UK Nonlinear
News, `February 1999`

# Recent thesis

### Lab. de Mathematiques et Physique theorique

#### Sebastien Neukirch

#### Supervisors: Professor Hector Giacomini

**Abstract**
This thesis is a study of differential equations
and
nonlinear dynamical systems in which dissipative dynamical systems
attractors are considered. 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 modelling Rayleigh-Benard convection.
The
importance of this system is that it was the first known example
exhibiting 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. The algebraic
structure of the integrals of motion is used 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 a Liénard system. Moreover, we obtain an
algebraic approximation and the multiplicity of each of the limit cycles.
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
system.

**Source**:
Sebastien Neukirch (`
seb@celfi.phys.univ-tours.fr`)

<< Move to `UK Nonlinear News` Issue
Fifteen Index Page
(February 1999).

Last Updated: 5th January 1999.

`uknonl@amsta.leeds.ac.uk`.