Chaos in the Lorenz Model

Read Garcia chapter 3.4 Borrow the Gleick book from me (footnote Garcia page 86)

Make sure you understand the physical meaning of the equations.

This project is a little open-ended. Ultimately, the goal here is to explore the idea of “sensitive dependence on initial conditions” in the context of “Lorenz Attractors”. You should look up those terms and understand what they mean. Figure 3.13 shows the Lorenz attractors you will be studying. Whether you develop the adaptive RK4 code Garcia talks about or stick with regular RK4 is up to you: all the code you need is there in Garcia (and therefore all on the web), if you want to copy and re-write, that’s okay, as long as you give Garcia credit.

Problems 22, 24, and 25 should give you some ideas of places to start. You should definitely verify that your trajectory lines never cross each other – that’s often very difficult to see, but consider: if you did have two trajectories that crossed, then you could start your calculations at the crossover point. How would the system know which trajectory to follow?

What I want you to explore are the following questions: 1. given the SDoIC, do your resulting trajectories have any physical meaning? 2. Can you find any kind of pattern to (a) when the trajectory switches between attractors and (b) how many times it goes around each attractor before switching?

Make sure you make plots of x, y, and z vs. time as well as 3-d parametric plots of x-y-z.

Ultimately, what I would like you to present to the class is the physical meaning of the equations, how you get chaotic behavior out of the equations (SDoIC), and yet there are more or less stable zones in the chaos (the attractors). However, how close any given trajectory will get to the attractor, how many times it goes around, and how often it switches back and forth remain essentially unpredictable. No two trajectories ever cross. You should also consider the implications of chaos on how much we can believe our trajectories. Explore those topics and present your conclusions to the class.

Other resources: http://en.wikipedia.org/wiki/Lorenz_attractor http://hypertextbook.com/chaos/21.shtml