Perturbations of a kind of degenerate quadratic Hamiltonian system with saddle-loop (Q1600022)

Bifurcations of limit cycles in a plane quadratic Hamiltonian system with small quadratic perturbations \[ \dot{x}=-\partial_{y}H(x,y)+\varepsilon X(x,y,\varepsilon), \qquad\dot{y}=\phantom{-}\partial_{x}H(x,y)+\varepsilon Y(x,y,\varepsilon), \] are investigated. Here, \(X\), \(Y\) are polynomials of \(x, y\), depending analytically on the small parameter \(\varepsilon\). The unperturbed system \((\varepsilon=0)\) is supposed to have at least one center surrounded by a compact component of the algebraic curve \(H(x,y)=h\). The main result concerns the maximal number of limit cycles of such systems with degenerated Hamiltonian \(-H(x,y)=x(y^2+3(x-1)^2)\) in the finite plane.