J. Plasma Fusion Res. SERIES, Vol. 8 (2009)

Experimental Synchronization in Coupled Spatially Extended Systems
Takao FUKUYAMA
Faculty of Education, Ehime University, Bunkyo-cho 3, Matsuyama, Ehime 790-8577, JAPAN

Yoichi WATANABE, and Yoshinobu KAWAIInterdisciplinary Graduate School of Engineering Sciences, Kyushu University,
Kasugakoen 6-1, Kasuga, Fukuoka 816-8580, JAPAN(Received: 1 September 2008 / Accepted: 15 October 2008)

Synchronization between two unstable waves generated by current-driven ion-acoustic instability is observed
experimentally, and the dynamical behavior of coupled systems is studied. When two spatially extended systems
interact, and the dc potential applied to either of the two systems is varied gradually while maintaining coupling,
the systems exhibit synchronization. Furthermore, it is found that the systems exhibit hysteresis phenomena aroundthe threshold of synchronization.
Keywords: Chaos, Instability, Coupled Oscillators, Synchronization, Spatiotemporal Structure, Hysteresis
When synchronization occurs in coupled oscillators,
the coupled systems exhibit various synchronized statessuch as lag synchronization [13], phase synchronization[14], and complete synchronization [2,3]. Lag
synchronization implies synchronization in which twosignals lock their phases and amplitudes but with a timelag. In phase synchronization, phases are locked while theamplitudes remain uncorrelated. Finally, complete
synchronization implies synchronization in which there isa perfect hooking of the chaotic trajectories of twosystems.
The behavior of coupled spatially extended systems,
i.e. nonlinear wave-wave interaction in plasma, isinvestigated, and it is of interest to investigate different
original phenomena that occur due to spatiotemporalnonlinearity in plasma in comparison with the universal
characteristics of chaos. Previously, the authors havereported synchronization in coupled oscillators due to ion-acoustic instability [15], and contiguously-developed
results are reported in this paper.

1. Introduction

Synchronization between oscillating systems is auniversal phenomenon observed in nature; and its ancientroots were presented in the publication of Huygens’s
pendulum [1]. Recently, synchronization of not only
periodic oscillators (limit cycle) but also chaotic oscillators,
namely, chaos synchronization [2,3], was reported. Overthe past decades, investigations on the synchronization of
two chaotic oscillators have attracted considerable
attention and have been reported in many branches ofscience [4-7]; these investigations were pioneered by
Winfree, who investigated coupled nonlinear oscillators [8].
According to the characteristic property of coupledoscillators, two oscillators can be synchronized through acoupling interaction. The behavior of coupled oscillators isa phenomenon of interest in plasma physics so as tounderstand the interaction and coupling between various
waves excited in plasma. Many types of waves whoseanalysis is essential from the viewpoint of nonlinear
phenomena exist and propagate in plasma because ofvarious instabilities in plasmas.
Plasma is a typical nonlinear dynamical system withlarge degrees of spatiotemporal freedom. Thus far, mostinvestigations on coupled oscillators have been performed
under the specified condition that each oscillator is afunction that depends on only time, i.e., spatial freedom is
not taken into account; however the spatiotemporal
structure has recently attracted considerable attention[9-12]. Then, in spatially extended systems such as waves
in plasma, an understanding in space and time is required.
When the behavior of nonlinear wave-wave interaction inplasma is studied from the viewpoint of coupled nonlinear
oscillators and each wave corresponds to an autonomous
oscillator, it should be considered that each oscillator is aspatially extended system and has spatial freedom.

2. Experimental Setup and Excitation of Instability

Experiments are performed using a double-plasma
device [16] with a diameter of 70 cm and length of 120 cm.
The chamber of the device contains two cages composedof multipole permanent magnets for surface plasma
confinement; tungsten filaments acting as cathodes arewound around the chamber wall. The chamber is divided at
the center into two regions by a separation grid maintained
at a floating potential. In this experiment, plasma isgenerated only in one region, which is the experimentalregion. The chamber is evacuated to 4.0 × 10 -7 torr, andargon gas is introduced into the chamber at a pressure of
4.0 × 10 -4 torr. Typical plasma parameters are as follows:electron density ne ~ 108 cm−3 and electron and ion

author’s e-mail:fukuyama@ed.ehime-u.ac.jp

91©2009 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchT. Fukuyama et al., Experimental Synchronization in Coupled Spatially Extended Systems

