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This work is devoted to study the existence of periodic solutions for afamily of discontinuous differential systems $Z(x,y;\epsilon)$ with many zones.We show that for $\epsilon$ sufficiently small the averaged functions at anyorder control the existence of crossing limit cycles for systems in thisfamily. We also provide some examples dealing with nonlinear centers.

Averaging theory at any order for computing limit cycles of discontinuous piecewise differential systems with many zones