### Author Topic: Phase portraits - Improper nodes  (Read 3324 times)

#### Yuan Bian

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##### Phase portraits - Improper nodes
« on: November 19, 2014, 11:02:48 AM »
Prof, how about direction of improper node?

#### Victor Ivrii

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##### Re: Phase portraits - Improper nodes
« Reply #1 on: November 19, 2014, 11:45:58 AM »
Prof, how about direction of improper node?
Don't hijack topics!

OK, consider canonical form of $\mathbf{x}'=A\mathbf{x}$:
\begin{equation*}
\begin{pmatrix} x' \\ y'\end{pmatrix}= \begin{pmatrix}r & 1 \\ 0 &r\end{pmatrix}\begin{pmatrix} x \\ y\end{pmatrix}.
\end{equation*}
Then $y= Ce^{rt}$, $x= (Ct+C_1)e^{rt}$. Right? Depending on $r<0$ and $r>0$ you get one of two pictures (stable and unstable, respectively).

Now you need to learn if it is clock-wise or counter-clock-wise. Again the sign of the top-right element of the matrix defines it (clock-wise iff it is positive)
« Last Edit: November 19, 2014, 04:12:06 PM by Victor Ivrii »