M ay 2 00 5 On the stability of limit cycles for planar differential systems . ∗

We consider a planar differential system ˙ x = P (x, y), ˙ y = Q(x, y), where P and Q are C 1 functions in some open set U ⊆ R 2 , and ˙ = d dt. Let γ be a periodic orbit of the system in U. Let f (x, y) : U ⊆ R 2 → R be a C 1 function such that P (x, y) ∂f ∂x (x, y) + Q(x, y) ∂f ∂y (x, y) = k(x, y) f (x, y), where k(x, y) is a C 1 function in U and γ ⊆ {(x, y) | f (x, y) = 0}. We assume that if p ∈ U is such that f (p) = 0 and ∇f (p) = 0, then p is a singular point. We prove that T 0 ∂P ∂x + ∂Q ∂y (γ(t)) dt = T 0 k(γ(t)) dt, where T > 0 is the period of γ. As an application, we take profit from this equality to show the hyperbolicity of the known algebraic limit cycles of quadratic systems.