engleski

Linearization and Holder Continuity for Nonautonomous Systems

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.

nonautonomous dynamics ; linearization ; Grobman-Hartman theorem

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