Control of Non-Linear Oscillations in Plasma Governed by a Van Der Pol Equation
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 is essential in 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 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.
GENERALITY ON THE DYNAMICAL SYSTEMS
The dynamical systems constitute a vast eld for science and study. Thus, their study is very important and it generates some interesting phenomena. And then, we will study some of the dynamical systems which are involved along the work.
A dynamical system is one of which the state is described by a vector ~x with n components and of which the evolution is governed by a simple differential equation of the type : _~ x = ?!F (~x) (1.1) For example, the Hamilton system is a dynamical system. In fact, in compact notation, we have: _~
x = J
= ?!F (~x): (1.2) with Fi(~x) = X+1 j=1 Jij @H @xj and
Where J is the Jacobian matrix and I the unit matrix of n order. The solution ~x(t) such that ~x(t = 0) = ?!x ?! 0 is called a good and it is written down like this: t( ?!x0) A x point (or equilibrium point or singular point) of a system described by the equation( 1.1) is a ?!xe which is solution of the equation: ?!F ( ?!xe) = ?!0 A x point ?!xe is called well if all the values of the Jacobian matrix of the ood linearized around this x point have their real part negative. GBEDO Yemalin Gabin, [email protected], AUST 2011 NOTION OF STABILITY 8 Similarly, if all the values of the Jacobian matrix have their real part positive, then the x point is called source. A phase space can help dene the state of a system by associating its coordinates such as its position and its speed.The trajectory of the phase space is a curve of the same space representing an evolution of the system. A set of trajectories constitutes a portrait of phase. An autonomous system is a system in which time doesn’t intervene in the equation of motion explicitly, i.e the system of the form for which independent variable doesn’t appears explicitly x = f(x; x_ ). On the contrary, we have to deal with a forced system. The self-maintained oscillations are oscillations in which the lost energy is recovered during the following cycle in order to maintain these oscillations. Hysteresis is a phenomenon during which there is a jump from a great amplitude to a smaller one for a solution and this, vice-versa. A system to approach a periodic behavior which will thus appear as a closed curve in phase space is called a limit cycle.
A limit cycle is a closed orbit in the phase space such that no other closed orbit can be found arbitrary close to it. It’s a characteristic for a periodic regime. A close trajectory C of a dynamical system which has nearby open trajectories spiraling towards it both from inside and outside as t ! 1 is called stable limit cycle. If they spiral towards it from one side and spiral out from the other side, it is semi – stable limit cycle. If nearby open trajectories spiral away from C on both side the C is unstable limit cycle. If nearby trajectories neither approach nor recede from C, it is Neutrally-stable limit cycle. An attractor is an invariable set towards which all the trajectories of the dynamical system are turned and by which they are attracted. It’s included in a eld of an existent volume which constitutes its attracting pool. Thus, we can have: single attractors (punctual attractors, periodic attractors, bi-periodic attractors or quasi-periodic attractors) and the strange attractors (non-periodic attractants, fractal attractants, chaotic attractants). The attracting point is a single point corresponding to a stationary solution of the equation of motion. The periodic behaviour is associated with a single steady attractor called limit cycle which is characterized by its amplitude and period. The third type of single attractor is the tore Tr(r 2) and it corresponds to a quasi-periodic regime having r frequencies of independent basis. The strange attractors characterized a chaotic movement. The attracting pool of an attractor is the location (setting) of the phase space formed by the set of initial conditions and from which this attractor is obtained.
1.2 NOTION OF STABILITY
The notion of stability is very important and fundamental in the study case of any system.A x point must satisfy certain criteria among which the one of stability. For a state to be observable, it must be stable, i.e this state must and an initial state after being subject to a GBEDO Yemalin Gabin