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 $M(t) = Q(t)\exp(tB)$. Show that if $B$ has an eigenvalue of the form $\lambda = i\frac{2k\pi}{mT}$ with $k \in \mathbb{Z}$ and $m \in \mathbb{N}^*$, then (2) has a non-zero $mT$-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 $M(t) = Q(t)\exp(tB)$.
Show that if $B$ has an eigenvalue of the form $\lambda = i\frac{2k\pi}{mT}$ with $k \in \mathbb{Z}$ and $m \in \mathbb{N}^*$, then (2) has a non-zero $mT$-periodic solution.