Le nombre de cycles limites des centres quadratiques perturbés

Abstract

In this work, we first study the limit cycles which can bifurcate from the periodic orbits of the quadratic isochronous centers 𝒙̇ = −𝒚 + 𝒙 𝟐 , 𝒚̇ = 𝒙 + 𝒙𝒚, and 𝒙̇ = −𝒚 + 𝒙 𝟐 − 𝒚 𝟐 , 𝒚̇ = 𝒙 + 𝟐𝒙𝒚, when they are perturbed inside the class of all discontinuous quadratic polynomial differential systems with the straight line of discontinuity y  0 . In the second part of this work, we use the averaging method to study the maximum number of limit cycles of the differential system 𝒙̇ = −𝒚 + 𝒙𝒚 + ∑𝜺 𝒌 ∑ 𝒂𝒊,𝒋 (𝒌) 𝒙 𝒊𝒚 𝒋 𝒊+𝒋=𝟐 𝟐 𝒌=𝟏 , 𝒚̇ = 𝒙 + 𝒚 𝟐 + ∑𝜺 𝒌 ∑ 𝒃𝒊,𝒋 (𝒌) 𝒙 𝒊𝒚 𝒋 𝒊+𝒋=𝟐 𝟐 𝒌=𝟏 , where  is a sufficiently small parameter. Keywords: Limit cycle, quadratic isochrone center, differential system, averaging method