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Research Papers

Global Bifurcations of Mean Electric Field in Plasma L–H Transition Under External Bounded Noise Excitation

[+] Author and Article Information
C. Nono Dueyou Buckjohn

Ph.D Student
e-mail: bucknono@yahoo.fr

M. Siewe Siewe

Senior Lecturer
e-mail: martinsiewesiewe@yahoo.fr

C. Tchawoua

Associate Professor
e-mail: ctchawa@yahoo.fr

T. C. Kofane

Professor
e-mail: tckofane@yahoo.com
Laboratoire de Mécanique,
Département de Physique,
Faculté des Sciences,
Université de Yaoundé I,
BP 812, Yaoundé, Cameroun

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the Journal of Computational and Nonlinear Dynamics. Manuscript received May 22, 2012; final manuscript received February 23, 2013; published online May 31, 2013. Assoc. Editor: Stefano Lenci.

J. Comput. Nonlinear Dynam 8(4), 041011 (May 31, 2013) (18 pages) Paper No: CND-12-1079; doi: 10.1115/1.4024025 History: Received May 22, 2012; Revised February 23, 2013

In this paper, global bifurcations and chaotic dynamics under bounded noise perturbation for the nonlinear normalized radial electric field near plasma are investigated using the Melnikov method. From this analysis, we get criteria that could be useful for designing the model parameters so that the appearance of chaos could be induced (when heating particles) or run out for quiescent H-mode appearance. For this purpose, we use a test of chaos to verify our prediction. We find that, chaos could be enhanced by noise amplitude growing. The results of numerical simulations also reveal that noise intensity modifies the attractor size through power spectra, correlation function, and Poincaré map. The criterion from the Melnikov method which is used to analytically predict the existence of chaotic behavior of the normalized radial electric field in plasma could be a valid tool for predicting harmful parameters values involved in experiment on Tokamak L–H transition.

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Figures

Grahic Jump Location
Fig. 1

3D upper threshold bound in (Ω2,η) plane with the associated threshold amplitude F2 versus noise intensity η for homoclinic bifurcation; analytic results for different values of Ω2

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Fig. 2

3D inverse upper threshold bound in (Ω2,η) plane with the associated threshold amplitude F2 versus noise intensity η for heteroclinic bifurcation; analytic results for different values of Ω2

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Fig. 3

(top) Plot of asymptotic growth rate Kc versus c for the system in Eq. (7) (Kc is around 0); and (bottom) the associated mean square displacement M(n) as a function of n. We used N = 20,000 data points here and 100 equally spaced values for c. F2≈1.2;η = 0.0 corresponding to regular dynamics (M(n) do not have a linear growth).

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Fig. 4

(top) Plot of asymptotic growth rate Kc versus c for the system in Eq. (7) (Kc is around 1.0); and (bottom) the associated mean square displacement M(n) as a function of n. We used N = 2000 data points here and 100 equally spaced values for c. F2≈1.31;η = 0.18 corresponding to chaotic dynamics (M(n) has a linear growth).

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Fig. 5

(top) Plot of asymptotic growth rate Kc versus c for the system in Eq. (7) (Kc is around 1.0); and (bottom) the associated mean square displacement M(n) as a function of n. We used N = 20,000 data points here, and 100 equally spaced values for c. F2≈1.31;η = 10.0 corresponding to noise induced chaotic dynamic (the linearity of M(n) is more pronounced).

Grahic Jump Location
Fig. 6

(top) Plot of asymptotic growth rate Kc versus c for the system in Eq. (7) and (bottom) the associated mean square displacement M(n) as a function of n. We used N = 20,000 data points here and 100 equally spaced values for c. F2 ≈5.0;η = 1.0 corresponding to noisy regular dynamics.

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Fig. 7

(top) Plot of asymptotic growth rate Kc versus c for the system in Eq. (7) and (bottom) the associated mean square displacement M(n) as a function of n. We used N = 2000 data points here and 100 equally spaced values for c. F2 ≈6.1;η = 0.1 corresponding to chaotic dynamics.

