Finitely smooth normal form of an autonomous system with two pure imaginary roots (Q881092)

The author considers the real autonomous system \[ \dot\xi=\frac{d\xi}{dt}=Q(\xi),\tag{1} \] where \(\xi, Q(\xi)\in\mathbb{R}^{n+2}\;(n>0).\) \(Q(\xi)\) is an analytic function in a neighborhood of the origin, \(Q(0)=0\) and the matrix \(Q'(0)\) has \(n\) eigenvalues \(\lambda_1,\dots,\lambda_n\) lying outside the imaginary axis and a pair of imaginary eigenvalues \(\lambda_{n+1}=iw,\,\lambda_{n+2}=-iw.\) Using the standard linear transformation, system (1) is reduced to the following form \[ \begin{matrix} \frac{dx_j}{dt}&=&\varepsilon_jx_{j-1}+\lambda_jx_j+f_j(x,y),\quad j=1,\cdots,n\\ \frac{dy_j}{dt}&=&\lambda_{n+j}y_{j}+g_j(x,y),\quad j=1,2 \end{matrix}\tag{2} \] where \(x_j\) is a complex hyperbolic coordinate and \(y_j\) is a complex degenerate coordinate. In system (2), it is assumed that the two-dimensional invariant central manifold has a focus (spirals around the singular point); this system is not analytic but of class \(C^{\infty}.\) The author proves three existence theorems of nondegenerate transformation of class \(C^\infty\) reducing system (2) to a normal form. The cases either resonance or nonresonance relations are studied separately.