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

Abstract

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 II n=0 in the Chinese classification of quadratic systems. Bifurcation sets δ = δ*( l, m) ( m > 2, l > 0) and δ = δ sep( l, m) (l 2 ≥ 4 if m = −1, ∨ m ≠ −1) are proved to exist corresponding to a semistable limit cycle and a separatrix cycle appearing in II n = 0 respectively. The asymptotic behaviour of δ* and δ sep 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.