I went back and revisited the circle-type equation, but kept the changes from my tangent equation.

The result amazed me. (Note I was watching this unfold at 1.9fps.)
...but it crashed again, just short of a complete cycle.

Music by nagz, called "hajlongvarajzolok".
I believe it translates into something like, "drawing long hair."
please correct me if I'm wrong, anyone who speaks hungarian. :)