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 $$X'(t) = A(t) X(t) \tag{2}$$ where $X$ is a function from $\mathbb{R}$ to $\mathbb{C}^n$, of class $\mathscr{C}^1$ on $\mathbb{R}$. Let $\mathscr{S}$ be the space of solutions in $\mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2). Let $(Y_1, Y_2, \ldots, Y_n)$ be a basis of $\mathscr{S}$. For $t \in \mathbb{R}$, we denote by $M(t)$ the matrix whose columns are $Y_1(t), \ldots, Y_n(t)$.
Show that the matrix $(M(t))^{-1} M(t+T)$ is independent of $t \in \mathbb{R}$.

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 $$X'(t) = A(t) X(t) \tag{2}$$ where $X$ is a function from $\mathbb{R}$ to $\mathbb{C}^n$, of class $\mathscr{C}^1$ on $\mathbb{R}$. Let $\mathscr{S}$ be the space of solutions in $\mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2). Let $(Y_1, Y_2, \ldots, Y_n)$ be a basis of $\mathscr{S}$. For $t \in \mathbb{R}$, we denote by $M(t)$ the matrix whose columns are $Y_1(t), \ldots, Y_n(t)$.
Deduce that there exists $B \in \mathscr{M}_n(\mathbb{C})$ such that: $$\forall t \in \mathbb{R}, \quad M(t+T) = M(t) \exp(TB).$$

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 $$X'(t) = A(t) X(t) \tag{2}$$ where $X$ is a function from $\mathbb{R}$ to $\mathbb{C}^n$, of class $\mathscr{C}^1$ on $\mathbb{R}$. Let $\mathscr{S}$ be the space of solutions in $\mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2). Let $(Y_1, Y_2, \ldots, Y_n)$ be a basis of $\mathscr{S}$. For $t \in \mathbb{R}$, we denote by $M(t)$ the matrix whose columns are $Y_1(t), \ldots, Y_n(t)$. There exists $B \in \mathscr{M}_n(\mathbb{C})$ such that $M(t+T) = M(t)\exp(TB)$ for all $t \in \mathbb{R}$.
Deduce that there exists an application $Q : \mathbb{R} \rightarrow \mathrm{GL}_n(\mathbb{C})$ continuous on $\mathbb{R}$ and $T$-periodic such that $$\forall t \in \mathbb{R}, \quad M(t) = Q(t) \exp(tB).$$ (This identity is called the normal form of the matrix $M$).

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 $\mathscr{S}$ be the space of solutions in $\mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2). Let $(Y_1, Y_2, \ldots, Y_n)$ be a basis of $\mathscr{S}$. For $t \in \mathbb{R}$, let $M(t)$ be the matrix whose columns are $Y_1(t), \ldots, Y_n(t)$, and let $M(t) = Q(t)\exp(tB)$ be the normal form. We admit that there exist two matrices $D$ and $N$ of $\mathscr{M}_n(\mathbb{C})$ such that $D$ is diagonalizable, $N$ is nilpotent and $B = D + N$ and $DN = ND$. There exists a matrix $P \in \mathrm{GL}_n(\mathbb{C})$ and a diagonal matrix $\Delta$ such that $D = P\Delta P^{-1}$.
For $t \in \mathbb{R}$, we denote by $Z_1(t), Z_2(t), \ldots, Z_n(t) \in \mathbb{C}^n$ the columns of the matrix $M(t)P$. Show that $(Z_1, Z_2, \ldots, Z_n)$ is a basis of the space $\mathscr{S}$.

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 $\mathscr{S}$ be the space of solutions in $\mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2). Let $(Y_1, Y_2, \ldots, Y_n)$ be a basis of $\mathscr{S}$. For $t \in \mathbb{R}$, let $M(t)$ be the matrix whose columns are $Y_1(t), \ldots, Y_n(t)$, and let $M(t) = Q(t)\exp(tB)$ be the normal form. We admit that $B = D + N$ where $D$ is diagonalizable, $N$ is nilpotent, and $DN = ND$. There exists $P \in \mathrm{GL}_n(\mathbb{C})$ and a diagonal matrix $\Delta$ such that $D = P\Delta P^{-1}$. Let $Z_1(t), Z_2(t), \ldots, Z_n(t)$ be the columns of $M(t)P$.
Let $\lambda_1, \ldots, \lambda_n$ be the complex numbers such that $\Delta = \operatorname{Diag}(\lambda_1, \lambda_2, \ldots, \lambda_n)$. For all $0 \leqslant i \leqslant n-1$, $1 \leqslant k \leqslant n$ and $t \in \mathbb{R}$, we denote by $R_{i,k}(t)$ the $k$-th column of the matrix $\frac{1}{i!} Q(t) N^i P$. Show that for all $k \in \{1, 2, \ldots, n\}$, we have $$Z_k(t) = e^{\lambda_k t} \left(\sum_{i=0}^{n-1} t^i R_{i,k}(t)\right)$$ and verify that the applications $R_{i,k}$ are continuous on $\mathbb{R}$ and $T$-periodic.

