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In this paper, we apply the averaging method via Brouwer degree in a class ofplanar systems given by a linear center perturbed by a sum of continuoushomogeneous vector fields, to study lower bounds for their number of limitcycles. Our results can be applied to models where the smoothness is lost onthe set $\Sigma=\{xy=0\}$. We also apply them to present a variant of Hilbert16th problem, where the goal is to bound the number of limit cycles in terms ofthe number of monomials of a family of polynomial vector fields, instead ofdoing this in terms of their degrees.

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