The number of limit cycles from a cubic center by the Melnikov function of any order

Abstract In this paper, we consider the system x ˙ = y ( 1 + x ) 2 − ϵ P ( x , y ) , y ˙ = − x ( 1 + x ) 2 + ϵ Q ( x , y ) where P ( x , y ) and Q ( x , y ) are arbitrary quadratic polynomials. We study the maximum number of limit cycles bifurcating from the periodic orbits by using the Melnikov function of any order. We prove that the upper bound for the number of limit cycles is 3 and reached.