We are given the homogeneous system $$y'=\begin{bmatrix}5 & -3 & 2 \\ 15 & -9 & 6 \\ 10 & -6 & 4\end{bmatrix}y$$ with the initial value $y(0)=(1,1,0)$ and are supposed to check whether the critical points are (asymptotically) stable.
I tried to directly apply the formal definitions of stability but failed to arrive at something useful.
Is this a use case for Lyapunov? If so how do I find an explicit Lyapunov function? Lyapunov has only just been introduced and I can't find any examples for a case as simple as this. Most use some physical intuition about energy...
It is worth practicing Jordan normal form, especially when they give you something where all numbers are integers, with the possible exception of a denominator needed for the inverse matrix. That is, I am constructing $P^{-1}A P = J,$ where your coefficient matrix is called $A.$ The minimal polynomial is $x^2,$ so there will be a 2 by2 block and a singleton. We start with the right hand column, calling it $w,$ where the only requirement is that $w$ not itself be an eigenvector. I like $$ w = \left( \begin{array}{c} 0 \\ 0 \\ 1 \end{array} \right) $$ For the middle column, we are required to take $v = A w,$ so $$ v = \left( \begin{array}{c} 2 \\ 6 \\ 4 \end{array} \right) $$
For the left hand column, any eigenvector not a multiple of $v$ is acceptable, and I like $$ u = \left( \begin{array}{c} 0 \\ 2 \\ 3 \end{array} \right) $$ for $$ P = \left( \begin{array}{ccc} 0 & 2&0 \\ 2 & 6 & 0 \\ 3 & 4 & 1 \end{array} \right) $$ Next $$ P^{-1} = \frac{1}{2} \left( \begin{array}{ccc} -3 & 1&0 \\ 1 & 0 & 0 \\ 5 & -3 & 2 \end{array} \right) $$ and we have $P^{-1}A P = J,$ with $$ J = \left( \begin{array}{ccc} 0 & 0&0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array} \right) $$ The reason for finding actual matrices $P$ and $P^{-1}$ is simply that you get explicit $$ P J P^{-1} = A $$ It is from that expression that we can find explicit $e^A$ and $e^At$