### Author Topic: Q6 TUT 0601  (Read 4383 times)

#### Victor Ivrii

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##### Q6 TUT 0601
« 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.
$$\mathbf{x}' =\begin{pmatrix} 0 &-5\\ 1 &\alpha \end{pmatrix}\mathbf{x}.$$

#### Guanyao Liang

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##### Re: Q6 TUT 0601
« Reply #1 on: November 17, 2018, 03:56:38 PM »

#### Zhiya Lou

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##### Re: Q6 TUT 0601
« Reply #2 on: November 17, 2018, 04:08:44 PM »
here is my solution
« Last Edit: November 19, 2018, 11:37:27 PM by Zhiya Lou »

#### Victor Ivrii

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##### Re: Q6 TUT 0601
« Reply #3 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 $\alpha =\pm \sqrt{20}$?

« Last Edit: November 25, 2018, 09:23:17 AM by Victor Ivrii »

#### Mengfan Zhu

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##### Re: Q6 TUT 0601
« Reply #4 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.

#### Zhiya Lou

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##### Re: Q6 TUT 0601
« Reply #5 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.