A note on linear differential equations with periodic coefficients

We consider linear homogeneous differential equations of the form ẋ=A(t)x where A(t)A(t) is a square matrix of C1C1, real and TT-periodic functions, with T>0T>0. We give several criteria on the matrix A(t)A(t) to prove the asymptotic stability of the trivial solution to equation ẋ=A(t)x. These criteria allow us to show that any finite configuration of cycles in RnRn can be realized as hyperbolic limit cycles of a polynomial vector field.