Vs is fixed at 65 V, as mentioned before. The time series
and X–Y plot of the two instabilities are shown in Fig. 2.
Left, middle, and right traces correspond to time series
signals on G1, time series signals on G2, X-Y plot,respectively. For 0 V ≤ Vm ≤ 4 V, only instability 2oscillates since instability 1 is not yet excited in thisrange of Vm, as shown Fig. 2(a). For 4 V ≤ Vm ≤ 11 V,instability 2 is suppressed slightly and the oscillation falls
into disorder, as shown Fig. 2(b). For 11 V ≤ Vm ≤ 23 V,
the oscillation regains order, as shown in Fig. 2(c). For 23V ≤ Vm ≤ 40 V, instability 1 also begins to get excited andthe two waves interact; thus, the wave resulting frominstability 1 shows intermittency with burst oscillation.
However, the two waves oscillate almost independently,
and the coupled system does not attain thesynchronization state, namely, non-synchronization, asshown in Fig. 2(d). For 40 V ≤ Vm ≤ 55 V, the oscillation
caused by instability 1 is suppressed and considerably
smaller than that by instability 2, as shown in Fig. 2(e).
For 55 V ≤ Vm ≤ 58 V, the oscillation caused by instability
2 is also suppressed; therefore, coupled systems almost
stops oscillating, as shown in Fig. 2(f). With increasing
Vm, for 58 V ≤ Vm ≤ 65 V, the oscillations of the twowaves gradually increase again, as shown in Fig. 2(g).
For 65 V ≤ Vm ≤ 100 V, complete synchronization, whichimplies hooking of the chaotic trajectories of two systems,
is observed in the coupled system, as shown in Fig. 2(h).
Second, the control parameter Vm is gradually
decreased after being increased up to 100 V in the firstexperiment; coupling is maintained at this time, and thebehavior of the coupled systems is examined. Here, thecontrol parameter Vs is still maintained at 65 V. The timeseries and X–Y plot of the two instabilities in this case areshown in Fig. 3. Left, middle, and right traces correspondto time series signals on G1, time series signals on G2,X-Y plot, respectively. For 100 V ≥ Vm ≥ 48 V, complete
synchronization is observed, as shown in Figs. 3(a) and(b). For 53 V ≥ Vm ≥ 48 V, the amplitudes of coupledsystems suddenly increase while maintaining complete
synchronization, as shown in Fig. 3(b). With decreasing
Vm, for 48 V ≥ Vm ≥ 46 V, the shape of the X–Y plot
gradually changes from a straight line to an ellipse, asshown in Figs. 3(c) and (d). This implies that thecomplete synchronization in coupled systems changes tolag synchronization, in which two signals lock theirphases and amplitudes but with a time lag, as mentioned
earlier. For 46 V ≥ Vm ≥ 30 V, the oscillation caused byinstability 1 is suppressed and considerably smaller thanthat by instability 2, as shown in Fig. 3(e). The wave
generated by instability 2 shows intermittency including
burst oscillation, which is similar to the oscillation caused
by instability 1, as shown in Fig. 2(d). For 30 V ≥ Vm ≥ 0V, only the oscillation caused by instability 2 are presentsince the oscillation caused by instability 1 almost
disappears in this range of Vm, as shown Fig. 3(f).

temperatures Te ~ 0.5–1.0 eV and Ti ~ Te/(10–15),
respectively.
Current-driven ion-acoustic instability is excited by
two parallel mesh grids G1 and G2 having the following
dimensions: diameter 6.0 cm; grid 50 mesh/ in; andinterval L 3.0 cm; these two grids are placed in ameasurement system where dc potential is applied to eitherof them [17]. The instability is caused by the interaction
between electron streaming, i.e., electron current, andbackground ion-acoustic waves in the plasma [18-23].
When dc potential is applied to either of the two meshgrids, an electric field is generated between them.
Electrons are accelerated according to the generated
electric field and the electron current flows between thetwo mesh grids. As mentioned above, instability occurs
due to the interaction between this electron current and thebackground ion-acoustic waves in the plasma. When thebiased grid potential exceeds a threshold, a current-driven
ion-acoustic instability is excited and oscillations occur[17].
The dc potential Vm is applied to G1 and Vs is applied
to G2. Vm and Vs are control parameters that govern
instability 1 and instability 2, respectively. Vs is fixed at 65
V and Vm is varied. It should be noted that unstable“traveling” waves are excited by instability 1 andinstability 2; therefore, each wave is directly coupled by aninteraction. Coupled spatially extended systems areschematically shown in Fig. 1. Time series signals foranalysis are obtained from the fluctuating components ofthe currents on the biased mesh grids on both sides (G1,G2) and are sampled with a digital oscilloscope.

