Lorenz System: Visualization by Ross Mentor Dr. Bruce Stewart

© 1984 By Aerial Press & U.S. Department of Energy This movie gives a visual overview of the behavior of the Lorenz System, a simplified model of what happens when a volume of fluid is heated from below. The warm fluid at the bottom expands, becomes less dense that the cooler fluid above, and wants to rise to the top. The video begins with simultaneous views of (1) the temperature and movement of the fluid, represented by a rectangle with marker particles moving with the flow, and (2) the same dynamics represented in phase space, shown as a representative point moving along a dynamical trajectory. The fluid is seen in a two-dimensional cross-section. The marker particles are colored according to temperature: red is warm, blue is cool. Note that the marker particles are mostly red along the bottom. The Lorenz model supposes that enough heat passes out the top of the rectangle to keep a steady temperature difference between top and bottom. Hence the color of marker particles near the top is usually bluish. The phase space view was first imagined for dynamical systems by Henri Poincare. In the Lorenz model, the phase space has three coordinates, and can be visualized in three dimensions. Each coordinate measures a partial description of the dynamical state of the fluid. The x-coordinate in phase space measures the direction and speed of rotating motion of the fluid. Positive x values (toward the letter X attached to the axis) mean clockwise rotating movement of the fluid; negative values are counter-clockwise. The size of the x value gives the speed of rotation. The y-coordinate measures left-to-right temperature difference across the rectangle. When the fluid rotates, warm fluid moves up on one side of the rectangle, while cooler fluid moves down on the opposite side, changing the value of y. Because of this connection between x and y, most of the time they have the same sign (both positive or both negative). The z-coordinate measures the strength of an inversion layer in the vertical temperature profile. If the fluid rotates fast enough, intermediate warm and cool layers may develop, so that temperatures from bottom to top go from warm on the bottom to cool to warm to cool at the top. In the middle, warm over cool is a temperature inversion, which further complicates the dynamics. Note that in the first animation, the fluid in the rectangle is moving, but the representative point in the phase space is held in a fixed location. This is against the dynamics, an artifice intended to show that each dynamical state involves rotation at a fixed speed, together with fixed spatial pattern of temperature. When the dynamics are switched on, the representative point in phase space traces out a dynamical trajectory, showing how the motion and temperature patterns in the rectangle evolve. The focus of the movie is dynamical trajectories, patterns of groups of dynamical trajectories, and how they change as a parameter R increases. The R value is a measure of how much heat is applied at the bottom. So the overall plot is: set a constant heating rate, look at dynamical trajectories, identify patterns; then bump up the heating rate and repeat. Occasionally, the patterns of dynamical trajectories will change in very dramatic ways, which represent bifurcations. It is sometimes pointed out that the Lorenz system is drastically simplified as a model of fluid in a rectangle, so that beyond R =2 the behaviors of the Lorenz system differ from reality. However, the Lorenz system has been shown to be a very good model of a fluid confined in a torus, or shape of an auto tire inner tube, heated from below. . Experiments with this convection loop or thermosyphon have shown that the complex dynamics of the Lorenz system are most definitely real physical behavior. Credits This movie was composed and filmed on 16mm at Brookhaven National Laboratory in 1984, using then-state-of-the-art computer graphics hardware and custom animation software. Thanks to Alvy Ray Smith for being a fan. Transfer from 16mm film to video was done by FurnaceTV. Additional computer graphic animations of the Lorenz system were mastered to 1-inch analog videotape at the Princeton Interactive Computer Graphics Laboratory for the PBS Nova episode "The Strange New Science of Chaos," first aired January 31, 1989.