Thu 11/6

MAIN POINTS
6.0 states that we will begin studying nonlinear systems. (That was fast...) 6.1 defines a vector field on a plane, where each point on the plane has a velocity vector. A phase point traces out a solution on the vector plane. Because analytically finding solutions is so difficult, we focus on qualitative behavior of nonlinear systems, i.e fixed points, closed orbits, stability. The chapter introduces Runge Kutte. 6.2 gives a theorem for the existence of solutions, which are guaranteed it f is continuously differentiable. DIfferent trajectoris do not intersect. There is also a theorem that a trajectory confined to a region without a fixed point will eventually reach a closed orbit. 6.3 looks at the stability near a fixed point by substituting a disturbance into the Taylor expansion to find the Jacobian matrix, which is similar to f'(x) but for many variables. The text defines various sorts of behavior— repellers, attractors, saddles, centers, and fixed points.

CHALLENGES
I would like to practice making and using a Jacobian matrix. Substitution of polar coordinates (used on p. 153) is something I need to brush up on, and going through the Runge Kutte would be nice, too! It's something I didn't really get in Scientific Computation. Also... an explanation of topological equivalence..

REFLECTION
So yeah, we did Runge Kutte in Comp 365. We also used Jacobian matrix then, but I was never sure of the definition of a Jacobian. I see the Taylor expansion show up a lot in proofs, but we have never had to use it in practice— will we ever?