Limit cycle bifurcations by perturbing a kind of quadratic Hamiltonian system having a homoclinic loop (Q6614238)

Given a planar quadratic Hamiltonian system with an annulus of periodic orbits enclosed by a homoclinic orbit, perturbations of this system can generate limit cycles. The authors studied the number of limit cycles arising from perturbations defined by polynomial functions on each half-plane. This problem is related to the weakened 16th Hilbert problem. Furthermore, the number of limit cycles is determined by the zeros of the Melnikov function.\N\NUsing the Chebyshev property of the Melnikov function, the authors derive an upper bound on the number of limit cycles that bifurcate from the annulus of periodic orbits. They then perform careful manipulations and reformulations of the Melnikov function around the homoclinic orbit. In particular, they demonstrate that the Melnikov function around the homoclinic orbit is a linear combination of linearly independent analytic functions. Finally, they obtain a lower bound on the number of limit cycles around the homoclinic orbit.