For a school project I am asked to analyze a vector field. To analyze it, we need to generate a vector field with several critical points(focus, spiral, saddle..)
I have two questions about it:
1. I can generate most of them by manipulating functions (such as the function of a circle at (x,-y) has a vector field that is classified as focus) but how can I create one with all of them at different points? Such as a focus at around -2,2 and a spiral at 1,1 and so on?
2. If I was given vector field, then I could analyze the critical points by the Jacobian method but the other way around is quite difficult for some classes of critical points. For instance a spiral is formed if the eigenvalues of the Jacobian is complex conjugates (with nonzero imaginary values), so how do I go about coming up with a function of sort?