Parameterization of invariant manifolds : the Lorenz manifold

Abstract

This work is composed of three different parts. First of all, a deep study of the Lorenz equations is done, beginning with its physical deduction, continuing with its dynamical properties and ending with the discussion of three typical properties of chaotic attractors (Volume contraction, Local instability and global stability and how they are illustrated by the Lorenz system. The second part is based on Taylor’s method as a numerical integration method for the Lorenz differential equation system. The order of the expansion and the step size are the parameters to determine in order to have an error below a certain tolerance and a high computational efficiency. The last part is the one which gives the title to this project. Once we have a deep understanding of the dynamical system and a way to integrate it we can proceed to find an approximation for the invariant stable manifold using the parameterization method. A general theorem for the analytic case is first introduced and then the method is adapted to the Lorenz model, and hence obtaining a plot of this manifold.