Toronto Math Forum
MAT2442018F => MAT244Tests => Quiz6 => Topic started by: Victor Ivrii on November 17, 2018, 03:56:07 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.
$$\mathbf{x}' =\begin{pmatrix}
0 &5\\
1 &\alpha
\end{pmatrix}\mathbf{x}.$$

This is my answer.

here is my solution

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 $\alpha =\pm \sqrt{20}$?

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.

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.