaotic oscillation with an interaction between f0 and f1 .
For 5.33 % ≤ Iˆ ≤ 8.09 %, the system shows a mode-locked
or chaotic oscillation. For Iˆ ≥ 8.09 %, the system shifts tothe external force f1 = 18 kHz, namely quenching occurs.
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Fukuyama T. et al., Chaotic Oscillations via Quasi-Periodicity Caused by Applying External Modulation in Ionization Waves[3] P.Y. Cheung, S. Donovan and A.Y. Wong, Phys. Rev.
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9, 4525 (2002).
[6] K. Ohe, Current Topics in the Phys. Fluids 1, 319 (1994).
[7] D. Ruelle and F. Takens, Commun. Math. Phys. 20, 167(1971).
[8] S. Pfau and A. Rutscher, Beitr. Plasmaphys. 8, (Germany,1968).
[9] P. Grassberger and I. Procaccia, Physica D9, 189 (1983).

Furthermore, every peak frequency in a quasi-periodic state
can be calculated as a combination of three frequencies f1, f2
and f 3. Thus, quasi-periodic route to chaos has been
experimentally observed in ionization waves system.
FUKUYAMA T. is supported by JSPS Postdoctoral
Fellowships for Research Abroad (Max-Planck-Institut,Germany) from April, 2004.

References[1] A. Komori et al., Phys. Fluids B 4, 3573 (1992).
[2] N. Ohno et al., Phys. Fluids B 5, 796 (1992).

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