The parameterization method for invariant manifolds of real-analytic dynamical systems

Abstract

[en] The main goal of this work is to devolope the parameterization method, which was introduced by X. Cabré, E. Fontich and R. de la Llave [HCF+16]. It is an important tool to study diverse invariant manifolds attached to fixed points in different contexts. To be acquainted with the parameterization method, we divide the work into two studies which are as follows. In the first study, we aim to prove the existence and regularity of invariant manifolds. Furthermore, we also demonstrate that the parameterization method in different contexts can reach to obtain different kinds of invariant manifold at fixed points. As a first simple application, the method allows us to give a quick proof of (un)stable manifolds theorems. For instance, the existence of a real-analytic one-dimensional stable manifolds at the origin for maps or a 2D stable manifolds for flows. Once, we proved the existence of the manifolds. The second study is to emphasize the computational aspects derived from the application of the parameterization method. We would like to work out the coefficients of the invariant manifold expanded in series and sketch the approximation of the invariant manifolds by using computer programs. We can get an efficient algorithm for numerical computation of invariant manifolds base on the parameterization method.