A necessary condition for a local smooth linearization of a system of differential equations (Q1386570)

The author considers systems \[ \dot x=Ax+\Phi(x)\tag{1} \] with \(\Phi\) of class \(C^\infty\) in a neighbourhood of zero, \(\|\Phi(x)\|=o(\| x\|)\), \(A\) having no eigenvalues on the imaginary axis. Necessary (and in certain cases sufficient) conditions are given for the existence of a \(C^k\) transformation (\(k\geq 1\)) \(x=H(y)\) in a neighbourhood of zero which reduces system (1) to the linear form \(\dot y=Ay\).