More on the parameterization method for center manifolds

TL;DR

This paper extends the parameterization method for center manifolds to various settings, including bifurcations, ODEs, and reaction diffusion equations, enabling detailed analysis of center dynamics.

Contribution

It generalizes the parameterization method to center manifolds in multiple contexts, including bifurcation points and reaction diffusion systems.

Findings

Parameter-dependent center manifolds near bifurcations identified

Method applied to reaction diffusion equations for qualitative analysis

Enhanced understanding of conjugate dynamics in normal form

Abstract

In a previous paper we generalized the parameterization method of Cabr\'{e}, Fontich and De la Llave to center manifolds of discrete dynamical systems. In this paper, we extend this result to several different settings. The natural setting in which center manifolds occur is at bifurcations in dynamical systems with parameters. Our first results will show that we can find parameter-dependent center manifolds near bifurcation points. Furthermore, we will generalize the parameterization method to center manifolds of fixed points of ODEs. Finally, we will apply our method to a reaction diffusion equation. In our application, we will show that the freedom to obtain the conjugate dynamics in normal form makes it possible to obtain detailed qualitative information about the center dynamics.