Résumé:
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
