### I'm interested in modeling and analyzing various real
world phenomena. Some of the topics I have worked on
and/or have an interest in working on are: Electronic
circuit dynamics, Chaotic scattering, Cancer modeling,
Particle Accelerator Physics, and Hydrodynamic Pilot Wave
Theory.

### 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.