Invariant manifolds of parabolic fixed points (II). Approximations by sums of homogeneous functions

TL;DR

This paper presents an algorithm to approximate invariant manifolds of parabolic fixed points using sums of homogeneous functions, applicable to both differentiable and analytic cases, and explores parameter dependence.

Contribution

The paper introduces a novel method for approximating invariant manifolds as sums of homogeneous functions, extending previous approaches to parabolic fixed points.

Findings

Algorithm successfully computes approximations as sums of homogeneous functions.

Approximations are valid for both differentiable and analytic cases.

Parameter dependence of invariant manifolds is analyzed.

Abstract

We study the computation of local approximations of invariant manifolds of parabolic fixed points and parabolic periodic orbits of periodic vector fields. If the dimension of these manifolds is two or greater, in general, it is not possible to obtain polynomial approximations. Here we develop an algorithm to obtain them as sums of homogeneous functions by solving suitable cohomological equations. We deal with both the differentiable and analytic cases. We also study the dependence on parameters. In the companion paper, Invariant manifolds of parabolic fixed points (I), these approximations are used to obtain the existence of true invariant manifolds close by. Examples are provided.