Linearization and H\" older Continuity for Nonautonomous 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.

Revised version. Accepted for publication in the Journal of Differential Equations

References in corpus (1)

• A generalized Grobman-Hartman theorem for nonautonomous dynamics

Cited by in corpus (5)

• Smooth Linearization of Nonautonomous Coupled Systems
• On the linearization of infinite-dimensional random dynamical systems
• Multiscale linearization of nonautonomous systems
• A Hartman-Grobman theorem for algebraic dichotomies
• Linearization and Hölder continuity of generalized ODEs with application to measure differential equations