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 $\mathscr{S}$ be the space of solutions in $\mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2). Using the notation and results of question 17b (in particular, $Z_k(t) = e^{\lambda_k t}\left(\sum_{i=0}^{n-1} t^i R_{i,k}(t)\right)$ where $R_{i,k}$ are continuous and $T$-periodic), deduce that if the real parts of the $\lambda_i$ for $1 \leqslant i \leqslant n$ are strictly negative and if $Y$ is any solution of (2), then $$\lim_{t \rightarrow +\infty} Y(t) = 0$$

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 $\mathscr{S}$ be the space of solutions in $\mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2). Let $(Y_1, Y_2, \ldots, Y_n)$ be a basis of $\mathscr{S}$. For $t \in \mathbb{R}$, let $M(t)$ be the matrix whose columns are $Y_1(t), \ldots, Y_n(t)$, and $M(t) = Q(t)\exp(tB)$ for some $B \in \mathscr{M}_n(\mathbb{C})$ and $T$-periodic $Q$. We admit that there exist two matrices $D$ and $N$ of $\mathscr{M}_n(\mathbb{C})$ such that $D$ is diagonalizable, $N$ is nilpotent and $B = D + N$ and $DN = ND$. There exists a matrix $P \in \mathrm{GL}_n(\mathbb{C})$ and a diagonal matrix $\Delta$ such that $D = P\Delta P^{-1}$.
For $t \in \mathbb{R}$, we denote by $Z_1(t), Z_2(t), \ldots, Z_n(t) \in \mathbb{C}^n$ the columns of the matrix $M(t)P$. Show that $(Z_1, Z_2, \ldots, Z_n)$ is a basis of the space $\mathscr{S}$.

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 $\mathscr{S}$ be the space of solutions in $\mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2). Let $(Y_1, Y_2, \ldots, Y_n)$ be a basis of $\mathscr{S}$. For $t \in \mathbb{R}$, let $M(t)$ be the matrix whose columns are $Y_1(t), \ldots, Y_n(t)$, and $M(t) = Q(t)\exp(tB)$ for some $B \in \mathscr{M}_n(\mathbb{C})$ and $T$-periodic $Q$. We admit that $B = D + N$ where $D$ is diagonalizable, $N$ is nilpotent, $DN = ND$, and $D = P\Delta P^{-1}$ with $\Delta = \operatorname{Diag}(\lambda_1, \lambda_2, \ldots, \lambda_n)$. Let $Z_1(t), \ldots, Z_n(t)$ be the columns of $M(t)P$.
For all $0 \leqslant i \leqslant n-1$, $1 \leqslant k \leqslant n$ and $t \in \mathbb{R}$, we denote by $R_{i,k}(t)$ the $k$-th column of the matrix $\frac{1}{i!} Q(t) N^i P$. Show that for all $k \in \{1, 2, \ldots, n\}$, we have $$Z_k(t) = e^{\lambda_k t} \left(\sum_{i=0}^{n-1} t^i R_{i,k}(t)\right)$$ and verify that the applications $R_{i,k}$ are continuous on $\mathbb{R}$ and $T$-periodic.

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 $\mathscr{S}$ be the space of solutions in $\mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2). Using the notation and results of question 17b, deduce that if the real parts of the $\lambda_i$ for $1 \leqslant i \leqslant n$ are strictly negative and if $Y$ is any solution of (2), then $$\lim_{t \rightarrow +\infty} Y(t) = 0.$$

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 $$X'(t) = A(t) X(t) \tag{2}$$ In this question, we suppose that (2) has a $T'$-periodic solution $X$ with $T' \notin \mathbb{Q} T$.
Show that for all $t \in \mathbb{R}$ and $u \in \mathbb{R}$, we have $$A(u) X(t) = A(t) X(t).$$ One may use without proof the fact that if $G$ is a subgroup of $(\mathbb{R}, +)$ which is not of the form $\mathbb{Z}a$ for $a \in \mathbb{R}$, then $G$ is dense in $\mathbb{R}$.

