PS6 is available, as is a combined handout on Phase portraits (in dvi, postscript, or pdf formats) and jump ropes (in dvi, postscript, or pdf formats). Both handouts include graphics which are not reproduced on the web.
I began by defining an autonomous system as one in which the function giving u' doesn't depend upon time. The first examples to consider are homogeneous linear: u' = Au. I gave as an example x' = -y, y' = x. Each point on the plane has an associated velocity vector. I drew the vector field, and explained that a solution to the equation is a parametrized curve in which the velocity at the moment you are at a point u is Au. This gives circles around 0 in this case. I drew some: a phase portrait. 0 is another solution. The arrow of time is part of the portrait, but no other time information is recorded.
Differential equations has three aspects: analytic, numerical, and qualitative. This is a chapter in the qualitative aspect. One important issue si the long-time behavior. We'll say that u'=Au is
Any homogeneous linear system exhibits one of these behaviors.
Following the handout on Phase Portraits, I described the six nondegenerate pictures: proper node, star, improper node, saddle, center, spiral. For several I used the Matlab facility pplane5.