I've been struggling to find a way to reverse engineer a differential equation based on knowing it's singular points. In this case, I'd like to create a flow on $[-1,1] \times [-1,1]$, which has saddles at (0,1/4) and (0/-1/4), with a source at (0,1/2), and a sink at (0,-1/2). It should also be vertical on the boundary.
I've been starting with the assumption that I need a system of ODEs, $\stackrel{.}{x} = f(x,y)$ and $\stackrel{.}{y} = g(x,y)$, where $f$ and $g$ are both zero at each of the points, and that the Jacobian would give me 4 conditions on the partial derivatives, but I seem to have too much data to make sense of this. Even when I think I have it for just one of the points, I have Sage graph it, and it's wrong.
Is there a more straightforward way?
I think this should be an example: $$ \begin{array}{c} \dot x = \left( {x}^{2}-1 \right) \left( {y}^{2}-1 \right) xy, \\ \dot y = \left( {x}^{2}+ \left( y-3/8 \right) ^{2}-{\frac {1}{64}} \right) \left( {x}^{2}+ \left( y+3/8 \right) ^{2}-{\frac {1}{64}} \right). \end{array} $$ Here plotted with vectors scaled for visual clarity.
The way I found it was by drawing the flow lines, then determining where $\dot x=0$ and where $\dot y=0$, where I could choose nice shapes like lines and circles for these zero sets, and then select the right sign to use.