ABSTRACT:

Solutions to autonomous systems of three ordinary differential equations are curves in R³ or S³. When the solutions are periodic, they are closed curves, and may be thought of as topological knots. We can use the infinite set of unstable periodic orbits that are the solutions to chaotic systems with strange attractors to characterize some of the topological properties of the solution set. We review all the relevant background and end with a deep and surprising result proved by Robert Ghrist a little over 10 years ago -- there exist chaotic systems whose solution sets contain every knot and kink type.