Local finitely smooth equivalence of real autonomous systems with two pure imaginary eigenvalues (Q1946448)

It is considered the real autonomous system \[ \dot{\xi}=\frac{d\xi}{dt}=Q(\xi),\tag{1} \] where \(\xi, Q(\xi)\in\mathbb{R}^{n+2}\), \(n>0\), \(Q\) is \(C^\infty\) in a neighbourhood of the origin, \(Q(0)=0\), and the matrix \(Q'(0)\) has \(n\) eigenvalues lying outside the imaginary axis and a pair of pure imaginary (conjugate) eigenvalues. It is proved that, for systems having a focus on the center manifold, the problem of finitely smooth equivalence is solved in terms of finite segments of the Taylor series of their right-hand sides.