Linearization and H\" older Continuity for Nonautonomous Systems

TL;DR

This paper establishes conditions for transforming nonautonomous systems into partially linearized forms and proves Hölder continuity of these transformations, extending previous results to more general settings.

Contribution

It generalizes existing linearization results for nonautonomous systems by providing broader conditions and includes Hölder continuity of the transformation and its inverse.

Findings

Provided a transformation to linearize nonautonomous systems under general conditions.

Proved Hölder continuity of the transformation and its inverse.

Extended results to discrete systems.

Abstract

We consider a nonautonomous system \[ \dot x=A(t)x+f(t,x,y),\quad \dot y = g(t,y)\] and give conditions under which there is a transformation of the form $H(t,x,y)=(x+h(t,x,y),y)$ taking its solutions onto the solutions of the partially linearized system \[ \dot x=A(t)x,\quad \dot y = g(t,y).\] Shi and Xiong \cite{SX} proved a special case where $g(t,y)$ was a linear function of $y$ and $\dot x=A(t)x$ had an exponential dichotomy. Our assumptions on $A$ and $f$ are of the general form considered by Reinfelds and Steinberga \cite{RS}, which include many of the generalizations of Palmer's theorem proved by other authors. Inspired by the work of Shi and Xiong, we also prove H\" older continuity of $H$ and its inverse in $x$ and $y$. Again the proofs are given in the context of Reinfelds and Steinberga but we show what the results reduce to when $\dot x=A(t)x$ is assumed to have an exponential dichotomy. The paper is concluded with the discrete version of the results.