Fig.1. Schematic of coupled spatially extended systems.

3. Results and Discussion

3.1 Dynamical Behavior of Coupled Systems
The behavior of coupled spatially extended systems
is investigated in the case that two oscillations caused by
the current-driven ion-acoustic instability interact by awave-wave interaction. The experiment consists of twoprocesses explained in the following paragraphs.
First, the control parameter Vm is gradually increased
while maintaining coupling and the behavior of thecoupled systems is examined. Here, the control parameter

92T. Fukuyama et al., Experimental Synchronization in Coupled Spatially Extended SystemsFig. 2. The time series and X–Y plot of the two instabilities are shown. Left, middle, and right traces correspond to timeseries signals on G1 (Vm is applied), time series signals on G2 (Vs is applied), X-Y plot, respectively. The control parameter
Vm is gradually increased. Here, the control parameter Vs is fixed at 65 V. (a) Vm = 0 V, (b) Vm = 8 V, (c) Vm = 20 V, (d) Vm =33 V, (e) Vm = 45 V, (f) Vm = 55 V, (g) Vm = 60 V, and (h) Vm = 85 V.

Fig. 3. The time series and X–Y plot of the two instabilities are shown. Left, middle, and right traces correspond to timeseries signals on G1 (Vm is applied), time series signals on G2 (Vs is applied), X-Y plot, respectively. The control parameter
Vm is gradually decreased. Here, the control parameter Vs is fixed at 65 V. (a) Vm = 55 V, (b) Vm = 50 V, (c) Vm = 48 V, (d)Vm = 47 V, (e) Vm = 46 V, and (f) Vm = 10 V.
3.2 Hysteresis in Coupled Systems
The behavior of coupled systems is observed when Vs
is fixed and Vm is varied. The observation results are listed
in Table 1. Vm is first increased and then decreased; then,for any given value of Vm, the behaviors of the coupledsystems with an increase in this Vm value are different
from those with a decrease in this value. This implies thatthe coupled systems exhibit hysteresis phenomena aroundthe threshold of synchronization, as shown in Fig. 4. Suchhysteresis phenomena related to synchronization barely93T. Fukuyama et al., Experimental Synchronization in Coupled Spatially Extended Systems

appear in coupled systems; that is, the behavior of asystem is uniquely decided by an individual control
parameter. Plasma is a medium with a strong
self-nonlinearity because of large degrees ofspatiotemporal freedom, and in most cases, phenomena
occurring due to a control parameter should beconsidered as non-Markovian processes. A synchronizedcoupled system is almost steady, and the history of its
stability is stored in plasma; therefore, hysteresis
phenomena are observed.
Range (V)0 ≤ Vm ≤ 23
23 ≤ Vm ≤ 40
40 ≤ Vm ≤ 55

Instability 1
Suppressed
ExcitedSuppressed

Instability 2
ExcitedExcitedExcited55 ≤ Vm ≤ 58

Suppressed

Suppressed

58 ≤ Vm ≤ 100
100 ≥ Vm ≥ 46
46 ≥ Vm ≥ 0

ExcitedExcitedSuppressed

ExcitedExcitedExcitedStateNo Synch.
No Synch.
No Synch.
No
Oscillation
Synch.
Synch.
No Synch.

Fig. 5. The behavior of coupled systems is shown: (a)typical non-synchronized and (b) synchronized states. Leftand right traces correspond to the time series and X–Yplot of the two instabilities, respectively.

Table 1. The behavior of coupled systems is observed
when Vs is fixed and Vm is varied. The observation results
are listed.

Fig. 6. Correlation between two oscillators is shown: (a)the phase difference | Φ1 − Φ 2 | and (b) amplitudedifference between two oscillators that are in a typicalnon-synchronized state.

Fig. 4. Hysteresis phenomena around the threshold ofsynchronization are exhibited.
3.3 Correlation of Phase and Amplitude between TwoOscillators in Coupled Systems
When synchronization occurs in coupled oscillators,
there exists a correlation between the phases of oscillations
and a correlation between the amplitudes of theseoscillations. Figures 5(a) and (b) show typicalnon-synchronized and synchronized states, respectively.
The time series and X–Y plot of the two instabilities areshown in this figure. Figures 6(a) and (b) show the phasedifference | Φ1 − Φ 2 | and amplitude difference between
two oscillators that are in a typical non-synchronized state.
Sampled time series are analyzed using a low-pass filter
that can pass frequencies of less than 100 kHz. Figure 6(a)shows that the phase difference increases with time.

