Stationary points of an ODE-system (again)

Question

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

Answer 1

For the stability you should evaluate the Jacobi matrix (see my comment here). The geometric type of linearization is determined by the eigenvalues of this matrix. You can start learning these matters, e.g., here.