Electronic circuit dynamics:

It has been observed that some circuits involving feedbacks act in an erratic manner.  In electrical engineering terms, they may cause logical contradictions and races.

We first modeled the R-S flip-flop as a discrete dynamical system in an ad-hoc manner.  We showed this system can be chaotic for certain initial conditions.  Here are some pretty pictures from the simulations of this model:

Not satisfied with an ad-hoc method, we decided to develop a discrete dynamical model from first principles, i.e. modeling each gate and combining the models in a logical manner to arrive at a model for the entire system.  This way, not only can we get more realistic results for the R-S flip-flop circuit, but also for any logical circuit we desire to model.  Since logical circuits with no feedbacks are stable, our model can predict the output given a certain input perfectly, but this is a trivial case.  Testing it against physical realizations of the R-S flip-flop and other circuits with feedback, such as various ring oscillators, really shows the versatility of our model.

Chaotic scattering:

Chaotic scattering has been studied from the early 70s and 80s in solitary wave collisions from
the Phi-Four equation (called Kink-Antikink collisions).  These were mainly numerical studies that gave insight into the phenomena.  However, since the equation is so difficult to work with there has been very little analysis done.  In more recent years reduction techniques have been used to approximate the Phi-Four PDE with a system of ODEs and also as an iterated map.

We have gone further and developed a mechanical analog (a ball rolling on a special surface) of chaotic scattering in Kink-Antikink collisions.  This was done in order to conduct experiments.  In addition to experiments we have analyzed the system thoroughly, including the dissipation
that comes from friction.  The experimental setup is shown bellow.

Distribution of metastases:

The proper prediction of how metastatic tumors are distributed can help save lives.

We used a model from Iwata et. al. (2000) and sought ways of simplifying it, improving upon it, and finding numerical solutions.  We saw that the Iwata model can be solved using an upwind scheme.  For the new models we focused on the affects of drugs on tumors, and simplifying the PDE into ODEs.  Drug affects was a major focus due to the lack of models taking drugs into account in the literature.  This is difficult to do, however, because drugs attack cells indiscriminately.  Probabilistically it has a bigger affect on larger tumors than smaller ones, but it seems as though a stochastic model is needed.

Alternate proof of Peixoto's theorem in 1-D:

Peixoto's theorem is one of the most important theorems in Dynamical Systems.  It was proved by Dr. Mauricio Matos Peixoto in 1962.  This proof is extremely involved - far too involved for most undergraduate students to follow.  We develop an alternate - pedagogical proof of the simpler 1-D case, with the goal of allowing senior undergraduate students to follow and understand the proof and consequently some of the ideas involved in the much bigger proof of the 2-D case.

Ferrofluids:

I have designed a system involving ferrofluids, and will model and analyze it once the pending papers on circuit dynamics are published.

Bubble logic:

I have designed the R-S flip-flop for the bubble logic system on paper.  I hope to implement it physically in the near future.

Particle Accelerator Physics:

We numerically simulate the beam dynamics of the Energy Recovery Linear Particle Accelerator (ERL) design for Argonne National Lab's (ANL) Advanced Photon Source (APS).  The code BI, created by Ivan Bazarov, is benchmarked against our own code for simpler Accelerators.  Then the full ERL is simulated and the results analyzed.  We conclude that the ERL, if built, would theoretically be stable.  Therefore, it would be pheasible to build it.

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