Limit Cycles in a Class of Generalized Relaxation Oscillators

This paper considers the Hopf bifurcation, double homoclinic loop bifurcation and Poincaré bifurcation of the generalized relaxation oscillators [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text]. The unperturbed system is a Hamiltonian system with a double homoclinic loop emerging through a nilpotent saddle. It is proved that the Hopf cyclicity at each center is 4. The system has at least 10 limit cycles bifurcating from the double homoclinic loop and at least 3 limit cycles bifurcating from the unbounded periodic annulus. The lower bound for the number of coexisting limit cycles bifurcating from the entire plane is at least 13. For the Poincaré bifurcation, it is proved that the system has at most 10 limit cycles bifurcating from each bounded periodic annulus and at most 11 limit cycles bifurcating from the unbounded periodic annulus.