MAT244--2018F > Quiz-6

Q6 TUT 0401

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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.

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

Guanyao Liang:
This is my answer.

Jingze Wang:
First, try to find the eigenvalues with respect to the parameter


$A=\begin{bmatrix}
\alpha&1\\
-1&\alpha\\
\end{bmatrix}$


$det(A-rI)=(\alpha-r)(\alpha-r)+1=0$

$r^2-2{\alpha}r+\alpha^2+1=0$

$r=\frac{2\alpha\pm\sqrt{-4}}{2}$

$r=\alpha\pm2i$       $\color{red}{r_\pm =\alpha \pm i\; V.I.}$


Notice there are always complex eigenvalues, and $\alpha=0$ is critical value since $\alpha=0, \alpha>0, \alpha<0$ have different phase portraits


When $\alpha=0$ , real parts of eigenvalues are 0


When value of $\alpha$ is slightly below 0
Then $\alpha<0$ , real parts of eigenvalues are negative


When value of $\alpha$ is slightly above 0
Then $\alpha>0$ , real parts of eigenvalues are positive

Jiacheng Ge:
My answer to c is slightly different.

Jingze Wang:
I am sorry, but what's the difference? I cannot find it. :)

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