grandes-ecoles

Q1a Second order differential equations Structure of the solution space View

Let $q : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous application, periodic with period $T > 0$. We consider the differential equation $$y'' + qy = 0. \tag{1}$$ Justify the existence of two solutions $y_1$ and $y_2$ in $\mathscr{C}^2(\mathbb{R}, \mathbb{C})$ to (1) such that: $$\left\{ \begin{array}{l} y_1(0) = 1 \\ y_1'(0) = 0 \end{array} \text{ and } \left\{ \begin{array}{l} y_2(0) = 0 \\ y_2'(0) = 1. \end{array} \right. \right.$$ Justify that $\operatorname{Vect}(y_1, y_2)$ is the set of solutions of (1) in $\mathscr{C}^2(\mathbb{R}, \mathbb{C})$.

Q1b Second order differential equations Structure of the solution space View

Let $q : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous application, periodic with period $T > 0$. We consider the differential equation $$y'' + qy = 0. \tag{1}$$ Let $y_1$ and $y_2$ be the solutions of (1) with initial conditions $y_1(0)=1, y_1'(0)=0$ and $y_2(0)=0, y_2'(0)=1$. Show that: $$\forall t \in \mathbb{R}, \quad y_1(t) y_2'(t) - y_1'(t) y_2(t) = 1.$$

Q2 Second order differential equations Floquet theory and periodic-coefficient second-order ODE View

Let $q : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous application, periodic with period $T > 0$. We consider the differential equation $$y'' + qy = 0. \tag{1}$$ Let $y_1$ and $y_2$ be the solutions of (1) with initial conditions $y_1(0)=1, y_1'(0)=0$ and $y_2(0)=0, y_2'(0)=1$. Show that if $y \in \mathscr{C}^2(\mathbb{R}, \mathbb{C})$ is a solution of (1), then the function $t \mapsto y(t+T)$ is also one. Deduce that for all $t \in \mathbb{R}$: $$y(t+T) = y(T) y_1(t) + y'(T) y_2(t).$$

Q3 Second order differential equations Floquet theory and periodic-coefficient second-order ODE View

Let $q : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous application, periodic with period $T > 0$. We consider the differential equation $$y'' + qy = 0. \tag{1}$$ Let $y_1$ and $y_2$ be the solutions of (1) with initial conditions $y_1(0)=1, y_1'(0)=0$ and $y_2(0)=0, y_2'(0)=1$. Let $\mu \in \mathbb{C}^*$, and let $\lambda \in \mathbb{C}$ such that $\mu = e^{\lambda T}$. Show that the following three assertions are equivalent.
(a) Equation (1) has a non-zero solution $y \in \mathscr{C}^2(\mathbb{R}, \mathbb{C})$ that satisfies: $$\forall t \in \mathbb{R}, \quad y(t+T) = \mu y(t).$$
(b) The complex number $\mu$ is a solution of the equation with unknown $x$: $$x^2 - \left(y_1(T) + y_2'(T)\right) x + 1 = 0.$$
(c) The differential equation (1) has a non-zero solution $y \in \mathscr{C}^2(\mathbb{R}, \mathbb{C})$ such that: $$\forall t \in \mathbb{R}, \quad y(t) = e^{\lambda t} u(t),$$ where $u \in \mathscr{C}^2(\mathbb{R}, \mathbb{C})$ is a $T$-periodic function.

Q4a Second order differential equations Floquet theory and periodic-coefficient second-order ODE View

Let $q : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous application, periodic with period $T > 0$. We consider the differential equation $$y'' + qy = 0. \tag{1}$$ Let $y_1$ and $y_2$ be the solutions of (1) with initial conditions $y_1(0)=1, y_1'(0)=0$ and $y_2(0)=0, y_2'(0)=1$. Let $\mu_1, \mu_2$ be the complex roots of the equation with unknown $x$: $$x^2 - \left(y_1(T) + y_2'(T)\right) x + 1 = 0.$$ Show that if $\mu_1 \neq \mu_2$ and if $\lambda$ is a complex number such that $\mu_1 = e^{\lambda T}$, then for any solution $y$ of (1), there exist two $T$-periodic functions $w_1$ and $w_2$, as well as two complex numbers $\alpha$ and $\beta$ such that $$\forall t \in \mathbb{R}, \quad y(t) = \alpha e^{\lambda t} w_1(t) + \beta e^{-\lambda t} w_2(t).$$

Q4b Second order differential equations Floquet theory and periodic-coefficient second-order ODE View

