Solutions périodiques de certaines classes d'équations différentielles perturbées

Abstract

In this work, we consider the limit cycles of a class of polynomial differential systems of the form {<K1.1/>┊ <K1.1 ilk="MATRIX" > u=v-ε(g₁¹(u)+f₁¹(u)v)-ε²(g₁²(u)+f₁²(u)v), v=-u-ε(g₂¹(u)+f₂¹(u)v)-ε²(g₂²(u)+f₂²(u)v), </K1.1> where g₁¹,g₁²,f₁¹,f₁²,g₂¹,g₂²,f_{2 }¹and f₂² are polynomials of a given degree and ε is a small parameter. We obtain the maximum number of limit cycles that bifurcate from the periodic orbits of a linear center u=v, v=-u, by using the averaging theory of first and second order. This work addresses a special case of the second part of Hilbert's 16th problem, which remains unsolved in general. The results contribute to the qualitative understanding of perturbed planar polynomial systems.