During the previous weeks we have seen two kinds of attractors; attracting rest points and limit cycles. This week we will use MAPLE to investigate our first strange attractor, the world famous Lorentz attractor. In 1962 Edward Lorentz was studying the following system of differential equations which provides a simplistic model of convection roles in the atmosphere:
Exercise 1. Using Trajectories to Trace Out Attracting Sets.
Given a general nonlinear dynamical system it is often quite difficult (if not impossible) to find all its attracting sets using pencil and paper. Finding rest points is simple enough but limit cycles and more complicated attracting sets are much harder to locate, especially in dimensions greater than two. To remedy this situation we can use the computer to help us on our way. The basic idea is this: if you follow a trajectory for a long enough time is will eventually tend towards an attracting set and once it is close to such a set it will travel along it and ``trace out'' some of its structure. As an example of this method we will use trajectories to find the limit cycle(s) of the following dynamical system:
Exercise 2. The Lorentz Attractor
Now we will use the method of Exercise 1 to get a glimpse of the structure of the Lorentz attractor. Enter the MAPLE worksheet as it appears in Fig.2. This yields a three dimensional plot of a trajectory for the Lorentz equations listed in the Introduction with parameters r = 28, s = 10, and b = 8/3. As you can see the trajectory quickly starts to travel along a mysterious butterfly shaped attractor. Change the initial conditions and verify that this is always the case.
The behaviour you are seeing does not occur for all values of the parameters. Change the value of r to 0, 5, 10, 15, 20 and back to 28 and see how the behaviour of the trajectory changes. See what happens when you change s and b as well.
Use the following instruction to plot just the variable x as a function of time.