Let $q : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous application, periodic with period $T > 0$. We consider the differential equation $$y'' + qy = 0. \tag{1}$$ Let $y_1$ and $y_2$ be the solutions of (1) with initial conditions $y_1(0)=1, y_1'(0)=0$ and $y_2(0)=0, y_2'(0)=1$. Let $\mu_1, \mu_2$ be the complex roots of the equation with unknown $x$: $$x^2 - \left(y_1(T) + y_2'(T)\right) x + 1 = 0.$$ Suppose that $\mu_1 = \mu_2$. Show that $\mu_1 = \mu_2 = \pm 1$ and that equation (1) admits a periodic solution in $\mathscr{C}^2(\mathbb{R}, \mathbb{C})$.

Q5a Applied differentiation Properties of differentiable functions (abstract/theoretical) View

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$.
Let $V$ be a convex open set of $E$ and $h$ a function of class $\mathscr{C}^1$ from $V$ to $E$. Suppose that there exists a real number $C \geqslant 0$ such that for all $x \in V$, $\|dh(x)\| \leqslant C$. Show that for all $x_1$ and $x_2$ in $V$, we have $\left\|h(x_2) - h(x_1)\right\| \leqslant C \left\|x_2 - x_1\right\|$.

Q5b Applied differentiation Properties of differentiable functions (abstract/theoretical) View

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$.
Show that there exists a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$

Q5c Applied differentiation Properties of differentiable functions (abstract/theoretical) View

Let $a \in E$ and let $U$ be an open set of $E$ containing $a$. Let $f : U \rightarrow E$ be an application of class $\mathscr{C}^1$ on $U$ such that $df(a) = \operatorname{Id}_E$. We fix a real number $r > 0$ such that $\overline{B(a,r)} \subset U$ and $$\forall x_1, x_2 \in \overline{B(a,r)}, \quad \left\|f(x_1) - f(x_2)\right\| \geqslant \frac{1}{2} \left\|x_1 - x_2\right\|.$$ Show that for all $x \in B(a,r)$, the linear map $df(x)$ is injective.

Q8a Groups Subgroup and Normal Subgroup Properties View

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ Justify that $\mathbb{C}[A]^*$ is an abelian subgroup of $\mathrm{GL}_n(\mathbb{C})$.

Q8b Groups Subgroup and Normal Subgroup Properties View

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ Show that $(\mathbb{C}[A])^* = \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C})$.

Q9 Groups Subgroup and Normal Subgroup Properties View

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ Show that $\exp(\mathbb{C}[A]) \subset (\mathbb{C}[A])^*$.

Q10a Proof Proof That a Map Has a Specific Property View

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ For $a \in \mathbb{R}$, we define the application $$\begin{array}{ccc} Z_a : [0,1] & \longrightarrow & \mathbb{C} \\ t & \longmapsto & t + iat(1-t). \end{array}$$ Show that the application $$\begin{array}{clc} ]0,1[ \times \mathbb{R} & \longrightarrow & \mathbb{C} \\ (t,a) & \longmapsto & Z_a(t) \end{array}$$ is injective.

Q10b Proof Existence Proof View

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ For $a \in \mathbb{R}$, we define the application $$\begin{array}{ccc} Z_a : [0,1] & \longrightarrow & \mathbb{C} \\ t & \longmapsto & t + iat(1-t). \end{array}$$ Let $M_1$ and $M_2$ be two elements of $(\mathbb{C}[A])^*$. Show that there exists $a \in \mathbb{R}$ such that $$\forall t \in [0,1], \quad M(t) = Z_a(t) M_1 + \left(1 - Z_a(t)\right) M_2 \in (\mathbb{C}[A])^*.$$

Q10c Proof Proof of Set Membership, Containment, or Structural Property View

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ For $a \in \mathbb{R}$, we define the application $$\begin{array}{ccc} Z_a : [0,1] & \longrightarrow & \mathbb{C} \\ t & \longmapsto & t + iat(1-t). \end{array}$$ Using the result of question 10b, deduce that $(\mathbb{C}[A])^*$ is path-connected.

Q11a Matrices Matrix Norm, Convergence, and Inequality View

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ Show that there exists an open set $U$ of $\mathbb{C}[A]$ containing $0$ and an open set $V$ of $\mathbb{C}[A]$ containing the identity matrix $I_n$ such that the exponential function induces a continuous bijection from $U \subset \mathbb{C}[A]$ to $V$ whose inverse is a continuous function on $V$.

