There is another exercise where I am given a non-linear system of ODEs and a question that confuses me:
Consider $x'=x(y+1)\\ \text{ and }\\ y'=xy+2.$
"Determine stationary points of the system, and linearize the system at the stationary points, determine the geometric type of the linearizations and the stability."
The only stationary point is $(x=2,y=-1)$. For the stability, I wrote $f(x)=x(y+1),\\ g(y)=xy+2,$ then differentiated: $f'(x)=y+1,\\ g'(y)=x.$ At $(-1,2)$, these values are $0$ and $2$. For $f$, I wrote that we cannot use linear terms in the expansion and hence would have to consider $f''$ and so on. For $g$, since it's greater than zero, the point is not linearly stable in $y$-direction.
But what is meant by "determine the geometric type of the linearizations"? What are types of linearizations? What am I missing here?
all the best,
Marie