Complex dynamics of a dc glow discharge tube: experimental modeling and stability diagrams

ABSTRACT We report a detailed experimental study of the complex behavior of a dc low-pressure plasma discharge tube of the type commonly used in commercial illuminated signs, in a

microfluidic chip recently proposed for visible analog computing and other practical devices. Our experiments reveal a clear quasiperiodicity route to chaos, the two competing frequencies

being the relaxation frequency and the plasma eigenfrequency. Based on an experimental volt-ampere characterization of the discharge, we propose a macroscopic model of the current flowing in

the plasma. The model, governed by four autonomous ordinary differential equations, is used to compute stability diagrams for periodic oscillations of arbitrary period in the control

parameter space of the discharge. Such diagrams show self-pulsations to emerge remarkably organized into intricate mosaics of stability phases with extended regions of multistability

(overlap). Specific mosaics are predicted for the four dynamical variables of the discharge. Their experimental observation is an open challenge. SIMILAR CONTENT BEING VIEWED BY OTHERS EARLY

TURBULENCE AND PULSATILE FLOWS ENHANCE DIODICITY OF TESLA’S MACROFLUIDIC VALVE Article Open access 17 May 2021 STABILIZATION OF LIQUID INSTABILITIES WITH IONIZED GAS JETS Article 31 March

2021 UNIFIED FRAMEWORK FOR LASER-INDUCED TRANSIENT BUBBLE DYNAMICS WITHIN MICROCHANNELS Article Open access 13 August 2024 INTRODUCTION An outstanding problem is to understand and control

the complex chaotic behaviors commonly observed in glow discharge plasma tubes. From an experimental point of view, the first observations of deterministic chaos1, mixed-mode oscillations

and homoclinic chaos in discharge tubes2 date back more than twenty years ago. Currently, much attention has been focused on the generic problem of phase synchronization using these devices.

In particular, phase synchronization has been demonstrated under different conditions3,4,5. More recently, the transition to chaos and constructive effects of noise, such as stochastic

resonance and coherence resonance, have been reported in plasma tubes6,7,8,9. All aforementioned works have in common the feature of using discharge tubes specifically built for research:

Geissler's tubes, Plücker's tubes and so on. Conversely, in this work, we present an application of nonlinear dynamics to study the behavior of a popular device not designed

specifically for research purposes. Due to the great impact of plasma as a component in modern image viewing devices, the investigation of plasma tubes is of considerable interest.

Additional applications of interest involve microfluidic chips proposed for visible analog computing10 and the ability of glow discharges to find the shortest way through a maze11.

Electrical discharges have been continuously the subject of innumerous works and much is known about them12,13,14,15,16. From a theoretical point of view, discharges have been traditionally

described taking into account the complex processes involved in the plasma recombination and electric conductivity. Such descriptions require the use of coupled partial differential

equations involving spatial and time variables, the transport of momentum and energy of plasma components, the continuity equation, diffusion equation, Poisson equation, etc. This means that

a realistic description is generally quite involved. However, a fair description of the discharge can be obtained by bypassing most of the aforementioned complexities and focusing solely on

the key feature, namely the discharge capacity of conducting electric current. In other words, one may consider a macroscopic volt-ampere characterization of the discharge, regarding then

the whole plasma simply as a nonlinear two-terminal component of an electric circuit. Clearly, such approach removes spatial dependencies and reduces the mathematical description to a

nonlinear set of coupled ordinary differential equations, a more easy problem to deal with. The quality of this approach, of course, is proportional to the quality of the volt-ampere

characterization of the plasma tube. In this work we explore the dynamical behavior of the plasma tube in the region of glow discharge. We present a phenomenological model based on the

macroscopic electrical features, leading to a two-dimensional nonlinear model. Additional considerations, related to the experimental observation of chaos, induce us to introduce additional

degrees of freedom. We then check experimentally and through numerical simulations that our model gives a fair representation of the discharge. Finally, we use the model to predict the

