Stability criterion for the resonance 1:3 (Q923193)

The paper is devoted to the investigation of fourth order systems \(\dot u=f(u)\), \(f(0)=0\), \(u\in {\mathbb{R}}^ 4\), where the matrix \((\partial f/\partial u)|_{u=0}\) has the two pairs of purely imaginary eigenvalues \(\pm i\omega\), \(\pm 3i\omega\). By passing to an appropriate factor system the author obtains a stability criterion which is slightly more general than a criterion given by \textit{L. G. Khazin} and \textit{Eh. Eh. Shnol'} [Prikl. Mat. Mekh. 44, 229-237 (1980; Zbl 0465.34033)]. Nevertheless, as the author regrets, the question of an exact analytic solution of the stability problem in this case remains open.