On a lemma of planar systems (Q1194500)

Consider the planar system \((*)\) \(dx/dt=P(x,y)\), \(dy/dt=Q(x,y)\), where \(P\) and \(Q\) are analytic at the origin \(O=(0,0)\). Assume that the origin \(O\) is an isolated critical point of (*). Let \(J(P,Q)\) denote the Jacobian of \(P\) and \(Q\). Cronin's Lemma says that: If \(J(P,Q)\geq 0\) in a neighborhood of the critical point \(O\) of \((*)\), then \(O\) has no hyperbolic sectors [see \textit{J. Cronin}, J. Differ. Equations 1, 150-170 (1965; Zbl 0145.334)]. In this paper the author considers the Liénard system \(dx/dt=y+\sqrt 2\;x^ 2\), \(dy/dt=-x^ 3\), which has \(J(P,Q)=3x^ 2\geq 0\) in the \(xy\)-plane. However the author proves that its critical point \(O\) has at least one hyperbolic sector. This fact implies that the above lemma does not hold.