Toronto Math Forum
MAT2442014F => MAT244 MathLectures => Topic started by: Yuan Bian on November 19, 2014, 11:02:48 AM

Prof, how about direction of improper node?

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 clockwise or counterclockwise. Again the sign of the topright element of the matrix defines it (clockwise iff it is positive)