distribution of self-pulsations in the control parameter space of the discharge. Before proceeding, we stress the fact that although our model is able to explain a number of experimentally

measured features of the discharge, it is just a rough approximation of the complex spatial processes underlying the plasma. Over the years, there have been numerous attempts of modeling

plasma discharges as circuits. Such models, however, aim to reproduce the plasma with a much higher accuracy than is needed here. For instance, instead of _ordinary_ differential equations,

more detailed models normally involve _partial_ differential equations. Detailed references about generation of low- and high-energy plasmas can be obtained, e.g., in the several specialized

books12,13,14,15,16. RESULTS The object of our investigations is a plasma tube device of the type commonly used in commercial neon signs and other applications10,11,12,13,14,15,16. The tube

is filled with neon at a low-pressure of the order of a few _Torr_ and has a length of 50 cm and 2.5 cm of diameter. Both electrodes are identical and can be used as anode or cathode

indifferently. The experimental setup is sketched in Fig. 1. An adjustable high-voltage dc source _V__bias_ (_Fluke 415B_) allows us to excite and drive the tube in the glow discharge

operating region. The _V__bias_ plays the role of control parameter. Here, _R__b_ and _R__l_ are a 150 kΩ ballast resistor and a 1 kΩ load resistor (nominal values), respectively. The

capacitor _C_ (2.4 nF), in parallel to the tube, makes the circuit behave as an electrical relaxation oscillator. With this setup, we performed preliminary measurements to determine the

electrical nonlinear characteristic of the tube. More explicitly, we measured the voltage drop _v__l_ through _R__l_ (using an Agilent 34401A multimeter) corresponding to several _V__bias_

values. From these data we calculated the discharge voltage across the tube (_v__t_ in our equations) and reconstructed the electrical volt-ampere characteristic curve using the following

equation The experimental data are plotted in Fig. 2 where we use the total current _i_ instead of _i_2, supposing that the current through _R__l_ and the tube are the same. The obtained

volt-ampere characteristic curve has a negative slope and is in good agreement with neon lamp curves found in literature13. The range of interest for _V__bias_, corresponding to the data

over the maximum peak plotted in Fig. 2, is 730 V < _V__bias_ < 2000 V. In this operating region we studied the dynamical behavior of the plasma tube. Two distinct signals were

experimentally acquired: the discharge voltage across the tube together with the corresponding light emission. A probe ×10 and a photodiode (_New Focus 1621_, not shown in Fig.1) were used.

Both voltage signals were recorded with a digital oscilloscope (LeCroy LT342). The recorded temporal sequences were used to construct bifurcation diagrams. Fixing _C_ = 2.4 nF, Fig. 3(a)

shows a bifurcation diagram where the maxima of the voltage signal are plotted versus the control parameter _V__bias_ and Fig. 3(b) shows a zoom of the range 730 V < _V__bias_ < 900 V.

This diagram hints to the presence of a sub-harmonic transition to chaos. Figure 3 (c) shows a return map, namely a plot of successive pairs of maxima of the signal. This return map

illustrates a typical period-2 oscillation. Then, by increasing _V__bias_, one observes a period-4 oscillation and another period-2 oscillation (see Figs. 3(d) and 3(e)), before and after an

interior crisis characterized by a sudden decrease in the relative amplitude of the voltage signal (see Fig. 3(a) around 1350 V). Lastly, Fig. 3 (f), shows a structure resembling somewhat

the familiar Hénon chaotic attractor. Investigating the _V__bias_ range 730 V–900 V allows us to localize a period-one solution at _V__bias_ = 730 V. Here a limit cycle at a frequency of 670

