Problem 1: Lorenz in phase space

 

Okay, time for you to play with this beast and start to untangle some of its properties and underlying chaos (and order). Do the following

1. Code up the Lorenz equations in Stella and check their qualitative behavior against some of the runs in table 1. Don't worry if they don't match wiggle, for wiggle. With these equations, that is impossible. If you have difficulty getting it to work, let me know I have a pre-coded version available. I have found that most of the behavior can be seen running for about a time of 25 with a time step of 0.025. Always use a 4th order Runge Kutta scheme.
2. Extra credit: Show by substitution that this problem has 3 fixed points (no motion), and two rolls with different directions.
3. Investigate the evolution of these equations in phase space. Phase space for this problem is three dimensional but you can see most of the action if you plot W vs. . Do the following
   1. set r=10 and make a phase plot comparing two initial conditions with and (0,-1,0). Watch the time series plot at the same time and try to understand the relationship between the Time-Series and the phase portrait. Try some other initial conditions. Do you think this solution is stable? What happens when you get near to a fixed point?
   2. increase r to 28 (Lorenz's famous run) and rerun the the problem from (0,1,0) (this is Lorenz's famous run, which he did on a Royal McBee LGP-30 Computing machine at about one second per time step. Surprisingly, these Macs aren't much faster). Plot both the time series for W and the phase portrait. Now what is the behavior of the fixed points (do they attract or repel?) Can you explain qualitatively what is happening? Can you guess when the solution will flip?
   3. Now explore the ``sensitivity to initial conditions'', do 3 runs with initial conditions (0,0.9,0), (0,1,0), (0,1.1,0) (use the sensi spec menu to automate this). Make a comparison plot for W vs. time and W vs. . How long does it take for the different solutions to go their separate ways? At what point would you consider the behavior chaotic? In phase space, however, note that all the solutions still fall within the funny butterfly shaped strange attractor. It's not a fixed point, it's not a periodic orbit, it's just strange....