On the limit cycles of the floquet differential equation

We provide sufficient conditions for the existence of limit cycles for the Floquet differential equations x(t) = Ax(t) + ε(B(t)x(t) + b(t)), where x(t) and b(t) are column vectors of length n, A and B(t) are n × n matrices, the components of b(t) and B(t) are T-periodic functions, the differential equation x(t) = Ax(t) has a plane filled with T-periodic orbits, and ε is a small parameter. The proof of this result is based on averaging theory but only uses linear algebra.