MAT244-2014F > MAT244 Math--Lectures

Phase portraits - Improper nodes

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Yuan Bian:
Prof, how about direction of improper node?

Victor Ivrii:

--- Quote from: Yuan Bian on November 19, 2014, 11:02:48 AM ---Prof, how about direction of improper node?

--- End quote ---
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)