 Hz is observed suggesting that we deal with a stationary regime of the plasma characterized only by the frequency of the relaxation oscillator imposed by the capacitor _C_. In absence of

the capacitor _C_, as in Ref. (6), the plasma is in a homogeneous state. In fact, a slightly increase of _V__bias_ leads to period-two solutions and to a narrow chaotic region. As an

example, at _V__bias_ = 750 V, the corresponding power spectrum reveals a dominant peak at 890 Hz, its near subharmonic at 450 Hz and a weak remnant peak at 670 Hz. This suggest that a two

frequency route to chaos or quasiperiodicy scenario is encountered. The second frequency, competing with the relaxation frequency, is peculiar of the investigated plasma. The nature of

transition to chaos was more clearly manifested when decreasing the dissipativity, i.e. when the capacitor value is increased from _C_ = 2.4 nF to _C_ = 4.8 nF. As can be seen in Fig. 4, in

this case the discharge displays an elusive quasi-periodicity route to chaos, a scenario which is corroborated by a careful analysis of a torus breaking phenomenon. The bifurcation diagram

in Fig. 4 (a) is markedly different from the one in Fig. 3 (a). The Poincaré sections indicate a transition from a limit cycle (Fig. 4 (c)) to a two dimensional torus (Fig. 4 (d)), as

characterized by the closed loop section. A further increase of the control parameter leads to a torus breaking (Fig. 4 (e)) and, successively, to a condition of developed chaos (Fig. 4

(f)). The temporal behavior and magnitude spectrum associated to the quasiperiodic doubling cascade obtained at _C_ = 2.4 nF and _V__bias_ = 1000 V are reported in Fig. 5(a) and 5(b)

respectively. Fig. 5(b) clearly displays two peaks, the lower one (plasma eigenfrequency) not being a subharmonic of the relaxation frequency at _f_ = 1610 Hz. Fig. 5(c) and 5(d) display the

behavior of the relaxation frequency and plasma eigenfrequency as a function of _V__bias_ for _C_ = 2.4 nF and _C_ = 4.8 nF, respectively. In the first case (Fig. 5 (c)), their ratio

(_f__eig_/_f__mod_) is near 0.5 while in the second case (Fig. 5(d)) is near 0.89. We now derive experimentally a macroscopic model of the nonlinear behavior of the plasma discharge. The

starting point are Kirchhoff's laws applied to the electrical circuit of Fig. 1. Denoting by _v__t_ and _i_2 the discharge voltage and the current through the tube, respectively, it is

not difficult to derive the following equations where we disregarded a term containing _R__l_ because . The last two equations yield An additional equation governs the voltage drop on the

loop where we introduced the spurious inductance _L_ of the tube and its voltage-current characteristic _G_(_i_2). The nonlinear function _G_(_i_2) is determined experimentally by measuring

voltage drop _v__l_ through _R__l_ (see Fig. 1) and calculating _v__t_ from Eq. (1). In order to obtain a dimension-less form of the previous equations we make the following simplifying

assumptions. The experimental electrical characteristics reaches an asymptotic value of about _v_ = 360 V on the right border of the normal glow region, we use this value to rescale our

equations and to define the first two dimensionless quantities _y_ and _g_ (hereafter, dropping the subscript, we replace _i_2 by _i_) Since the order of magnitude of the current _i_ is 10−3

 A, we introduce a convenient scale factor α = 10−3 A and a dimensionless variable _x_ such that As shown in Fig. 5(a), observing the temporal sequences we can note that the interspikes

interval between two successive peaks is almost invariant with respect to our control parameter _V__bias_ and that it assumes the value of about β1 = 0.5 · 10−3 s. We use this value to

introduce the dimensionless time variable _t_ → β1 · _t_ (maintaining the same symbol _t_). Using two additional definitions, _V__bias_ ≡ _v_ · _y__o_ and β2 ≡ _R__b__C_, from Eq. (5) we

obtain where we used the abbreviations _A_0 = _y__o_β1/β2, _A_1 = β1/β2 and _A_2 = _A_1_R__b_α/_v_.Similar arguments allows us to rewrite Eq. (6) as where the parameter μ = (α · _L_)/(_v_ ·

