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 an application continuous 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 such that $M(t+T) = M(t)\exp(TB)$ for all $t$. 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 an application continuous 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 such that $M(t+T) = M(t)\exp(TB)$ for all $t$.
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.