MAT244--2018F > Quiz-6

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.

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