β1) is connected to the spurious inductance _L_ which cannot be bypassed experimentally. We assume for _G_(_i_) = _v_ · _g_(_i_) a simple mathematical form accounting for the two regions of

the characteristic curve, namely redefining _k_1 = _K_1 · α and _k_2 = _K_2 · α we arrive at the following dimensionless form The values of the parameters _y__c_, _a_, _k_1 and _k_2 are

obtained by fitting the experimental data (Fig. 2). Such fits give _y__c_ = 0.977, _a_ = 0.9425, _k_1 = 0.4662, _k_2 = 26.1, with the quality coefficient of fit being , indicating a quite

good fit to the data17. Experimentally we observe some windows of chaotic behavior. Since the Poincaré–Bendixson theorem forbids chaos in two-dimensional systems (such the relaxation

oscillator described by Eqs. (15) and (16)), to be able to describe the experimental results we need extend the dimensionality of the model by introducing additional degrees of freedom

related to the plasma eigenfrequencies. Observing the temporal sequences, we note that in chaotic regime the spikes amplitudes change significantly, in an oscillatory way, but their

interspikes intervals are unchanged. This behavior can be reproduced by the following damped oscillator This plasma oscillation is driven by the current and it is rapidly damped when the

current decreases to zero. We suggest an oscillatory modulation around the working point to reproduce the experimental time series. In this case, the new electrical characteristic is where

we fix the parameters _y__c_ = 1, _a_ = 1, _k_1 = 0.5 and _k_2 = 26. With this, our macroscopic model of the discharge tube is now complete: The parameters _A_0, _A_1, _A_2 are related to

the experimental configuration adopted. Namely, _A_0 is connected to the bias voltage, plays the role of control parameter and varies in the range [3, 7.8]. Parameters _A_1, _A_2 depend of

the capacitor contained in the electrical circuit, while μ, β, ω and γ are free parameters. Concerning the parameter μ, we add the following. Since α = 10−3 A, _v_ = 360 V and β1 = 0.5 ·

10−3 s, from the definition of μ in Eq. 8 we get Since it is highly unlikely to have spurious inductances _L_ of the order of 1 H, one sees that μ is lower than 10−3. Note, however, that μ

can be used to externally control a time-scale of the circuit. DISCUSSION The experimentally derived macroscopic model in Eqs. (13)–(16) can be used to predict numerically the distribution

of stable self-generated complex oscillatory patterns in the discharge. Such study serves a few good purposes. On one hand, one may derive a wide-ranging phase diagrams providing a

systematic classification of the oscillatory states, of their relative abundance and of the boundary separating oscillatory phases. Since it is much harder to perform real-life experiments

over extended parameter domains than computer simulations, numerically obtained phase diagrams allow experiments to be planed and performed for more promising parameter regions in control

space. The availability of numerical predictions provides data against which to compare the reliability of the theoretical description and to improve it where needed. Using the model in Eqs.

(13)–(16) we computed the bifurcation diagram shown in Fig. 6 (a) and three return maps as indicated in the figure. These plots should be compared with the corresponding ones seen in Fig.

3. Note the larger interval of _A_0 in Fig. 6 (a). As one sees, while there is a fair overall agreement between Fig. 3 and 6, the bifurcations seen for higher values of _A_0 in Fig. 3 (a)

display a reverse doubling scenario which is not seen in Fig. 6. This means that the agreement between measurements and modeling deteriorates as _A_0 increases. We have also attempted to

locate an adequate region of the model to reproduce unambiguously the quasiperiodicity route observed experimentally. While the model can provide signs of quasiperiodicity, clear evidence

could not be found. Our model was also used to compute the _isospike diagrams_18,19,20,21,22 shown in the four panels of Fig. 7, i.e. phase diagrams depicting for every point in control

parameter space the number of spikes contained in one period of the regular oscillations. The colors of the individual panels depict the number of spikes contained in one period of the

