Global Centers in Piecewise linear Differential Equations in the Cylinder

We characterize global centers (all solutions are periodic) of the piecewise linear equation $x^{\mathrm{\prime }}=a(t)\mathrm{\mid }x\mathrm{\mid }+b(t)$ when the coefficients $a,b$ are trigonometric polynomials, under some generic hypotheses. We prove that the global centers are those determined by the composition condition on $a,b$. That is, the equation has a global center if and only if there exist polynomials $P,Q$ and a trigonometric polynomial $h$ such that $a(t)=P(h(t))h^{\mathrm{\prime }}(t)$, $b(t)=Q(h(t))h^{\mathrm{\prime }}(t)$.