I'm missing something here, I don't know how to derive the vector forces for this system of differential equations show in the picture. It is explained in the picture, but I don't get the parts that I marked in orange (... which implies that k < k* ...). Also, how do you know c_dot = 0 is a vertical line and not a horizontal one. How do you know are A2 is where c_dot > 0? Where is the unstable arm?
What they say at the bottom of the page that K=capital <-> k = K/L in the steady state is that is decreasing at the same rate as the labor force (=n), I guess?
Is it possible to linearize these differential equations and derive the solutions analytically assuming log utility (e.g. as a percentage deviation from the steady state)? How would you start this?