# The parameterization method for center manifolds

## TL;DR

This paper extends the parameterization method to center manifolds at non-hyperbolic fixed points, providing a new proof of their existence and regularity, along with error bounds for approximations.

## Contribution

It generalizes the parameterization method to non-hyperbolic cases, enabling simultaneous computation of the center manifold and its conjugate system with error estimates.

## Key findings

- New proof of existence and regularity of center manifolds.
- Error bounds for approximations of the center manifold.
- Simultaneous computation of the center manifold and conjugate system.

## Abstract

In this paper, we present a generalization of the parameterization method, introduced by Cabr\'{e}, Fontich and De la Llave, to center manifolds associated to non-hyperbolic fixed points of discrete dynamical systems. As a byproduct, we find a new proof for the existence and regularity of center manifolds. However, in contrast to the classical center manifold theorem, our parameterization method will simultaneously obtain the center manifold and its conjugate center dynamical system. Furthermore, we will provide bounds on the error between approximations of the center manifold and the actual center manifold, as well as bounds for the error in the conjugate dynamical system.