On a Dulac function for the Kukles system (Q610362)

Consider the planar system \[ \dot x=y,\;\dot y= h_0(x)+ h_1(x)y+ h_2(x) y^2+ h_3(x) y^3,\tag{\(*\)} \] where the functions \(h_i\) are sufficiently smooth and \(h_0(0)= 0\). The goal of the authors is to estimate the number of limit cycles of system \((*)\) and their location by means of a generalized Dulac function. The authors describe an analytic way (based on solving linear differential equations) and a numerical approach (based on solving linear programming problems) to construct such functions in some strip of the phase plane. The proposed methods are applied to some examples.