Les attracteurs des systèmes dynamiques dissipatifs de Lorenz et de Liénard : nombre, forme et localisation

Abstract : This thesis deals with the study of differential equations and nonlinear dynamical systems. A study of dissipative dynamical systems' attractors is presented. In particular, the chaotic Lorenz attractor and the limit cycles of Liénard systems are studied. The first part is dedicated to the Lorenz system. This system is obtained when simplifying the Boussinesq equation involved in the Rayleigh-Bénard convection. The importance of the Lorenz systems lies in the fact that it is the first one to exhibit a chaotic flow. We make use of transverse sections (surfaces or curves that are crossed by the flow in only one direction) to gain information on the chaotic attractor of the system. We use the algebraic structure of the integrals of motion to find the equations of the transverse sections. These transverse sections allow us to give algebraic bounds to the spread of the attractor when it exists but also to give ranges of values of the parameters for which no chaotic behavior is possible. The second part introduce a simple algorithm which gives the number of limit cycles in Liénard systems. Moreover, we obtain an algebraic approximation and the multiplicity of each of the limit cycle. This algorithm is not perturbative as it does not need a small parameter to work. In fact it changes the initial problem of solving differential equations into searching the number of roots of a one variable polynomial. Furthermore we obtain, thanks to this algorithm, algebraic approximations to the bifurcation curves (Hopf, saddle-node, heteroclinic) of the Liénard systems.
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Theses
Physics [physics]. Université Pierre et Marie Curie - Paris VI, 1998. French


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Submitted on : Tuesday, March 2, 2004 - 1:34:32 PM
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Sebastien Neukirch. Les attracteurs des systèmes dynamiques dissipatifs de Lorenz et de Liénard : nombre, forme et localisation. Physics [physics]. Université Pierre et Marie Curie - Paris VI, 1998. French. <tel-00005184>

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