Mathematical Modeling of Social Phenomena

Dr. Courtney Brown

Assignment #12

Use Phaser for this assignment. You are going to investigate the most famous example of chaos in continuous time, the Lorenz attractor. (The most famous example of chaos in discrete time is the logistic map.) Since the Lorenz attactor is a continuous time model with three dimensions (i.e., variables), you will have to use a Runge-Kutta to work with this model. The three equation model is,

dx/dt = s(y - x)

dy/dt = rx - y - xz

dz/dt = xy - bz

You can find the model in Phaser's 3D library. Originally, the variables were related to air currents, heat, and convection. (Lorenz was a mathematician turned meteorologist.) However, I would like you to play with the definition of the variables to see if you can come up with some social interpretation of the dynamics captured by the model algebra. (In general, we never reinterpret other people's models ... unless we are working with general statistical models. We are doing it here only to give you practice in thinking about interpretations of continuous time models with interaction terms.) Your reinterpretation should consume no more than two pages of this assignment (one is better). In the remainder of the assignment, you are to produce three phase portraits of the Lorenz attractor. The first will be a picture with one initial conditioni with the z axis perpendicular to the screen (and thus invisible). The second phase portrait will be the same thing but with two (close) initial conditions. This is to demonstratie how history diverges dramatically overtime with chaos, even if the initial conditions are very close. Produce three parallel time series plots for this phase portrait as well. Each time series plot will be for a separate variable, but containing two initial conditions. (Since there are three variables, you need three time series plots!) The third phase portrait should be the same as the first phase portrait, but this time rotated on its axes in order to show the z axis. For this plot, make sure the y-axis remains vertical! Use the same parameter values that are given in Phaser. This assignment will help you understand the sensitivity to initial conditions as well as the seemingly randomness of chaos. Try a smaller step size than 0.1 to get smooth plots.