# Toronto Math Forum

## MAT244-2014F => MAT244 Math--Lectures => Topic started by: Yuan Bian on November 19, 2014, 11:02:48 AM

Title: Phase portraits - Improper nodes
Post by: Yuan Bian on November 19, 2014, 11:02:48 AM
Prof, how about direction of improper node?
Title: Re: Phase portraits - Improper nodes
Post by: Victor Ivrii 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)