Bifurcation theory of limit cycles by higher order Melnikov functions and applications (Q6559409)

For planar near Hamiltonian systems, the authors studied the maximum number of limit cycles that can emerge from Hopf and homoclinic bifurcations using higher order Melnikov functions. They applied the theoretical results to investigate a symmetric 8-loop Hamiltonian with perturbations up to the cubic nonlinearity. For the Hopf bifurcation, they found that the maximum number of limit cycles in the neighbourhood of two centers is six. This result is not new. For the homoclinic bifurcation, they found that the maximum number of limit cycles in the neighbourhood of the double homoclinic loop or one of the two symmetric homoclinic loops is eight. This result is new. As the total cyclicity of period annuli surrounding centers is known to be at most nine, it remains open whether nine limit cycles can appear in the symmetric 8-loop system considered in this paper.