Autumn 2023

A. Pasieka: Lorenz Attractor

Thursday, Sept 14, 21  (2pm, Nc 356)

Mike Sullivan

Geometric Lorenz Attractor

Video of Day 1
Video of Day 2

Here are links sent by Mike:

http://galileo.math.siu.edu/Courses/405/F22/MATH405Section9_8.pdf

http://galileo.math.siu.edu/Courses/405/F22/Lorenz.mp4  (18 min video)

http://galileo.math.siu.edu/Courses/DiffEq/GLA.pdf

http://galileo.math.siu.edu/Courses/DiffEq/video1384802557.mp4 (2 min video)

Plus, this is a series of well done videos on chaos theory. Number 7 covers some of the topics covered in my seminar talks.

http://galileo.math.siu.edu/Courses/405/F22/ChaosVideoSeries.html

Abstract of the talk:  The Lorenz equations, developed and studied by the meteorologist Edward Lorenz in the 1960s, were one of the first examples of a chaotic system. They are a set of three nonlinear ordinary differential equations in three variables. The trajectories move toward a bounded subset of $\mathbb R^3$ and then seem to oscillate about, but never truly repeat. Mathematicians began studying these equations using both quantitative and qualitative methods. The former proved very difficult. The latter yielded results. Guckenheimer and Williams developed a geometric model with behavior similar to the Lorenz equations. Then Birman and Williams used topological methods to analyze this Geometric Lorenz Attractor. Decades later it was proved that the two systems are equivalent.

Participants should have had basic undergraduate courses in differential equations and linear algebra. We will begin with a review of systems of ordinary differential equations. This should last about 20 minutes. Then we will cover the construction of the Geometric Lorenz Attractor. That should take about 30 minutes. Next week we will cover Birman and Williams' proofs of the knot types of periodic orbits in the Geometric Lorenz Attractor.

This material was originally covered last Fall in my MATH 405 course. For now, you may wish to amuse yourself with this video.