stable oscillation of each of the four variables _x_, _y_, _z_, _w_, as indicated in the figure caption. Black represents “chaos”, i.e. parameters for which it was not possible to detect any

periodicity. Specific details about how these stability diagrams were computed are given below in the _Computational Methods_. The stability diagrams in Fig. 7 allow one to recognize the

rich and intricate interplay between the continuous spike-adding and spike-doubling mechanisms responsible for the complexification of periodic oscillation of the electric discharge. Each

variable produces a complex mosaic of colors, showing how the number of spikes self-organize in control space. A significant feature in Fig. 7 is that, although all phase diagrams display

the same structure, independently of the variable used to construct them, the individual phases vary in a way that is quite hard to summarize in any means other that by displaying them

graphically. Figure 8 shows bifurcation diagrams for the four variables of the model, computed by varying two parameters simultaneously along the diagonal straight line segments shown in

Fig. 7. While the overall structure of the diagrams is essentially the same, independent of the variable used to count spikes, the number of spikes in the several branches changes

considerably, corroborating the sequences recorded in the stability diagram of Fig. 7. Noteworthy are the many jumps observed in the bifurcation diagrams, which signal abundance of

multistability in the region, i.e. the possibility of stabilizing distinct attractors according to the initial conditions used. This behavior stresses the richness of the dynamical states

supported by the discharge. In this paper, we reported an experimental study of a low-pressure electrical discharge recording for it the standard doubling scenario as well as a remarkable

elusive route to chaos by quasiperiodicity. By characterizing the discharge through a volt-ampere characteristic, we developed a simple model reproducing its basic features. Based on this

model, we performed a detailed classification of the oscillatory behaviors, periodic or not, supported by the discharge. By computing stability diagrams for all model variables, we

characterized both the size and shape and the unexpected sequential ordering underlying the organization of stability phases. Our diagrams show precisely where the number of spikes changes

as a function of the variables used to count them. We found a plethora of stability islands which are simply too complicated to be classified systematically or to be described by other means

than purely graphically. Incidentally, we mention that currently there is no method to locate analytically stability phases for nonlinear oscillations of _arbitrary periods_ so that the

only way to find them is through direct numerical computations. Note that the information in our phase diagrams allows one to effectively _control the dynamics_, namely to select the final

dynamical state precisely by performing just a single change of parameters. This is in sharp contrast with conventional methods of controlling the dynamics23,24, which require

pre-investigating unstable orbits, do not include a prescription for the precise selection (targeting) of the final state and require the permanent application of external perturbations. In

summary, although relatively simple, the macroscopic discharge model reported here can reproduce basic experimental observations and reveals rich and unexpected dynamical facts. Future work

should tell if the complex stability mosaics predicted by this model can be also found in experimental diagrams or in predictions derived taking into account spatial phenomena of the

discharge as described, e.g. by more sophisticated discharge models based on sets of partial differential equations. METHODS COMPUTATIONAL METHODS: The isospike diagrams18,19,20,21,22 in

Fig. 7 were obtained by solving the model equations numerically for the following set of parameters: μ = 3 × 10−4, β = 8 × 10−3, γ = 4.2, _A_1 = 1.4 and _A_2 = 0.6, where the last two values

refer to _C_ = 2.4 nF. To this end, we used the standard fourth-order Runge-Kutta algorithm with fixed-step, _h_ = 5 × 10−6, over a mesh of 400 × 400 equally spaced points. For each value

of ω, we started the numerical integrations at _A_0 = 3 from the arbitrarily chosen initial condition (_x_, _y_, _z_, _w_) = (1, 1, 0.01, 0.01) and then proceed by _following the attractor_,

using the last obtained values of the variables to start every new integration involving infinitesimal changes of parameters. The first 400 × 106 time-steps were discarded as transient time

needed to reach the final attractor. The subsequent 100 × 106 iterations were then used to compute the number of spikes contained in one period of the oscillations, by recording up to 800

extrema (maxima and minima) of the time series of the variable under consideration, together with the instant when they occur and recording repetitions of the maxima. As indicated by the

colorbar in the figures, a palette containing 17 colors was used to represent “modulo 17” (i.e. recycling colors) the number of peaks (maxima) contained in one period of the oscillations.

