Four limit cycles in quadratic two-dimensional systems with a perturbed first-order weak focus (Q606563)

Consider the planar system \[ \begin{cases} {dx\over dt} &= x^2+ xy+ y,\\ {dy\over dt} &= a_2x^2+ b_2xy+ c_2y^2+ {x\over \varepsilon}.\end{cases}\tag{\(*\)} \] The author reduces \((*)\) to a Liénard system and proves under some conditions on the coefficients \(a_2\), \(b_2\) and \(c_2\) that for sufficiently small \(\varepsilon>0\) system \((*)\) has four limit cycles: one to the left of the straight line \(x=-1\) and two to the right of it, the fourth is located in a small neighborhood of the origin.