Obstacles to local smooth linearization of systems of ordinary differential equations (Q1916591)

The paper is devoted to the problem of the existence of a nondegenerate invariable diffeomorphism \(\xi= H(\eta)\) of the class \(C^k\), \(k\geq 1\), which reduces the system \(\dot\xi= A\xi+ F(\xi)\), \(|F(\xi)|= o(|\xi|)\) into the linear system \(\dot\eta=A \eta\). For the system \(\dot x= x\), \(\dot y= 2y\), \(\dot z= -z\), \(\dot w= 2w+ xy^2 z^3\) it is shown that \(k= 2\) is the greatest value for which such a diffeomorphism exists.