TWO GENERAL CENTRE PRODUCING SYSTEMS FOR THE POINCARÉ PROBLEM

Abstract

We consider the polynomial system $\td{x}{t}=-y-ax^{s+3}y^{n-s-3}-bx^{s+1}y^{n-s-1},$\, $\td{y}{t}=x+cx^{s+2}y^{n-s-2} + dx^sy^{n-s}$ where $n \ge 3$ is an odd integer and $s=0, \dots, n-3$ is an even integer. We calculate the first three nonzero Lyapunov coefficients for the system and obtain a Gr\obner basis for the ideal generated by them. Potential centre conditions for the system are obtained by setting the basis elements equal to zero and solving the resulting system. This gives five basic solutions and within this set we find two well known classes of centres and three new centre producing systems. One of the three is a variant of one of the other new systems, so we obtain two general independent systems which produce multiple centre conditions for each $n \ge 5.$