|Judith Hubbardemail@example.com||(607) 272-7562|
My work has focused on the full system. I have been researching periodic solutions, with and without canards, and the symbolic dynamics involved therein.
This contains a very brief introduction to the forced van der pol equation and explains what I am looking at.
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.
This paper examines a symmetry-breaking bifurcation on the bifurcation diagram of period 3 orbits and follows that asymmetric curve.
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:
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.
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.
These solutions are for epsilon = 0.0001, omega = 1.35.
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.
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.