FE-5

[Victor Ivrii]

For the system of differential equations
\begin{equation*}
\left\{\begin{aligned}
&x' =\tan (y) - \frac{1}{2}\tan (x)  \,,\\
&y' = \tan (x) - \frac{1}{2}\tan (y) \,.
\end{aligned}\right.
\end{equation*}

(a) Linearize the system at a critical point $(x_0 ,y_0)$ of your choice;

(b) Describe the type of the critical point $(x_0,y_0)$ of the linearized and of the original system;

(c) Sketch the phase portraits of the linearized and of the  original system near this critical point $(x_0,y_0)$.

[Michal Staszewski]

Solution attached.

[Victor Ivrii]

Solution attached.

Few remarks.
1. Field is obviously $\pi$-periodic with respect to both $x$ and $y$ and it is singular as $x=(m+\frac{1}{2})\pi$  or $y=(n+\frac{1}{2})\pi$ with $m,n\in \mathbb{Z}$ so one needs to consider only square $\{ -\frac{\pi}{2}<x < \frac{\pi}{2},  -\frac{\pi}{2}<y < \frac{\pi}{2}\}$ where $(0,0)$ is an only equilibrium point.

2. Missing: eigenvectors (so directions of separatrices have not been found.

3. Global phase portrait would be appreciated.

[Michal Staszewski]

A follow-up.

[Michal Staszewski]

And the computer-generated phase portrait.

[Victor Ivrii]

Now it is OK. Note that vector field breaks (on the picture you see the change of direction)  at lines I mentioned