Toronto Math Forum
MAT244--2018F => MAT244--Tests => Quiz-6 => Topic started by: Victor Ivrii on November 17, 2018, 03:54:02 PM
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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.
$$\mathbf{x}' =\begin{pmatrix}
\alpha &1\\
-1 &\alpha
\end{pmatrix}\mathbf{x}.$$
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This is my answer.
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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
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My answer to c is slightly different.
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I am sorry, but what's the difference? I cannot find it. :)
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I think the difference is the direction of rotation.
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Our graphs are all clockwise :)
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Our graphs are all clockwise :)
Why?
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The potraits are clockwise, Since b > 0 and c < 0
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Our graphs are all clockwise :)
Isn't Guanyao's 2nd graph counterclockwise?
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Our graphs are all clockwise :)
Why?
If we choose a unit vector $\begin{bmatrix}1 \\ 0\end{bmatrix}$ and do matrix vector multiplication with the matrix $\begin{bmatrix} \alpha > 0 & 1 \\ -1 & \alpha > 0 \end{bmatrix}$, the vector $\begin{bmatrix}\alpha > 0 \\ -1\end{bmatrix}$ follows the phaseportrait CW. $\alpha$ > 0 means the phaseportrait points outward and is unstable.
If we choose a unit vector $\begin{bmatrix}1 \\ 0\end{bmatrix}$ and do matrix vector multiplication with the matrix $\begin{bmatrix} \alpha < 0 & 1 \\ -1 & \alpha < 0 \end{bmatrix}$, the vector $\begin{bmatrix}\alpha < 0 \\ -1\end{bmatrix}$ also follows the phaseportrait CW. $\alpha$ < 0 means the phaseportrait points inward and is stable.
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Michael, I asked why it is clockwise. Not about stability.
I especially made an announcement. For not providing explanation about clockwise/counter-clockwise rotation on Test (and Exam) the mark will be reduced
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Thank you for posting the announcement about the explanation of CW/CCW. I found the textbook a little confusing to read and myself and a few others found this video helpful: https://www.youtube.com/watch?v=dpbRUQ-5YWc
At time 19:42 they display a technique to determine CW vs CCW using generic vectors and matrix A. I think it might be more intuitive but not as rigorous as the explanation you gave. And you explanation you gave in the announcement was very helpful!