Is there an algorithmic way of drawing phase portraits by hand?

[mjordan2nd]

I have trouble understanding how to draw phase portraits. Is there a good pedagogical overview of their basics (such as elliptical vs hyperbolic orbits, separatrices etc.), along with an algorithmic way of how to draw them when it is not easy/possible to find v(x)? Basically something explaining how to draw phase portraits directly from the physics of the problem without having to do much math.

 

[Simon Bridge]

Nope - you just have to do the math.
Once you have got used to how they tend to turn out, you can use that experience to make educated guesses - where you only need a sketch.