# Toronto Math Forum

## MAT244--2018F => MAT244--Tests => Quiz-6 => Topic started by: Victor Ivrii on November 17, 2018, 03:56:07 PM

Title: Q6 TUT 0601
Post 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.
$$\mathbf{x}' =\begin{pmatrix} 0 &-5\\ 1 &\alpha \end{pmatrix}\mathbf{x}.$$
Title: Re: Q6 TUT 0601
Post by: Guanyao Liang on November 17, 2018, 03:56:38 PM
Title: Re: Q6 TUT 0601
Post by: Zhiya Lou on November 17, 2018, 04:08:44 PM
here is my solution
Title: Re: Q6 TUT 0601
Post by: Victor Ivrii on November 19, 2018, 05:38:28 AM
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 $\alpha =\pm \sqrt{20}$?

Title: Re: Q6 TUT 0601
Post by: Mengfan Zhu on November 20, 2018, 12:45:04 AM
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.
Title: Re: Q6 TUT 0601
Post by: Zhiya Lou on November 20, 2018, 08:55:58 AM
when $\alpha$ = $\sqrt{20}$  (it's positive, outward, unstable)
Similar for $\alpha$ = $-\sqrt{20}$, it's negative, inward, stable, counterclockwise still.