engleski

Smooth Linearization of Nonautonomous Coupled Systems

In a joint work with Palmer \cite{; ; BDP}; ; 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.

linearization ; differentiability ; nonautonomous coupled systems

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