tues sep 23

MAIN POINTS
Some fixed points must exist for all values of a given parameter— however, the nature of their stability may change as the parameter changes. Transcritical bifurcation is the model for this. This is given by the equation x'=rx-x^2. As r varies the parabola defined by x vs. x' shifts from the left to the right, but always crosses the axis at x=0. Thus, there is no disappearance of the fixed point as there is with saddle bifurcation.

The next chapter discusses an application— lasers. Energy excites a material, which emits light. After a certain threshold, the energy is released in phase as a laser. The text shows that the differential equation for the number of photons, n'=gain-loss, can be rewritten in a form similar to the transcritical bifurcation. As pump strength increases, the fixed point goes from being a stable fixed point at 0 to an unstable fixed point at 0. This means that the number of photons increases given any slight prodding. A new stable fixed point appears for some positive value of n. The point at which the origin fixed point becomes unstable is called the laser threshold.

CHALLENGES
Some practice drawing bifurcation diagrams would be nice... it seems like you just have to draw a vector field for many values of a parameter, and pretend that they're cross-sections of the bifurcation diagram.

Is there a graphical way to envision Fig. 3.2.1 on the computer?

REFLECTION
3.3 provided a way about thinking about physics that I never thought about. It really is a new dimension of physics when you can state behaviors in terms of their derivatives rather than explicitly...

Also, the emphasizes that first order systems are interesting because of the way we can change parameters... Wouldn't this be similar to simply adding another variable, and making the system second-order?