Q6 TUT 0601

[Victor Ivrii]

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.
$$\mathbf{x}' =\begin{pmatrix}
0 &-5\\
1 &\alpha
\end{pmatrix}\mathbf{x}.$$

[Guanyao Liang]

This is my answer.

[Zhiya Lou]

here is my solution

[Victor Ivrii]

Zhiya  is right but one needs to justify counterclockwise direction of rotation (complex roots) and find directions of eigenvectors (real roots).

Also, what happens as $\alpha =\pm \sqrt{20}$?

[Mengfan Zhu]

Hello everyone, this is my answer for quiz 6.
To analyze part(c), I think that we can divide this question into two parts: the real roots and complex roots.
But I am not sure, do we need to consider other conditions?
Is there anyone can share your opinions below?
Thank you very much.

[Zhiya Lou]

Updates:
Rotation direction for complex roots: look at original matrix, since b= -5 <0 and c=1>0, so it is counterclockwise.
When the real part is positive, spiral outward, unstable; when the real part is negative, spiral inward, stable.

when $\alpha$ = $\sqrt{20}$  (it's positive, outward, unstable)
It is in repeated root case, but only has one independent eigenvector, therefore, we could graph in the direction of this eigenvector, following the direction of counterclockwise as the same for complex roots.
Similar for $\alpha$ = $-\sqrt{20}$, it's negative, inward, stable, counterclockwise still.