# Transition To Chaos By A Cascade Of Period Doubling Bifurcations In Plasma

M. Aflori1, C. Ionita2, S. Chiriac1, D. G. Dimitriu1, R. Schrittwieser2

1Faculty of Physics, “Al. I. Cuza” University of Iasi, 11 Carol I Blvd., RO-700506 Iasi, Romania

e-mail: dimitriu@uaic.ro

2Institute of Ion Physics and Applied Physics, Leopold-Franzens University of Innsbruck, 25 Technikerstr., A-6020

Innsbruck, Austria; e-mail: Roman.Schrittwieser@uibk.ac.at

Abstract. New experimental results are presented on the transition to chaos of a plasma conductor when a multiple double

layer structure performs a complex nonlinear dynamical evolution. The dynamics of a simple double layer, which

starts at high values of the voltage applied to the electrode, consists in periodic disruptions and re-aggregations of the

structure in front of the electrode. At every disruption of the double layer structure, a bunch of electrons is released,

which can trigger an ion-acoustic instability in the plasma conductor. When a multiple double layer structure transforms

into a dynamic state, a more complex behaviour is observed. The constituent double layers start their own dynamics one

by one with increasing voltage on the electrode. The whole dynamic of the plasma conductor is a superposition of these

individual dynamics. Simultaneously with every new dynamical start of a double layer, a period doubling bifurcation is

recorded. The FFT’s of the recorded oscillations of the current collected by the electrode show the appearance of new

sub-harmonics of the fundamental frequency simultaneously with every period doubling bifurcation. Thus it is a Feigenbaum

scenario of transition to chaos by cascade of period doubling bifurcations which occurs in the plasma. A comprehensive

nonlinear dynamic analysis of the recorded signals was performed, offering an excellent insight into the

physical mechanism of the investigated phenomena.

Keywords: Multiple Double Layers, Period Doubling Bifurcation, Chaos

PACS: 05.45.-a, 52.25.Gj, 52.40.Kh

INTRODUCTION

Multiple double layers are complex nonlinear structures in plasmas, consisting of two or more concentric double

layers attached to the anode of a glow discharge [1-3] or to a positively biased electrode immersed into plasma [4].

They appear as several bright and concentric shells attached to the electrode. The axial profile of the plasma potential

has a stair step shape, with potential jumps close to the ionization potential of the used gas. The successive double

layers are precisely located at the abrupt changes of luminosity between two adjacent plasma shells. At high values

of the voltage applied to the electrode the multiple double layers structure evolves into a dynamic state, consisting

of periodic disruptions and re-aggregations of the constituent double layers.

Here we present experimental results which show that the plasma conductor passes into a chaotic state through a

cascade of period doubling bifurcations when the voltage on the electrode increases. The FFT’s of the current oscillations

collected by the electrode show the appearance of new sub-harmonics of the fundamental frequency, simultaneously

with every start of the dynamics of a constituent double layer.

EXPERIMENTAL RESULTS AND DISCUSSION

The experiments were performed in a plasma diode, schematically presented in Fig. 1. Plasma is created by an

electrical discharge between the hot filament (marked by F in Fig. 1) as cathode and the grounded tube (made of non-magnetic stainless steel) as anode. The plasma was pulled away from equilibrium by gradually increasing the

voltage applied to a tantalum disk electrode (marked by E in Fig. 1) with 1 cm diameter, under the following experimental

conditions: argon pressure p = 5⋅10–3 mbar, plasma density npl = 108 – 109 cm–3. When the voltage on the

electrode reaches VE ≅ 180 V, a multiple double layers structure consisting of 8 luminous plasma shells (photo in

Fig. 2) appears. The number of shells depends on the background gas, its pressure, the electrode voltage and the discharge

current [5]. By a further increase of the voltage on the electrode, the current collected by it becomes time dependent

(see the oscillations of the current, the FFT’s of them and the attractor of the system dynamic in the reconstructed

state space in Fig. 3a, 3b and 3c, respectively). These oscillations appear because the multiple double layer

structure passes into a dynamic state, characterized by periodic disruptions and re-aggregations of the constituent

double layers. The double layers pass into the dynamic state one by one, starting with the inner one because the current

density is higher through it [4]. When the second double layer starts its own dynamics, a phenomenon of period

doubling bifurcation is observed (see the oscillations of the current, the FFT’s of them and the attractor of the system

dynamics in the reconstructed state space in Figs. 3d, 3e and 3f, respectively). By further increasing the electrode

voltage, a cascade of period doubling bifurcations appear (see Figs. 3g-3r, respectively), simultaneously with the

transition of each double layer into a dynamic state. Finally, the plasma system transforms into a chaotic state, characterized

by uncorrelated oscillations with different frequencies (see Figs. 3s-3u, respectively).

The stability of a double layer is assured by the balance between the production of electrons and positive ions

through electron-neutral impact ionizations and excitations and the particle losses by recombination and diffusion

[4]. At high values of the current through the structure, this equilibrium is lost and the double layer passes into a dynamic

state. When the double layer disrupts, the initially trapped particles (electrons and positive ions) are released

and move towards the electrodes as bunches of particles. In the case of a multiple double layer, the free particles

have to pass through the others double layers and can affect their stability. Now, when two double layers (being constituents

of a multiple double layer structure) pass into dynamic states, there will be two pairs of particle beams

through the rest of the structure (two electron beams and two ion beams). Then the dynamic of the multiple double

layer structure becomes more complex and a period doubling bifurcation appears (a first sub-harmonic in the current

oscillation spectrum and a bifurcation of the limit cycle in the reconstructed state space are observed). Further on,

every new start of the dynamic of a double layer occurs simultaneously with a new period doubling bifurcation, new

sub-harmonics appear in the current oscillation spectrum and new bifurcations of the limit cycle in the reconstructed

state space take place. This is the well-known Feigenbaum scenario [6] of transition to chaos by cascades of period

doubling bifurcations. The final plasma system state is a chaotic one, consisting of uncorrelated oscillations with a

broad spectrum and many peaks present, which correspond to the unstable periodic dynamics of the multiple double

layer structure.

Experimental results are presented emphasizing the Feigenbaum scenario of transition to chaos by cascades.

ACKNOWLEDGMENTS

This work was supported in part by the Leopold-Franzens University of Innsbruck and by the Romanian Ministry

of Education – National Authority for Scientific Research, under the excellence grant cod ET 69.

REFERENCES

1. C. Chan and N. Hershkowitz, Phys. Fluids 25, 2135-2137 (1982).

2. L. Conde and L. Leon, Phys. Plasmas 1, 2441-2447 (1994).

3. O. A. Nerushev et al., Phys. Rev. E 58, 4897-4902 (1998).

4. C. Ionita, D. G. Dimitriu and R. Schrittwieser, Int. J. Mass Spectrom. 233, 343-354 (2004).

5. M. Aflori et al., IEEE Trans. Plasma Sci. 33, 542-543 (2005).

6. M. J. Feigenbaum, Los Alamos Science 1, 4-27 (1908).

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