Differential Equations Direction Field Problem

Question

Can anyone do this problem? I'm struggling with it:

Answer 1

For part a: As we discussed in the comments The spiraling inwards tells you that you have eigenvalues of the form $a\pm bi$, by some scaling of a rotation matrix (this is from euler's identity).

For part b: I think the idea here is this is not a hyperbolic fixed point, which tells you that linearization is not a very good approximation of solution behavior. You can tell this qualitatively here as there is no stable and unstable manifold to be found, i.e. nothing that in two dimensions looks hyperbolic or like a saddle. I think, although the quality of the graph makes it hard to tell, that the part bounded by the solution curve in green is a center manifold, although this may be a problem for the answer in part a as it would imply that $a=0$, or the eigenvalues are purely imaginary in a center manifold.

For part c: This is just asking you to notice that the vectors along the x axis have zero x component, so the change in x along the x axis is zero, and there is an equilibrium in the x direction.

edit: looking more closely at the graph, I think what the question wants you to conclude for part a is that real part is 0 (guessing from the green circle, not spiral, at the center), which is good news for our answer to part b.