Let $M \in M_{n}(\mathbf{R})$. We denote by (S) the differential system: $$(\mathrm{S}) \quad X' = MX$$ where $X$ is a function from the variable $t$ in $\mathbf{R}$ to $\mathbf{R}^{n}$, differentiable on $\mathbf{R}$. We assume $n = 2$, we then denote $X = (x; y)$ where $x$ and $y$ are two functions differentiable from $\mathbf{R}$ to $\mathbf{R}$ and we set $z = x + iy$. Let $M \in M_{2}(\mathbf{R})$ be semi-simple. Give a necessary and sufficient condition, concerning the real and imaginary parts of the eigenvalues of $M$, for every solution of (S) to have each of its coordinates tend to 0 as $+\infty$.
Let $M \in M_{n}(\mathbf{R})$. We denote by (S) the differential system:
$$(\mathrm{S}) \quad X' = MX$$
where $X$ is a function from the variable $t$ in $\mathbf{R}$ to $\mathbf{R}^{n}$, differentiable on $\mathbf{R}$.
We assume $n = 2$, we then denote $X = (x; y)$ where $x$ and $y$ are two functions differentiable from $\mathbf{R}$ to $\mathbf{R}$ and we set $z = x + iy$.
Let $M \in M_{2}(\mathbf{R})$ be semi-simple. Give a necessary and sufficient condition, concerning the real and imaginary parts of the eigenvalues of $M$, for every solution of (S) to have each of its coordinates tend to 0 as $+\infty$.