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.