Welcome to Lana Panasyuk website

Chaos came to my life when I decided to participate in preparation and assisted teaching for the Geosystems class - part of my MIT EAPS Department new Professional Master's Degree Program.

I want to shear with you two models of chaos I explored: Driven Pendulum and Lorenz Attractor.

In spring 1996, listening the introduction of the chaos in solar system by Jack Wisdom in terms of pendulums, I decided to build a simple model of Driven pendulum. My older son, Sanya , offered his Lego sets and took an active part building the model: here is photo of Sanya and the pendulum.

The pendulum (the gray stick at the bottom of the colorful column) is driven through the string (the black pipe connecting the pendulum and the blue stick at the top of the colorful colunm) by the sinusoidal vertical motion. We simulated the sinusoid by attaching the blue stick to the wheel which rotates by the Lego motor through the set of gears (you won't see them because they are hidden at the top of the colorful column.)
The system displays all major stages of the chaotic system. We regulate the driving frequency by changing the angular speed of the wheel through the Lego motor controls. In the low input energy (slow wheel rotation) the pendulum returns to the vertical hang from any initial conditions. At the high energy - it rotates with full circle around and around (another steady state). At the moderate energy the pendulum swings back and forth, up and down. Sometimes it looks like ready to stop, but next moment it swings even more. So the chaos is there!

It's hard to describe, so stop by Jack Wisdom's office (we gave this "toy" to him for his class) and look yourself! Or try to build it - it's fun!

Lorenz Attractor is a chaotic system first described by Edward Lorenz of MIT in 1963. The system is ideal for investigation of a simple chaotic system behavior!

I coded it in MATLAB to facilitate the chaos exploration: it integrates the three coupled nonlinear differential equations that define the 'Lorenz Attractor' and shows the demo. As the integration proceeds you will see: a point moving in 3-D parameter space (top left plot), a point in the time-temperature axes, or a tent map (top right plot), and temperature accociated with the advection: difference between actual temperature and temperature in the absence of convection (center plot).
Lorenz Attractor Interface

Depending on the initial conditions and Rayleigh number value, the system will show most stages of a chaotic behavior.

By changing the Rayleigh number relative to its critical value (which equals to 657.5 for the onset of thermal convection in a fluid layer heated from below), you will find regimes of:

no motion (conductive heat transfer);
unstable zero attractor together with stable clockwise or counter-clockwise flows;
unstable transient chaos together with stable clockwise or counter-clockwise flows;
three possible stable attractors: chaotic attractor, clockwise, and counter-clockwise flows;
clockwise and counter-clockwise flows become unstable but the chaos is stable;
Lorenz Chaotic Attractor is stable;
limit cycles appear at high Rayleigh numbers.

References

Lorenz, E.N., Deterministic non-periodic flow, J. Atmos. Sci., 20, 130-141,1963
Ott, Edward, Chaos in dynamical systems, Cambridge University Press,1993

This code was modified from lorenz.m MATLAB-file by me for educational purposes of MIT class on GeoSystems, 1996. Since so many people have contacted me asking for the code, I've decided to place it here. You could use the code as is, but please honour the authorship.