Separatrix Cycles and Multiple Limit Cycles in a Class of Quadratic Systems

In this paper a class of quadratic systems is studied. By quadratic systems we mean autonomous quadratic vector fields in the plane. The class under consideration is class IIn=0 in the Chinese classification of quadratic systems. Bifurcation sets δ = δ*(l, m) (m > 2, l > 0) and δ = δsep(l, m) (l2 ≥ 4 if m = −1, m ≠ −1) are proved to exist corresponding to a semistable limit cycle and a separatrix cycle appearing in IIn = 0 respectively. The asymptotic behaviour of δ* and δsep is investigated if (l, m) tend to the boundary of its domain of existence. Especially the case of large parameters, which is related to singularly perturbed differential equations (relaxation oscillations), is considered. After a blowing up of the variables the problem is studied with the use of Pontryagin-integral techniques for bifurcation of limit cycles from Hamilton systems.