Linear estimate for the number of limit cycles of a perturbed cubic polynomial differential system

Abstract

Perturbing the cubic polynomial differential systems x ̇ = − y ( a 1 x + a 0 ) ( b 1 y + b 0 ) , y ̇ = x ( a 1 x + a 0 ) ( b 1 y + b 0 ) having a center at the origin inside the class of all polynomial differential systems of degree n , we obtain using the averaging theory of second order that at most 17 n + 15 limit cycles can bifurcate from the periodic orbits of the center.