Fig. 7. Correlation between two oscillators is shown: (a)the phase difference | Φ1 − Φ 2 | and (b) amplitudedifference between two oscillators that are in a typicalsynchronized state.
Thus, there is no correlation between the phases, and thetwo waves are not phase synchronized. Figure 6(b) showsthat the oscillation of the amplitude difference is almost
disorder. Thus, there is no correlation between amplitudes,
and the two waves are not synchronized with respect totheir amplitudes. Figures 7(a) and (b) show the phasedifference between two oscillators and their amplitudedifference, respectively, when the oscillators are in the94

T. Fukuyama et al., Experimental Synchronization in Coupled Spatially Extended Systems

typical synchronized state. Sampled time series areanalyzed using the same low-pass filter. Figure 7(a) showsthat the phase difference between the two coupledoscillators remains practically constant at approximately
zero with time. Thus, the phases of the coupled oscillators
are synchronized in the process of coupling. Figure 6(b)shows that the oscillation of the amplitude difference isalmost order. Thus, there exists a correlation between
amplitudes, and the amplitudes of the two waves aresynchronized.

[16] R. J. Taylor, K. R. Mackenzie and H. Ikezi: Rev. Sci.
Instrum. 43, 1675 (1972).
[17] K. Taniguchi, H. Kuwae, N. Hayashi, and Y. Kawai, Phys.
Plasmas 5, 401 (1998).
[18] M. Yamada and M. Raether, Phys. Fluids 18, 361 (1975).
[19] K. Nishikawa and C.-S. Wu, Phys. Rev. Lett. 23, 1020
(1969).
[20] D. B. Fenneman, M. Raether, and M. Yamada, Phys.
Fluids 16, 871 (1973).
[21] D.-I. Choi and W. Horton, Jr., Phys. Fluids 17, 2048
(1974).
[22] H. Tanaca, A. Hirose, and M. Koganei, Phys. Rev. 161,94(1967).
[23] Y. Kawai, Ch. Hollenstein, and M. Guyot, Phys. Fluids 21,970 (1978).

4. Conclusion

The dynamical behavior of coupled spatiallyextended systems, caused by current-driven ion-acoustic
instability, is investigated experimentally. When twowaves interact, and the control parameter governing
either of the two systems is varied gradually, the systems
exhibit synchronization in a certain region. It is foundthat the coupled systems exhibit hysteresis phenomena
around the threshold of synchronization, account of thestrong nonlinearity of plasma. When synchronization
occurs in coupled oscillators, the two oscillators possesscorrelation with respect to both their phase and amplitude.

References
C. Hugenii, Horoloquim Oscilatorium (Parisiis, France,1673).
[2] H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32(1983).
[3] L. M. Pecora and T. L. Carroll, Phys. Rev. Lett. 64, 821(1990).
[4] K. Coffman, W. D. McCormick, and Harry L. Swinney,Phys. Rev. Lett. 56, 999 (1986).
[5] R. Roy and K. S. Thornburg, Jr, Phys. Rev. Lett. 72, 2009
(1994).
[6] H. Yamazaki, T. Yamada, and S. Kai, Phys. Rev. Lett. 81,4112 (1998).
[7] T. Fukuyama and Y. Kawai, J. Phys. Soc. Jpn. 71, 1809
(2002).
[8] T. Winfree, J. Theor. Biol. 16, 15 (1967).
[9] S. Boccaletti, J. Bragard, F. T. Arecchi, and H. Mancini,Phys. Rev. Lett. 83, 536 (1999).
[10] N. Baba, A. Amann, E. Schoell, and W. Just, Phys. Rev.
Lett. 89, 074101 (2002).
[11] R. Neubecker and B. Guetlich, Phys. Rev. Lett. 92,154101 (2004).
[12] T. Fukuyama, R. Kozakov, H. Testrich, and C. Wilke,
Phys. Rev. Lett. 96, 024101 (2006).
[13] M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys.
Rev. Lett. 78, 4193 (1997).
[14] M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, Phys.
Rev. Lett. 76, 1804 (1996).
[15] T. Fukuyama and Y. Kawai, J. Phys. Soc. Jpn. 71, 1809
(2002).
[1]95