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 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 $t$.
We assume in this question 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 invariant 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.

Consider that complex-valued functions $p(x)$ and $q(x)$ satisfy the simultaneous ordinary differential equations below: $$\frac{\mathrm{d}p(x)}{\mathrm{d}x} = -ib\, q(x)\exp(-2iax),$$ $$\frac{\mathrm{d}q(x)}{\mathrm{d}x} = -ib\, p(x)\exp(2iax).$$ Here, $i$ is the imaginary unit, and $a$ and $b$ are real constants.
Derive the simultaneous ordinary differential equations for complex-valued functions $f(x)$ and $g(x)$, based on the change of variables $f(x) = p(x)\exp(iax)$ and $g(x) = q(x)\exp(-iax)$.

Consider that complex-valued functions $p(x)$ and $q(x)$ satisfy the simultaneous ordinary differential equations below: $$\frac{\mathrm{d}p(x)}{\mathrm{d}x} = -ib\, q(x)\exp(-2iax),$$ $$\frac{\mathrm{d}q(x)}{\mathrm{d}x} = -ib\, p(x)\exp(2iax).$$ Here, $i$ is the imaginary unit, and $a$ and $b$ are real constants. Let $f(x) = p(x)\exp(iax)$ and $g(x) = q(x)\exp(-iax)$.
Show that the value of $|f(x)|^{2} + |g(x)|^{2}$ is independent of $x$, where $|A|$ denotes the absolute value of a complex number $A$.

Consider that complex-valued functions $p(x)$ and $q(x)$ satisfy the simultaneous ordinary differential equations below: $$\frac{\mathrm{d}p(x)}{\mathrm{d}x} = -ib\, q(x)\exp(-2iax),$$ $$\frac{\mathrm{d}q(x)}{\mathrm{d}x} = -ib\, p(x)\exp(2iax).$$ Here, $i$ is the imaginary unit, and $a$ and $b$ are real constants. Let $f(x) = p(x)\exp(iax)$ and $g(x) = q(x)\exp(-iax)$.
Let $a = 0.8$ and $b = 0.6$. Solve the simultaneous ordinary differential equations derived in Question II.1 using the initial values $f(0) = 1$ and $g(0) = 0$, and obtain $f(x)$ and $g(x)$.

Let $t$ be a real independent variable, and let $a ( t ) , x ( t )$ and $y ( t )$ be real-valued functions. Answer the following questions.
(1) Let $a ( t )$ be a continuous and periodic function of $t$ whose period is $T$. Find the initial-value-problem solution $x ( t )$ of the ordinary differential equation
$$\frac { \mathrm { d } } { \mathrm {~d} t } x ( t ) = a ( t ) x ( t ) ,$$
where $x ( 0 ) = x _ { 0 } \neq 0$.
(2) Find the necessary and sufficient condition on $a ( t )$ such that the solution $x ( t )$ in Question (1) is a periodic solution with a period $T$.
(3) Find the initial-value-problem solutions $x ( t )$ and $y ( t )$ of the simultaneous ordinary differential equations
$$\begin{aligned} \frac { \mathrm { d } } { \mathrm {~d} t } x ( t ) & = - k x ( t ) + \sin ( t ) \cos ( t ) y ( t ) \\ \frac { \mathrm { d } } { \mathrm {~d} t } y ( t ) & = ( - k + \sin ( t ) ) y ( t ) \end{aligned}$$
where $k$ is a real constant, $x ( 0 ) = x _ { 0 } \neq 0$ and $y ( 0 ) = y _ { 0 } \neq 0$.
(4) Explain briefly how $x ( t )$ and $y ( t )$ converge as $t \rightarrow \infty$ when $k > 0$ in Question (3).
(5) Draw a schematic graph of the solution trajectory in Question (3) on the $y x$-plane where $k = 0 , x _ { 0 } = 2$ and $y _ { 0 } = 1$.