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