Grahic Jump Location
Fig. 8

(top) Plot of asymptotic growth rate Kc versus c for the system in Eq. (7) and (bottom) the associated mean square displacement M(n) as a function of n. We used N = 20,000 data points here, and 100 equally spaced values for c. F2 ≈6.1;η=10.0 corresponding to noise induced chaotic dynamics.

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Fig. 9

Asymptotic growth rate Kc versus F2 for Eq. (7); homoclinic orbits

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Fig. 10

Asymptotic growth rate Kc versus F2 for Eq. (7); heteroclinic orbits

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Fig. 11

(top) Time history, (middle) the associated autocorrelation functions and (bottom) fast Fourier transform for time series data from the system in Eq. (7): F2≈1.0;η = 5.0 corresponding to noisy motion

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Fig. 12

(top) Percentage of false nearest neighbor; and (bottom) autocorrelation function, for time series data from the system in Eq. (7), with embedding dimension m in [0,...,10]: F2≈1.31;η=0.18 corresponding to chaotic motion

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Fig. 13

(top) Time history, (middle) the associated autocorrelation functions and (bottom) fast Fourier transform for time series data from the system in Eq. (7)—(heteroclinic orbit): F2≈5.0;η=5.0 corresponding to noise-induced chaotic motion

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Fig. 14

(top) Time history, (middle) the associated autocorrelation functions and (bottom) fast Fourier transform for time series data from the system in Eq. (7)— (heteroclinic orbit): F2≈6.1;η=0.1 corresponding to chaotic motion

Grahic Jump Location
Fig. 15

(top) Time history, (middle) the associated autocorrelation functions and (bottom) fast Fourier transform for time series data from the system in Eq. (7)—(heteroclinic orbit): F2≈6.1;η=1.0

Grahic Jump Location
Fig. 16

(top) Time history, (middle) the associated autocorrelation functions and (bottom) fast Fourier transform for time series data from the system in Eq. (7)—heteroclinic orbit): F2≈6.1;η=5.0

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Fig. 17

Poincaré map for homoclinic orbits with different noise amplitude: F2≈1.3 (first row) η=0.18; (second row) η=1.0; (third row) η=5.0

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Fig. 18

(bottom) Poincaré map for homoclinic orbits with (middle) the associated phase portrait and (top) time history (noise-free system, under the critical threshold): F2=1.1;η=0.0

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Fig. 19

Noisy bifurcation diagram for heteroclinic orbits with the associated zoom for F2 ∈ [0,...,4.5] (noisy system, under the critical threshold): η=0.18

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Fig. 20

(top) Poincaré map for heteroclinic orbits with (middle) the associated phase portrait and (bottom) time history (noise-free system, under the critical threshold): F2=5.9;η=0.0

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Fig. 21

(top) Poincaré map for heteroclinic orbits with (middle) the associated phase portrait and (bottom) time history (noisy system, F2 under the critical threshold): F2=5.9;η=5.0

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Fig. 22

Poincaré map for heteroclinic orbits (noisy system, F2 at the critical threshold): F2=6.1;η=0.1

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Fig. 23

Poincaré map for heteroclinic orbits (noisy system, F2 at the critical threshold): F2=6.1;η=1.0

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Fig. 24

Poincaré map for heteroclinic orbits: F2=8.0;η=0.0

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Fig. 25

Poincaré map for heteroclinic orbits: F2=8.0;η=1.0

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Fig. 26

Scaled power spectrum of u(t) oscillations in the system in Eq. (7)—(homoclinic orbits) for: F2=1.0 and noise intensity η=5.0

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Fig. 27

Scaled power spectrum of u(t) oscillations in the system in Eq. (7)—(homoclinic orbits) for: F2≈1.3 and noise intensity η=0.18

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Fig. 28

Scaled power spectrum of u(t) oscillations in the system in Eq. (7) —(heteroclinic orbits) for: F2=5.9 and noise intensity η=0.0

Grahic Jump Location
Fig. 29

Scaled power spectrum of u(t) oscillations in the system in Eq. (7) —(heteroclinic orbits) for: F2=6.1 and noise intensity η=0.1

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Fig. 30

Scaled power spectrum of u(t) oscillations in the system in Eq. (7) —(heteroclinic orbits) for: F2=6.1 and noise intensity η=5.0

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