Limit cycles for some families of smooth and non-smooth planar systems

We apply the averaging method in a class of planar systems given by linear center perturbed sum continuous homogeneous vector fields, to study lower bounds for their number limit cycles. Our results can be applied models where smoothness is lost on set Σ = { x y 0 } . They also motivate consider variant Hilbert 16th problem, goal bound cycles terms monomials family polynomial instead doing this degrees.