Let $T > 0$ be a real number and $n \in \mathbb{N}^*$ be a natural integer. Let $A : \mathbb{R} \rightarrow \mathscr{M}_n(\mathbb{C})$ be a continuous application on $\mathbb{R}$ and $T$-periodic. We consider the differential system $$X'(t) = A(t) X(t) \tag{2}$$ Let $B \in \mathscr{M}_n(\mathbb{C})$ be the matrix from the normal form $M(t) = Q(t)\exp(tB)$. We assume that there does not exist a vector subspace $V \subset \mathbb{C}^n$, different from $\{0\}$ and $\mathbb{C}^n$, such that, for all $t \in \mathbb{R}$, $V$ is stable under $A(t)$. Give a necessary and sufficient condition on $A$ and on $B$ for (2) to have at least one non-zero periodic solution.
Let $T > 0$ be a real number and $n \in \mathbb{N}^*$ be a natural integer. Let $A : \mathbb{R} \rightarrow \mathscr{M}_n(\mathbb{C})$ be a continuous application on $\mathbb{R}$ and $T$-periodic. We consider the differential system
$$X'(t) = A(t) X(t) \tag{2}$$
Let $B \in \mathscr{M}_n(\mathbb{C})$ be the matrix from the normal form $M(t) = Q(t)\exp(tB)$.
We assume that there does not exist a vector subspace $V \subset \mathbb{C}^n$, different from $\{0\}$ and $\mathbb{C}^n$, such that, for all $t \in \mathbb{R}$, $V$ is stable under $A(t)$. Give a necessary and sufficient condition on $A$ and on $B$ for (2) to have at least one non-zero periodic solution.