I'm taking an introductory course in nonlinear dynamics inspired by the lectures of Steven Strogatz on the subject. The main goal of the course is to provide tools that enable one to qualitatively analyze an otherwise unsolvable (at least analytically) nonlinear system.
We are now wrapping up the study of two dimensional systems, and know how to classify most types of fixed point on the the plane via linearization, but when dealing with fixed points that have a handedness to them like spirals, centers and degenerate nodes, the only way to find it (as far as we were taught) is to plot for nearby points and deduce if the trend is CW or CCW.
Now, a natural thing to come up with when dealing with vector fields is to take its curl about a point. That seems to be a more clever way of finding the handedness around such fixed points, but in our lectures and even in Steven Strogatz lectures there is no mention of the usage of curl or even divergence for that matter as tools for understanding the behaviour of the vector field around a fixed point.
So the question is - Is there a subtle point that I'm missing which prevents me from using divergence and curl in the study of the behaviour of nonlinear systems ?
*I am aware that beyond 3 dimensions the usage of curl is problematic.
Thanks for any help