– Control of Non-Linear Oscillations in Plasma Governed by A Van Der Pol Equation –
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ABSTRACT
This thesis deals with the control of non-linear oscillations in plasma governed by a classical Van der Pol equation. The main interest devoted to such an investigation is that non-linear oscillations in plasma are essential in the industry.
In chapter 1, we present some generality on the dynamical systems. Chapter 2 is focussed on the analytical study of the Van der Pol equation in the autonomous
regime. The amplitude and the phase of the stable limit cycle are derived using the Averaging Method . In chapter 3, we investigate the oscillations in plasma.
TABLE OF CONTENTS
Dedication 1
Acknowledgements 2
Abstract 5
General introduction 6
1 GENERALITY ON THE DYNAMICAL SYSTEMS 7
1.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 NOTION OF STABILITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.1 GLOBAL APPROACH . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.2 LOCAL APPROACH . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 The concept of bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Notion of chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Notion of the Lyapunov exponent . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5.1 Application to a logstic map . . . . . . . . . . . . . . . . . . . . . . . . 12
2 ANALYTICAL STUDY OF THE VAN DER POL EQUATION IN THE
AUTONOMOUS REGIME 14
2.1 Descripion Van der Pol oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Analytical study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 Fixed Points and Stability . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 Existence of the limit cycles . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Oscillations in plasma 19
3.1 The classical Van der Pol equation . . . . . . . . . . . . . . . . . . . . . . . . 19
General conclusion 23
Bibliography 24
INTRODUCTION
The dynamical systems constitute a very large field of science [1, 2, 3, 4, 5]. They are generally represented by non-linear equations. A phenomenon is non-linear when his evolution doesn’t obey a linear Mathematics law.
The non-linear electric oscillators are those whom the one of the constitutive elements to the characteristics intensity – tension is of the non-linear form. We can mention (diode, transistor, operational amplier, etc) The presence of the non-linear components is in the beginning of many observed phenomena.
We can also mention (hysteresis, resonance [6, 7]) Another important phenomenon resulting of the presence of non-linear components is the apparition of chaos, curious phenomenon, rich and complex met almost in all branches of instruction: electronic, astronomy, biology, chemistry, economy, etc Particularly in electronic, it is known the works of a Dutch electrical engineer Balthazar Van der Pol on an oscillator presenting a various mode[8] whom equation is.
BIBLIOGRAPHY
[1] A. H. NAYFEH AND D. T. MOOK , NONLINEAR OSCILLATIONS (WILL , NEW YORK , 1979).
[2] J. J. COLLINS AND I. N. STEWART , J. MATH. BIOL. 30 , 827 (1992)
[3] J. J. COLLINS AND I. N. STEWART , J. NONLINEAR Sciences 3 , 349 (1993)
[4] J. J. COLLINS AND I. N. STEWART , J. BIOL. CYBERN. 68 , 287 (1993)
[5] I. PASTOR , VICTOR M. PEREZ-GARCIA , F. ENCINAZ-SANZ , and J. M. GUERRA, PHYS. REV. E48 , 171 (1993)
[6] JAMES GLEICK , theory of chaos” , ALBIN MICHEL S. A. (1989)
[7] I. PERCIVAL NEW SCIENTIST 21 (1989)
[8] B. Van der Pol , PHILOS. MAG. 43 , 700 (1922) ; 7-2 , 978 (1926) ; 7-3 , 65 (1927)
[9] Husch,M.W,and SMALE ,S. , 1974 , Dierential equations , Dynamical systems and linear algebra “,ACADEMIC PRESS .
[10] HASLER M.J. Electrical circuit with chaotic behavior ” PROC.IEEF ,75 , PP.1009 – 10021 , (1987) .
[11] B. Van der Pol , Phil. Mag. 3 , 64 (1927) ,
[12] ALI HASSAN NAYFEH , Introduction to perturbation techniques .
[13] J.Proud and al . , Plasma Processing of Materials : Scientic opportunities and Technologies Challenges . National academy press , Washington D . C (1991)
[14] Richard Dendy , Plasma Physics : An introduction course Cambridge University Press (1993)
[15] B. E. Keen and W. H. Fletcher , Phys , Rev. Lett 23 , 14 (1969) GBEDO
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