I want to find all critical points of the following nonlinear system: $$\def\b{\begin{pmatrix}}\def\e{\end{pmatrix}}$$ $$\b y_1' \\ y_2'\e = \b 5y_2 -15 \\y_2^2 - y_1 ^2\e$$
Then use linearisation to find the type and stability of the critical points.
So first of all, finding the critical points: $$\b 5y_2 -15 \\y_2^2 - y_1 ^2\e = \b 0\\0 \e$$
Gives us $y_2=3$ and $y_2 = \pm y_1$
Giving us the critical points $(3,3)$ and $(-3,3)$.
Now I need to use linearisation to find the type and stability of the critical points, but I have no idea how to do this, and I have looked at two textbooks, but don't understand.
Can someone please explain this (likely simple) concept to me?
It's easier notationally and conceptually to talk about a general system of the form $y' = f(y)$. The idea is to approximate the system near a critical point by the closest linear one - that is, approximate $f$ near a critical point by the closest linear function. This is exactly what the first-order Taylor expansion does: if $f(y_0) = 0$, then $$f(y_0 + h) = \nabla f (y_0)\cdot h+\text{higher order terms.}$$
Thus we can approximate the nonlinear system near $y_0$ using the linear system $$h' = \nabla f(y_0) \cdot h.$$ In coordinates this is $$h_i ' = \sum_j \frac{\partial f_i }{\partial y_j} \Big|_{y_0}h_j.$$ So the linear approximation at a critical point is described by the Jacobian matrix of $f$ at that critical point. Once you have computed these matrices you can determine stability, etc as in the linear case - check the eigenvalues.