MA 226: LAB 4

The Lorenz Equations

This lab is due Thursday, April 22, 2004, BEFORE CLASS BEGINS. No late labs will be accepted. You should use the software called "Lorenz Equations" and "Butterfly Effect" for this 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 it is expensive to print the color pictures from the software. 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

Here R is a parameter that you will vary between 0 and 30. R is called the Rayleigh number for the system.

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.

First 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 open the tool Lorenz Equations. 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 should use the xz-view. 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 Butterfly Effect. In this tool you will see the solutions for two different initial values, one with x = 0, the other with x = 0.001 for R = 28. You will choose the y and z-values. Use the yz-view of the 3 dimensional phase space to view the solutions. You will see both the phase space and the y(t) graphs of your chosen solution. By clicking on different -values, you will see different plots.

First explain what the moving straight white line in the yz-plane menas. Next explain what the grey shadings in the y(t)-graphs mean. Then, relate this to the motions of the two fluid particles in their boxes to the right. 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.