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PhD thesis
 thèse de doctorat

Title
''Dissipative dynamical systems : the shape of the Lorenz chaotic attractor and the number of Li\'enard limit cycles''

Date
November 1998, 178 pages

Advisor
Hector Giacomini, professor at the university de Tours, France

Honors
'Félicitations du jury'

Keys words

Differential equations

Lorenz system

Dynamical systems

Liénard system

Chaos

Attractor

Limit cycle

Algorithm

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.

Jury

Referee (see report): Freddy Dumortier, Professeur
Limburgs Universitair Centrum
Universitaire Campus B-3590
Diepenbeek,Belgique
32 11 26 80 04
fdumorti@luc.ac.be

Referee (see report): Jaume Llibre, Professeur
Departament de Matematiques
Universitat autonoma de Barcelona
08193 Bellaterra, Barcelona, Espagne
jllibre@mat.uab.es

Javier Chavarriga, Professeur
Departament de Matematica
Universitat de Lleida
Palca Victor Siurana, 1
25003 Lleida, Espagne
34 73 70 21 15
dinamics@eup.udl.es

Bernard Derrida, Professeur
Laboratoire de Physique Statistique
Ecole Normale Superieure
24, rue Lhommond
75006 Paris
Bernard.Derrida@lps.ens.fr

Hector Giacomini, Professeur
Laboratoire de matématiques et physique théorique
Faculté des Sciences, Université de Tours
37002 Tours, France
02 47 36 69 41
giacomin@celfi.phys.univ-tours.fr

Miguel A. F. Sanjuan, Professeur
Escuela Superior de Ciencias Experimentales y Tecnología
Universidad Rey Juan Carlos
Camino de Humanes 63 Fax : (34) 916476404
28936 Mostoles, Madrid Tfno.: (34) 916476193
España-Spain
msanjuan@escet.urjc.es

Laurent Véron, Professeur
Laboratoire de matématiques et physique théorique
Faculté des Sciences, Université de Tours
37002 Tours, France
02 47 36 69 46
veron@univ-tours.fr

 

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