equilibrium solutins and stabity

Question

I am asked to find all equilibrium solutions to this system of differential equations:

$$\begin{cases} x ' = \tan(-x+y) \\ y'= x(y+1) \end{cases} $$ and to determine if they are stable.

I do not know what to do, I have found equilibrium solutions $(0,k \pi)$ and $(-1-k \pi,-1)$. Is it okey? What about stability. I tried to do translation and linearization but I fail.

Answer 1

The Jacobian is given by

$$Df(\vec{x}) = \begin{pmatrix} -\sec^2(-x+y) & \sec^2(-x+y) \\ y+1 & x \end{pmatrix}$$

which gives us two sets of linearized matrices

$$Df(0,k\pi) = \begin{pmatrix} -1 & 1 \\ k\pi + 1 & 0 \\ \end{pmatrix}$$

$$Df(-1+k\pi,-1) = \begin{pmatrix} -1 & 1 \\ 0 & -1 + k\pi \\ \end{pmatrix}$$

Can you take it from here? What are the eigenvalues of these matrices?