Date of Award

Fall 2008

Project Type


Program or Major

Mathematics: Applied Mathematics

Degree Name

Master of Science


In recent years, several methods of controlling chaotic systems have been developed and implemented. The main idea in each method is to stabilize on an orbit around a chaotic attractor, which generally has a dense set of unstable periodic orbits. One such control scheme repeatedly applies a sequence of controls to a double scroll oscillator. Most control sequences result in the stabilization of an approximate unstable periodic orbit regardless of initial condition. These stabilized periodic orbits are called chaotic unstable periodic orbit-lets (cupolets). Due to the nature of cupolets, it is possible to switch between cupolets, and thus periodic orbits, by changing from one control sequence to another. Switching between orbits is a continuous and smooth transition, but may involve significant chaotic transients.

We will present three methods of transitioning between cupolets and suggest some applications of this procedure. The first method involves applying the second control sequence at a location on the first orbit. The second method is a zero-length transition which can be used if two cupolets intersect. The third method is applicable when transitioning between non-intersecting cupolets. This method switches between intermediate cupolets in an efficient controlled manner in getting from one cupolet to the next.