Upper bounds for the number of limit cycles through linear differential equations (Q997109)

Consider the differential system \[ \dot x= y,\quad\dot y= h_0(x)+ h_1(x) g+ h_2(x) y^2+ y^3\tag{\(*\)} \] in the plane, where the functions \(h_i\), \(i= 0,1,2\), are smooth. To any natural number \(n\geq 1\), the authors associate with \((*)\) two functions \(f_n(x,y)\) and \(M_n(x)\), where \(M_n\) is a solution of a linear differential equation of order \(n+1\) which does not change sign. Under some additional assumptions on \(f_n\) and \(M_n\) it is proved that any limit cycle of \((*)\) does not cut the curve \(f_n(x,y)= 0\) and that the number of limit cycles of \((*)\) in an \(\ell\)-connected component of \(\mathbb{R}^2\setminus\{f_n(x, y)= 0\}\) is at most \(\ell\), and that all these limit cycles are hyperbolic. The proof is based on the Bendixson-Dulac criterion for \(\ell\)-connected sets. Some concrete examples are treated.