On the pseudonormal form of real autonomous systems with two pure imaginary eigenvalues (Q1946429)

The author considers the real autonomous system \[ \dot{\xi}=\frac{d\xi}{dt}=Q(\xi),\tag{1} \] where \(\xi, Q(\xi)\in\mathbb{R}^{n+2}\), \(n>0\), \(Q\) is a function of class \(C^{\infty}\) in some neighbourhood of the origin, \(Q(0)=0\), and the matrix \(\tilde{A}=Q'(0)\) has \(n\) eigenvalues outside the imaginary axis and a pair of pure imaginary eigenvalues. The aim of the article is to define what the pseudonormal form looks like and to construct a transformation reducing system (1) to its pseudonormal form in some neighbourhood of the origin. The notion of resonance is refined, and the notions of removable and irremovable resonances are introduced.