Bifurcations of limit cycles in Kukles systems of arbitrary degree with invariant ellipse (Q712585)

The paper deals with an upper bound of limit cycles for a particular class of Kukles polynomial systems of arbitrary degree \(n\) \[ \dot{x}=-y, \; \dot{y}= -ae^2 + x(1-e^2)+(x^2+y^2-e^2(x+a)^2)\Big(-\frac{1}{a}+ \sum_{i=1}^{n-2} (q_{i0}x^i+q_{0i}y^i)\Big) \] with invariant ellipse \(x^2+y^2-e^2(x+a)^2=0\). It is proved that, for certain values of the parameters, the system has one algebraic limit cycle. Writing the system as a perturbation of a Hamiltonian system, the authors show that the first Poincaré-Melnikov integral of the system is a polynomial whose coefficients are the Lyapunov quantities. By using this approach, the maximal number of rest limit cycles is obtained.