Smooth Linearization of Nonautonomous Coupled Systems

TL;DR

This paper establishes conditions for smooth transformations that linearize nonautonomous coupled systems, extending previous work by removing spectral gap and nonresonance restrictions.

Contribution

It provides general conditions for smooth linearization of coupled systems without relying on spectral gap or dichotomy assumptions.

Findings

Existence of smooth invertible transformations for coupled systems.

Extension of previous linearization results to higher regularity.

No spectral gap or nonresonance conditions required.

Abstract

In a joint work with Palmer we have formulated sufficient conditions under which there exist continuous and invertible transformations of the form $H_n(x,y)$ taking solutions of a coupled system \begin{equation*} x_{n+1} =A_nx_n+f_n(x_n, y_n), \quad y_{n+1}=g_n( y_n), \end{equation*} onto the solutions of the associated partially linearized uncoupled system \begin{equation*} x_{n+1} =A_nx_n, \quad y_{n+1}=g_n( y_n). \end{equation*} In the present work we go one step further and provide conditions under which $H_n$ and $H_n^{-1}$ are smooth in one of the variables $x$ and $y$. We emphasise that our conditions are of a general form and do not involve any kind of dichotomy, nonresonance or spectral gap assumptions for the linear part which are present on most of the related works.