论文标题
非自主耦合系统的平滑线性化
Smooth Linearization of Nonautonomous Coupled Systems
论文作者
论文摘要
在与帕尔默的联合合作中，我们已经制定了足够的条件，在该条件下，$ h_n（x，y）的形式存在连续且可逆的转换，采用耦合系统的解决方案\ begin \ begin {equation*} x_ {n+1} = a_nx_n+f_n+f_n+f_n（x_n，y_n，y_n），\ quad y__n（y_n），y_n y_ y _ = n+1} \ end {equation*}上关联的部分线性化的未耦合系统\ begin \ begin {equation*} x_ {n+1} = a_nx_n，\ quad y__ {n+1} = g_n（y__n）。 \ end {equation*}在当前的工作中，我们更进一步，并提供了$ h_n $和$ h_n^{ - 1} $的条件，在其中一个变量中，$ x $和$ y $是平滑的。 我们强调的是，我们的条件是一种通用形式，不涉及对大多数相关作品中存在的线性部分的任何形式的二分法，非共振或光谱差距假设。
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.
