If I've got the following differential linear homogenous system of constant coefficients
$$x_1'(t)=a_{11}x_1+a_{12}x_2+...+a_{1n}x_n $$ $$x_2'(t)=a_{21}x_1+a_{22}x_2+...+a_{2n}x_n$$ $$... $$ $$x_n'(t)=a_{n1}x_1+a_{n2}x_2+...+a_{nn}x_n.$$
Can I say that its unique critical point is the point $(0,...,0) \in \mathbb{R}^n$ ?
Substituting, we can see clearly that $(0,...,0) \in \mathbb{R}^n$ is a critical point. But my doubt is, is it the only one?
Could someone help me?
That depends on the coefficients. You can rewrite the problem in vector-notation as follows: \begin{align*} \mathbf{x}' = A\mathbf{x} \end{align*} with the coefficient matrix $A$. Then every $\mathbf{x}\in \ker(A)$ is a critical point. So wether $0\in \mathbb{R}^n$ is the only critical point depends on the rank of $A$.