Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We fix $t _ { 0 } \in \mathbb { R }$ and set $E ( t ) = [ U ( t ) , V ( t ) ]$ where $U, V$ are the solutions of (IV.1) with $U(t_0) = \binom{1}{0}$, $V(t_0) = \binom{0}{1}$. $B \in \mathrm { GL } _ { 2 } ( \mathbb { C } )$ is the unique matrix such that $E(t+T) = E(t)B$ for all $t$. The Floquet multipliers of (IV.1) are the eigenvalues of $B$. Let $\rho \in \mathbb { C }$ be a Floquet multiplier of (IV.1) and $Z \in \mathbb { C } ^ { 2 }$ be an eigenvector of $B$ associated with this eigenvalue. We denote $Y : \begin{aligned} & \mathbb { R } \rightarrow \mathbb { C } ^ { 2 } \\ & t \mapsto E ( t ) Z \end{aligned}$. a) Prove that $\forall t \in \mathbb { R } , Y ( t + T ) = \rho Y ( t )$. b) Prove that there exists a complex number $\mu$ and a function $S : \left\lvert \, \begin{aligned} & \mathbb { R } \rightarrow \mathbb { C } ^ { 2 } \\ & t \mapsto S ( t ) \end{aligned} \right.$ non-zero and $T$-periodic such that $\forall t \in \mathbb { R } , Y ( t ) = \mathrm { e } ^ { \mu t } S ( t )$.
Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system
$$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$
We fix $t _ { 0 } \in \mathbb { R }$ and set $E ( t ) = [ U ( t ) , V ( t ) ]$ where $U, V$ are the solutions of (IV.1) with $U(t_0) = \binom{1}{0}$, $V(t_0) = \binom{0}{1}$. $B \in \mathrm { GL } _ { 2 } ( \mathbb { C } )$ is the unique matrix such that $E(t+T) = E(t)B$ for all $t$. The Floquet multipliers of (IV.1) are the eigenvalues of $B$.
Let $\rho \in \mathbb { C }$ be a Floquet multiplier of (IV.1) and $Z \in \mathbb { C } ^ { 2 }$ be an eigenvector of $B$ associated with this eigenvalue. We denote $Y : \begin{aligned} & \mathbb { R } \rightarrow \mathbb { C } ^ { 2 } \\ & t \mapsto E ( t ) Z \end{aligned}$.
a) Prove that $\forall t \in \mathbb { R } , Y ( t + T ) = \rho Y ( t )$.
b) Prove that there exists a complex number $\mu$ and a function $S : \left\lvert \, \begin{aligned} & \mathbb { R } \rightarrow \mathbb { C } ^ { 2 } \\ & t \mapsto S ( t ) \end{aligned} \right.$ non-zero and $T$-periodic such that $\forall t \in \mathbb { R } , Y ( t ) = \mathrm { e } ^ { \mu t } S ( t )$.