Q11b Matrices Matrix Norm, Convergence, and Inequality View

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ Using the result of question 11a, deduce that $\exp(\mathbb{C}[A])$ is an open set of $\mathbb{C}[A]$.

Q12 Matrices Matrix Norm, Convergence, and Inequality View

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ Show that $\exp(\mathbb{C}[A])$ is a closed set of $(\mathbb{C}[A])^*$.

Q13a Sequences and Series Matrix Exponentials and Series of Matrices View

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ We want to show that $\exp(\mathbb{C}[A]) = (\mathbb{C}[A])^*$. We suppose that $\exp(\mathbb{C}[A]) \neq (\mathbb{C}[A])^*$ and we fix $M_1, M_2 \in (\mathbb{C}[A])^*$ such that $M_1 \in \exp(\mathbb{C}[A])$ and $M_2 \notin \exp(\mathbb{C}[A])$.
Show that there exists a continuous map $f$ from $(\mathbb{C}[A])^*$ to $\{0,1\}$ such that $f(M_1) = 0$ and $f(M_2) = 1$.

Q13b Sequences and Series Matrix Exponentials and Series of Matrices View

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ We want to show that $\exp(\mathbb{C}[A]) = (\mathbb{C}[A])^*$. We suppose that $\exp(\mathbb{C}[A]) \neq (\mathbb{C}[A])^*$ and we fix $M_1, M_2 \in (\mathbb{C}[A])^*$ such that $M_1 \in \exp(\mathbb{C}[A])$ and $M_2 \notin \exp(\mathbb{C}[A])$.
Using the result of question 13a, conclude that $\exp(\mathbb{C}[A]) = (\mathbb{C}[A])^*$.

Q14 Sequences and Series Matrix Exponentials and Series of Matrices View

Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. Using the result $\exp(\mathbb{C}[A]) = (\mathbb{C}[A])^*$, conclude that $\exp(\mathscr{M}_n(\mathbb{C})) = \mathrm{GL}_n(\mathbb{C})$.

Q15 Systems of differential equations View

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 $\mathbb{R}$.
Show that there exists $\mu \in \mathbb{C}^*$ and a non-zero solution $Y \in \mathscr{C}^1(\mathbb{R}, \mathbb{C}^n)$ of (2) such that $$\forall t \in \mathbb{R}, \quad Y(t+T) = \mu Y(t).$$

Q16a Systems of differential equations View

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 for every real number $t$, $M(t) \in \mathrm{GL}_n(\mathbb{C})$ and $M'(t) = A(t) M(t)$.

Q16b Systems of differential equations View

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 $t \in \mathbb{R}$.

Q16c Systems of differential equations View

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: $$\forall t \in \mathbb{R}, \quad M(t+T) = M(t) \exp(TB).$$

Q16d Systems of differential equations View

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 $M$).

Q17a Systems of differential equations View

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 $\mathscr{S}$.

Q17b Systems of differential equations View

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.

Q17c Systems of differential equations View

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 $$\lim_{t \rightarrow +\infty} Y(t) = 0.$$

Q18a Second order differential equations Floquet theory and periodic-coefficient second-order ODE View

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.

Q18b Second order differential equations Floquet theory and periodic-coefficient second-order ODE View

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 $t$.
Suppose that there exists $m \in \mathbb{N}^*$ such that (2) has a non-zero $mT$-periodic solution. Show that $\exp(TB)$ has an eigenvalue that is an $m$-th root of unity.

Q19 Systems of differential equations View

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 $\mathbb{R}$.

Q20 Systems of differential equations View

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.

Q21 Second order differential equations Second-order ODE with initial or boundary value conditions View

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. Let $B \in \mathscr{M}_n(\mathbb{C})$ be the matrix such that $M(t+T) = M(t)\exp(TB)$ for all $t$. Consider the differential system $$X'(t) = A(t) X(t) + b(t) \tag{3}$$ where $b : \mathbb{R} \rightarrow \mathbb{C}^n$ is a continuous function on $\mathbb{R}$ and $T$-periodic. We assume that $1$ is not an eigenvalue of $\exp(TB)$. Show that (3) possesses a unique $T$-periodic solution.

Q22 Second order differential equations Reduction of a differential system to a second-order ODE View

Solve the differential system $$\left\{\begin{array}{l} x'(t) = x(t) - \cos(t) y(t) \\ y'(t) = \cos(t) x(t) + y(t) \end{array}\right.$$ and determine its normal form (see question 16d).

grandes-ecoles

Papers (191)

2024 x-ens-maths-b__mp