Hopf and homoclinic bifurcations for near-Hamiltonian systems (Q729887)

For perturbed planar Hamiltonian systems of the form \[ \dot x= H_y +\varepsilon f (x, y, a), \quad \dot y =-H_x +\varepsilon g(x, y, a) \] the authors provide a method to compute, under some conditions, all the coefficients of the development of the first order Melnikov function close to a homoclinic loop. Such coefficients allow to study the number of zeros of the Melnikov function and, as a consequence, the number of limit cycles bifurcating from periodic orbits. Two cases are considered: the case of a homoclinic loop surrounding a center and the case of a double homoclinic loop surrounding two centers. Finally, the method is applied to determine the number of limit cycles for the perturbed Liénard system \[ \dot x= y +\varepsilon f (x, y), \quad \dot y = x-x^3 +\varepsilon g(x, y), \] where \(f\) and \(g\) are polynomials which make the system centrally symmetric, according to the degree of such polynomials.