1982 A model for plasma instabilities

Christophe LETELLIER

In certain plasma experiments, the plasma is maintained as long as the instability saturation implies a small growth rate of instabilities ; perturbations are still mainly governed by a few linear modes. Chaos was produced with an elementary model based on qaudratic interaction between two modes [1]

Here the nonlinear decay of a linearly unstable high-frequency wave into a linearly damped low-frequency one. These two waves are governed by a growth and a damping rates and the quadratic coupling V. There is a frequency mismatch

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The complex mode amplitudes are governed by the equations

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After a coordinate transformation, the equations governing the wave amplitudes x and y, and the relative phase z are [2]

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When the mismatch is

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and varying the ratio between the growth and the damping rates, a period-doubling cascade is observed as the route to chaos (Fig. 1).

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Fig. 1. Bifurcation diagram versus the ratio between growth and the damping rates.

With the ratio

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a chaotic attractor can be obtained (Fig. 2). Its shape is very sharp and it is useful to use a differential embedding to represent the dynamics. When induced by variable z (left panel of Fig. 2), the state portrait looks like a Rössler attractor represented in a differential embedding induced by variable z. A better representation of the dynamics can be obtained by using the variable s = log (z) as shown in the right panel of Fig. 2.

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Fig. 2. Chaotic attractor produced by the simple model with quadratic interaction of waves.

This attractor can be characterized by a first-return map to the Poincaré section

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which is smooth and unimodal as expected after a period-doubling cascade (Fig. 3).

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Fig. 3. Smooth unimodal first-return map to a Poincaré section of the chaotic attractor.

The original purpose of the work by Claude Meunier, Marie-Noëlle Bussac and Guy Laval, by then at the Ecole Polytechnique (Palaiseau, France), was to show that intermittency was possible in plasma. This is can be easily checked with

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that is, just before the period-3 window. A typical time series of variable z is shown in Fig. 4.

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Fig. 4. Time series of variable {z} in a type-I intermittent regime.

The existence of a small channel between the first bisecting line and a return map can be exhibited by using a third-return map to the Poincaré section, as shown in Fig. 5.

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Fig. 5. Third-return map to the Poincaré section in the type-I intermittent regime, just before the period-3 window.

[1] S. Y. Vyshkind, Occurrence of stochasticity in the case of nondegenerate wave interaction in media with amplification. Radiophysics and Quantum Electronics, 21 (6), 600-604. 1978.

[2] C. Meunier, M. N. Bussac & G. Laval, Intermittency at the onset of stochasticity in nonlinear resonant coupling processes, Physica D, 4 (2), 236-243, 1982.

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