Toronto Math Forum
MAT2442018F => MAT244Tests => Quiz6 => Topic started by: Victor Ivrii on November 17, 2018, 03:54:02 PM

The coefficient matrix contains a parameter $\alpha$.
(a) Determine the eigenvalues in terms of $\alpha$.
(b) Find the critical value or values of $\alpha$ where the qualitative nature of the phase portrait for
the system changes.
(c) Draw a phase portrait for a value of $\alpha$ slightly below, and for another value slightly above,
each critical value.
$$\mathbf{x}' =\begin{pmatrix}
\alpha &1\\
1 &\alpha
\end{pmatrix}\mathbf{x}.$$

This is my answer.

First, try to find the eigenvalues with respect to the parameter
$A=\begin{bmatrix}
\alpha&1\\
1&\alpha\\
\end{bmatrix}$
$det(ArI)=(\alphar)(\alphar)+1=0$
$r^22{\alpha}r+\alpha^2+1=0$
$r=\frac{2\alpha\pm\sqrt{4}}{2}$
$r=\alpha\pm2i$ $\color{red}{r_\pm =\alpha \pm i\; V.I.}$
Notice there are always complex eigenvalues, and $\alpha=0$ is critical value since $\alpha=0, \alpha>0, \alpha<0$ have different phase portraits
When $\alpha=0$ , real parts of eigenvalues are 0
When value of $\alpha$ is slightly below 0
Then $\alpha<0$ , real parts of eigenvalues are negative
When value of $\alpha$ is slightly above 0
Then $\alpha>0$ , real parts of eigenvalues are positive

My answer to c is slightly different.

I am sorry, but what's the difference? I cannot find it. :)

I think the difference is the direction of rotation.

Our graphs are all clockwise :)

Our graphs are all clockwise :)
Why?

The potraits are clockwise, Since b > 0 and c < 0

Our graphs are all clockwise :)
Isn't Guanyao's 2nd graph counterclockwise?

Our graphs are all clockwise :)
Why?
If we choose a unit vector $\begin{bmatrix}1 \\ 0\end{bmatrix}$ and do matrix vector multiplication with the matrix $\begin{bmatrix} \alpha > 0 & 1 \\ 1 & \alpha > 0 \end{bmatrix}$, the vector $\begin{bmatrix}\alpha > 0 \\ 1\end{bmatrix}$ follows the phaseportrait CW. $\alpha$ > 0 means the phaseportrait points outward and is unstable.
If we choose a unit vector $\begin{bmatrix}1 \\ 0\end{bmatrix}$ and do matrix vector multiplication with the matrix $\begin{bmatrix} \alpha < 0 & 1 \\ 1 & \alpha < 0 \end{bmatrix}$, the vector $\begin{bmatrix}\alpha < 0 \\ 1\end{bmatrix}$ also follows the phaseportrait CW. $\alpha$ < 0 means the phaseportrait points inward and is stable.

Michael, I asked why it is clockwise. Not about stability.
I especially made an announcement. For not providing explanation about clockwise/counterclockwise rotation on Test (and Exam) the mark will be reduced

Thank you for posting the announcement about the explanation of CW/CCW. I found the textbook a little confusing to read and myself and a few others found this video helpful: https://www.youtube.com/watch?v=dpbRUQ5YWc
At time 19:42 they display a technique to determine CW vs CCW using generic vectors and matrix A. I think it might be more intuitive but not as rigorous as the explanation you gave. And you explanation you gave in the announcement was very helpful!