Judith Hubbard | judithh@its.caltech.edu | (607) 272-7562 |

This webpage documents the work I have been doing in the Cornell Math REU of 20002. I and four other students have been working with Dr. Guckenheimer and Dr. Weckesser in studying the Forced Van der Pol equation. For a more comprehensive review of the system, go to Katy Bold and Chantal Edward's website.

My work has focused on the full system. I have been researching periodic solutions, with and without canards, and the symbolic dynamics involved therein.

Research notes:

- periodic canards (ps) 7/12/02
This contains a very brief introduction to the forced van der pol equation and explains what I am looking at.

- periodic canards continued (ps) 7/15/02
This paper is simply a continuation of the previous one; it examines certain portions of the period 3 bifurcation diagram more closely to clear up questions which are listed at the beginning of the paper.

- asymmetric canards of period 3 (ps) 7/17/02
This paper examines a symmetry-breaking bifurcation on the bifurcation diagram of period 3 orbits and follows that asymmetric curve.

- patterns in period doubling cascade (ps) 7/23/02
Here I study a period doubling cascade coming off of the asymmetric curve mentioned in the previous paper and makes conjectures about patterns found in the doubled-period trajectories, where we see symbolic dynamics.

Pictures and figures not in the notes:

- Bifurcation diagram for eps = 0.001
This is a plot of the period doubling bifurcations for parameter values omega = 1.55 and epsilon = 0.001. All the curves I have found so far are on it, up through period 12.

- Bifurcation diagram of period 3 symmetric solutions for various epsilon, in black and white
- Same as above, only in rainbow color
- Magnified view of the first (b&w) diagram, in the complicated region where a is small
- Same as above, only in rainbow color
These pictures contain many plots of the period 3 symmetric solutions. In the color diagram, the purple curve is for epsilon = 0.0001, and then epsilon increases down the rainbow, so blue = 0.0003, green = 0.0005, yellow = 0.001, orange = 0.002, and red = 0.004. For all of these, omega = 1.55.

- Period 5 bifurcation diagram
These solutions are for epsilon = 0.0001, omega = 1.35.

- Period 3 bifurcation diagram for different omega, with doubling
These solutions are for epsilon = 0.001, omega = 1.388. This shows all of the period three symmetric bifurcation diagram, the asymmetric period three diagram, and one period six curve.

- Magnified view of previous figure
This magnified view only contains part of the symmetric period three diagram, but all of the asymmetric one and the period 6 curve mentioned above.

Postscript figures from the research notes above: Postscript figures

For more detail about the program, and more relevant links, see Dr. Weckesser's website about the REU.