MA 226: LAB 5

The Lorenz Equations

This lab is due Tuesday, December 11, 2001, BEFORE CLASS BEGINS. No late labs will be accepted. I suggest that you use the IDE software in the Mac Lab. You will be graded on exactly what is asked for in the instructions below. You need not turn in any additional data, graphs, paragraphs, etc. You should submit only what is called for, and in the order the questions are asked. It is perfectly acceptable to hand in hand-drawn figures, since you cannot print pictures using IDE. Remember that you will be graded on your use of English, including spelling, punctuation, logic, as well as the mathematics.

IMPORTANT: The work you submit should be your own and nobody else's. Any exceptions to this will be dealt with harshly. Copying or paraphrasing another student's lab report is strictly forbidden.

Lab. In this lab you will investigate the famous Lorenz system of differential equations. This is a three dimensional system of equations given by

The Lorenz equations represent a simplified set of equations for describing convection rolls in the atmosphere---just a small piece of what makes up weather patterns. You can think of a container that is filled with a fluid. The fluid is heated from below and cooled from above. Consequently, the "warmer" fluid tends to rise, while the "cooler" fluid descends. This sets up a rolling pattern in the container. In the Lorenz system, this is vastly oversimplifed by thinking of a single "particle" of fluid that rises and falls in the container.

Your goal in this lab is to begin to understand these equations and to understand the related notion of chaos. Be forewarned that, to this day, nobody has been able to completely understand the Lorenz equations, so you should think of this lab as being open-ended.

Before going to the lab, work out the following questions.

1. Find the formulas for all equilibrium points for the Lorenz system. Your formulas will depend on the parameter R. Remember that R lies between 0 and 30. Then answer the question: at which R-value(s) do you expect to see a bifurcation of equilibrium points?

Now go the lab and open the IDE tool Lorenz Equations: Phase Plane 0 < R < 30. This will allow you to view the phase portrait for the Lorenz system. Since this picture is three dimensional, you must view it in a projection to one of the three planes: xy, yz, or xz. You can also view any or all of the x(t), y(t), and z(t) graphs. The little box shows an animation of the aforementioned fluid particle. Now adddress the following questions.

2. For which parameter values do all solutions seem to tend to equilibrium solutions? What happens at the bifurcation(s) that you discovered above? Explain in a paragraph or two with pictures.

3. By looking at both the phase plane (in the xz-view) and the fluid particle, answer the following questions in a sentence or two. What do the equilibrium points correspond to in terms of the fluid particle? What do you think the variable x corresponds to in terms of the motion of the particle?

4. Are there other "bifurcations" (i.e., major changes in what happens) in the Lorenz system? Are there other values of R for which solutions do not tend to equilibrium points? For what (approximate) value of R does this first occur? Explain what you see in an essay of not more than one page.

5. Now open the tool Lorenz Equations: Discovery 1963. In this tool you will see the x(t) plot for R = 28 and various initial values. You will mimic the discovery of Lorenz by watching a particular x(t)-graph out to time T-time units. Then you will watch another graph whose initial value consists of (x(T), y(T), z(T)), but where each of these numbers are rounded to 3 decimal places. (The blue curve and letters represent the solution corresponding to the rounded off initial conditions.) By clicking on different -values, you will see different plots.

What you see has been called "sensitive dependence on initial conditions" by mathematicians (and "the Butterly Effect" in the popular press). In a short essay, explain what you think sensitive dependence on initial conditions means given what you see on the screen.