This week I have been mostly wondering what simple systems of differential equations look like. The pictures in the books often either have too many arrows or too few.  There’s something aesthetically unappealing about it. Being a humanities person, I want to see the stories of individual points. Just thought I’d share them.

Here’s a stable set of equations:

[begin{equation} dot{x}=-2x+y \ dot{y}=x-2yend{equation}]

The lines on the graph represent $$dot{x}=0, dot{y}=0$$ . Hence, where they meet is an equilibrium point, which may be stable or unstable.

Then there’s it’s more awkward cousin:

[begin{equation} dot{x}=10+2x-3y \ dot{y}=9+x-2yend{equation}]

Zooming out a bit (not using arrow method), but before it turns into a straight line zipping off to infinity:

I particularly like this damnably simple, stable spiral:

[begin{equation} dot{x}=-x+y \ dot{y}=-x-yend{equation}]

And here’s its badly behaved cousin, flinging points out all over the show:

[begin{equation} dot{x}=-2x+y \ dot{y}=x+2yend{equation}]

Code on github, should you wish to play.