Instability of isolated equilibria of dynamical systems with invariant measure in spaces of odd dimension (Q1974358)

Consider an autonomous system of differential equations in the space \({\mathbb R}^n\), where \(n\) is an odd number. It is assumed that the system has an invariant measure with smooth density and that \(x=0\) is an isolated rest point. It was conjectured by \textit{V. V. Ten} [Mosc. Univ. Mech. Bull. 52, No. 3, 1-4 (1997); translation from Vestn. Mosk. Univ., Ser. I 1997, No. 3, 40-43 (1997; Zbl 0916.34050)] that in this case, the rest point \(x=0\) is unstable. The authors prove this conjecture in the case where the system has a quasihomogeneous truncation for which \(x=0\) is an isolated rest point. They also construct an example of a system of the class \(C^\infty\) for which the conjecture fails.