The computation of stability diagrams is numerically a quite demanding task that we performed with the help of 1536 processors of a SGI Altix cluster having a theoretical peak performance of

16 Tflops. While it is possible to observe period-doubling routes, most of the times what happens is just the addition of a new peak to the waveform (without a corresponding doubling the

period). Eventually, after adding several peaks, one reaches a situation where the period roughly doubles a previously observed value. The bifurcation diagrams in Fig. 8 were obtained by

plotting the local maximum values (spikes) of the four variables along the lines shown in the four panels of Fig. 7, namely when tuning _A_0 and ω simultaneously along the line ω = 1.222_A_0

– 0.844. In all diagrams, both axis were divided into 600 equally spaced values. As done for the stability diagrams, computations were started at the minimum value of _A_0 from the initial

condition (_x_, _y_, _z_, _w_) = (1, 1, 0.01, 0.01) and continued by following the attractor using the same integrator and integration step. The first 16 × 106 steps were discarded as

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Google Scholar  Download references ACKNOWLEDGEMENTS E.P. and R.M. thank Professor F.T. Arecchi for many discussions during this work. They also thank K.G. Mishagin, O. Kanakov and K.

Al-Naimee for helpful discussions during the early stage of the experiment and its modelization and _Fondazione Ente Cassa di Risparmio di Firenze_ for financial support. J.G.F. was

supported by the Post-Doctoral grant SFRH/BPD/43608/2008 from FCT, Portugal. J.A.C.G. thanks support from CNPq, Brazil. All bitmaps were computed at the CESUP-UFRGS clusters, in Porto

Alegre, Brazil. This work was supported by the Max-Planck Institute for the Physics of Complex Systems, Dresden, in the framework of the Advanced Study Group on Optical Rare Events. AUTHOR

INFORMATION AUTHORS AND AFFILIATIONS * Istituto Nazionale di Ottica, Consiglio Nazionale delle Ricerche, Largo E. Fermi 6, 50125, Firenze, Italy Eugenio Pugliese, Riccardo Meucci, Stefano

Euzzor & Jason A. C. Gallas * Departamento de Física, Universidade Federal da Paraíba, 58051-970, João Pessoa, Brazil Riccardo Meucci, Joana G. Freire & Jason A. C. Gallas * CELC,

Departamento de Matemática, Faculdade de Ciências, Universidade de Lisboa, Lisboa, 1749-016, Portugal Joana G. Freire & Jason A. C. Gallas * Instituto de Altos Estudos da Paraíba, Rua

Infante Dom Henrique 100-1801, João Pessoa, 58039-150, Brazil Jason A. C. Gallas * Max-Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, Dresden, 01187, Germany Jason

A. C. Gallas Authors * Eugenio Pugliese View author publications You can also search for this author inPubMed Google Scholar * Riccardo Meucci View author publications You can also search

for this author inPubMed Google Scholar * Stefano Euzzor View author publications You can also search for this author inPubMed Google Scholar * Joana G. Freire View author publications You

can also search for this author inPubMed Google Scholar * Jason A. C. Gallas View author publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS E.P. and R.M.

conceived the experiments. E.P. and S.E. performed the experiments. J.G.F. and J.A.C.G. performed the simulations. All authors discussed the results and wrote the manuscript. ETHICS

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discharge tube: Experimental modeling and stability diagrams. _Sci Rep_ 5, 8447 (2015). https://doi.org/10.1038/srep08447 Download citation * Received: 27 October 2014 * Accepted: 20 January

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