Toronto Math Forum
MAT2442013S => MAT244 MathTests => Final Exam => Topic started by: Victor Ivrii on April 17, 2013, 03:05:12 PM

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 $(x_0,y_0)$.

Solution attached.

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.

A followup.

And the computergenerated phase portrait.

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