Monday, September 3, 2012

Notes from the Chaos Cookbook, Chapters 2 and 3

The remaining portion of chapter 2 in the Chaos Cookbook involves properties of the logistic equation logistic map, which was first used by biologists and ecologists to model population growth. This function displays chaotic behavior as its parameter k is varied between 2.7 and 4, roughly.

There are regions of relative stability in the logistic equation's map's bifurcation plot of the possible output values vs. k, followed subsequently by a period doubling cascade. There is also self-similarity in the plots (I've added one of my own below).

A strange attractor for the logistic equationmap is the set of all possible output values for given k. It is strange because the values do not appear in a predictable or meaningful order.

It's been proved that a period 3 system (three possible output values) indicates the onset of chaos.

Feigenbaum's constant is roughly 4.669. It applies generally to many chaotic systems.

Chapter 3 concerns differential equations, their phase space plots and features, and numerical methods for solving them. I am largely familiar with all of this information, so I only loosely read this chapter.

Of special note in this chapter was a reminder to myself that phase space trajectories for a system of differential equations may not overlap.

Also of note was that, for these systems, a strange attractor is a phase space trajectory that does not settle into some periodic limit cycle and often shoots around to different parts of phase space.

Python code for some of these projects may be found at my personal website: