The number of limit cycles from a cubic center by the Melnikov function of any order (Q2279552)

For the planar system \[ \dot{x}=y(1+x)^2-\varepsilon P(x,y),\;\; \dot{y}=-x(1+x)^2-\varepsilon Q(x,y), \] where \(P, Q\) are quadratic polynomials and \(\varepsilon\) is small, the authors obtain the upper bound three on the number of limit cycles. This is done by investigating the first four coefficients \(M_k(h)\) in the expansion with respect to \(\varepsilon\) of the displacement map related to the period annulus \(x^2+y^2=2h\in (0,1)\). Namely, \(M_1, M_2, M_3\) can produce up to two and \(M_4\) up to three limit cycles. If all they vanish, it is shown that the system becomes integrable, with a center manifold having 4 strata of co-dimension 6, 7 or 9.