Tues 10/7

MAIN POINTS
Chapter four considers differential equations solved for points on a circle, rather than as a vector field on a line. This seems to be the simplest case for which periodic behavior is possible. Example 4.1.1 solves for the very simple Theta'=sin(Theta), which has one stable FP and one unstable FP on opposite sides of the circle. It is simpler than considering the equation as on a line. 4.2 defines an oscillator. It is simply constant motion around the circle. If you subtract two oscillators you get a phase difference equation, which shows how the two oscillators go in and out of phase with each other.

4.3 defines a nonuniform oscillator. The value of the parameter determines whether or not there's a fixed point. The low values of the differential equation correspond to a "bottleneck," which is where the FP is most likely to occur as the parameter changes. These are due to ghosts, which are ghosts of bifurcation points.. hm. THey use the square-root scaling law somehow to calculate the time spent in the bottleneck.

CHALLENGES
What's the difference between a ghost and a bottleneck? They sort of seem like the same thing. How did they arrive at the integral that they use to calculate the oscillation period in 4.3? What is the relevance of this square-root scaling law?

REFLECTIONS
I know that earlier in the book they defined an oscillator, just not on the circle. There seems to be an interesting relationship between defining something on a straight line and defining it on the circle. For instance, defining the sinx function on the straight line means giving infinite solutions, while you only give one solution when you define it on the circle.