Invariant manifolds of maps and vector fields with nilpotent parabolic tori

TL;DR

This paper studies invariant manifolds of analytic maps and vector fields with nilpotent parabolic tori, providing conditions for their existence, analyticity, and algorithms for their approximation, with applications in dynamical systems.

Contribution

It introduces new conditions for the existence and analyticity of invariant manifolds near nilpotent parabolic tori and offers effective computational methods.

Findings

Invariant manifolds exist under specific nonlinear coefficient conditions.

Invariant manifolds are analytic away from the torus.

Algorithms for approximating invariant manifolds are developed.

Abstract

We consider analytic maps and vector fields defined in $\mathbb{R}^2 \times \mathbb{T}^d$, having a $d$-dimensional invariant torus $\mathcal{T}$. The map (resp. vector field) restricted to $\mathcal{T}$ defines a rotation of frequency $\omega$, and its derivative restricted to transversal directions to $\mathcal{T}$ does not diagonalize. In this context, we give conditions on the coefficients of the nonlinear terms of the map (resp. vector field) under which $\mathcal{T}$ possesses stable and unstable invariant manifolds, and we show that such invariant manifolds are analyitic away from the invariant torus. We also provide effective algorithms to compute approximations of parameterizations of the invariant manifolds, and present some applications of the results.