Global Centers in Piecewise Linear Equations in the Cylinder

We characterize global centers (all solutions are periodic) of the piecewise linear equation $$x'=a(t)|x| + b(t)$$ x ′ = a ( t ) | x | + 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 $$a(t)=P(h(t))h'(t)$$ a ( t ) = P ( h ( t ) ) h ′ ( t ) , $$b(t)=Q(h(t))h'(t)$$ b ( t ) = Q ( h ( t ) ) h ′ ( t ) .