Consider real planar polynomial differential systems of the form \[ {dx\over dt}= P(x,y),\quad {dy\over dt}= Q(x,y)\tag{1} \] with \(P(0,0)= Q(0,0)= 0\). The authors present an algorithm consisting of six steps to construct the boundaries of a Poincaré-Bendixson region to establish existence and location of a limit cycle to (1). The boundaries consists of curves described by the relations \[ f(x,y):= 1+ s_1x+ s_2y+ s_3 x^2+ s_4 xy+ s_5 y^2= 0, \] where the real coefficients \(s_i\), \(1\leq i\leq 5\), are to determine. The first two steps consist in constructing a local solution of (1) to any initial value \((a_0,b_0)\) in the form \[ \widetilde x(t)= a_0+ \sum^N_{i=1} a_i t^i,\quad\widetilde y(t)= b_0+ \sum^N_{i=1} b_i t^i \] with \(N\geq 4\) and in gluing the curve \(f(x,y)= 0\) to this solution.
Several examples are given to demonstrate the applicability of the presented algorithm.