A note on linear differential equations with periodic coefficients

Abstract

We consider linear homogeneous differential equations of the form x ̇ = A ( t ) x where A ( t ) is a square matrix of C 1 , real and T -periodic functions, with T > 0 . We give several criteria on the matrix A ( t ) to prove the asymptotic stability of the trivial solution to equation x ̇ = A ( t ) x . These criteria allow us to show that any finite configuration of cycles in R n can be realized as hyperbolic limit cycles of a polynomial vector field.