LAUSR

Abstract When the first average function is non-zero we provide an upper bound for the maximum number of limit cycles bifurcating from the periodic solutions of the center x ˙ = − y ( ( x 2 + y 2 ) / 2 ) m and y ˙ = x ( ( x 2 + y 2 ) / 2 ) m with m ≥ 1 , when we perturb it inside a class of discontinuous piecewise polynomial vector fields of degree n with k pieces. The positive integers m , n and k are arbitrary. The main tool used for proving our results is the averaging theory for discontinuous piecewise vector fields.