Melnikov function and limit cycle bifurcation from a nilpotent center (Q924306)

The near-Hamiltonian planar system \(\dot{x}=H_y+\varepsilon f(x,y)\), \(\dot{y}=-H_x+\varepsilon g(x,y)\) is considered, where \(H\), \(f\) and \(g\) are polynomials in \(x,y\) and \(\varepsilon>0\) is a small parameter when the unperturbed system has a nilpotent center. It is assumed that the equation \(H(x,y)=h\) defines a closed curve \(L_h\) surrounding the origin for \(h\) in an interval \(I\subset\mathbb{R}\), which intersects the positive \(x\)-axis at the point \(A(h)=(a(h),0)\). Let \(B(h,\varepsilon)\) be the first intersection point of the positive orbit of the perturbed system starting at \(A(h)\) with the positive \(x\)-axis. Then \(H(B)-H(A)=\int_{AB}dH=\varepsilon[M(h)+O(\varepsilon)]\), where the Abel integral \(M(h)=\oint_{L_h}(g\,dx-f\,dy)\) is the first-order Melnikov function of the system. The aim of the article is to give an algorithm for computation the first coefficients of the expansion of \(M(h)\). Application of this algorithm is given and the number of limit cycles for a cubic system is obtained.