A sufficient condition for a polynomial centre to be global (Q1192794)

The problem of global centre is studied for the differential system in the plane \((S)\) \(\dot x=P(x,y)\), \(\dot y=Q(x,y)\) where \(P\), \(Q\) are real polynomials of degree \(\leq n\), \(P(x,y)=\sum_{j=0}^ n p_ j(x,y)\), \(Q(x,y)=\sum_{j=0}^ n q_ j(x,y)\), in which \(p_ j\) and \(q_ j\) are homogeneous polynomials of degree \(j\). It is assumed that \(p_ n^ 2+q_ n^ 2\not\equiv 0\) and that \(P\) and \(Q\) have no non-constant common factors. An isolated critical point \(O\) of \((S)\) is said to be a centre if every orbit in a neighborhood of \(O\) is a cycle. It is a global centre if every orbit different from \(O\) is a non-trivial cycle. A sufficient condition for a centre to be a global centre is given. Theorem. If \(O\) is a centre of \((S)\), and \((\tilde S)\), the Poincaré extension of \((S)\) to the two dimensional sphere, has no critical point other than \(O^ +\) and \(O^ -\), then \(O\) is a global centre. A couple of corollaries are obtained. One applies to the cubic system \((S_ \varepsilon)\) \(\dot x=y+Ax^ 3+Bx^ 2y+Cxy^ 2+Dy^ 3\), \(\dot y=-\varepsilon x+kx^ 3+Lx^ 2y+Mxy^ 2+Ny^ 3\) with \(\varepsilon=1\) or 0. Corollary 1. If \(O\) is a centre of \((S_ \varepsilon)\) and the polynomial \(H_ 3=xq_ 3(x,y)-yP_ 3(x,y)\) is negative definite, then \(O\) is a global centre. If \(H_ 3\) has a positive value, then \(O\) is not a global centre. This Corollary is easily extended to a class of polynomial systems of odd degree: Corollary 2. If \(O\) is a centre, the polynomials \(H_ j=xq_ j(x,y)-yp_ j(x,y)\), \(j=1,2,\dots,n\), have the same sign, and \(H_ n\) has no zeros, then \(O\) is a global centre.