Page is under construction.... (and research is ongoing!)
Dynamical systems, and their applications to physical problems, is the main theme of my research. This webpage provides an overview that is more informal that a research statement. (Links in appropriate places coming soon - first to my papers on the "Publications" webpage, then to others' papers and websites related to the topic... Oh, the self-centered universe....)
Nontwist maps are ubiquitous in studying physical systems ranging from stellar dynamics to oceanography. My dissertation(link...) contains an extensive bibliography with references to these applications.
These maps show a rich variety of bifurcations. The periodic orbits collisions are the local bifurcations while reconnection of separatrices lead to a global change in topology. We have uncovered how the interactions between these leads to many new phase space structures. (links: Wurm-Apte-Morrison04, Wurm-Apte-Fuchss-Morrison05, Apte-Llave-Petrisor06)
A thorough understanding of these bifurcations is essential to study the breakup of invariant circles in nontwist maps. Unlike twist maps, some of the invariant circles can undergo the same bifurcation as the "periodic orbit collision" and simply disappear, while others can go through criticality. The circles "at the point of collision" are called shearless circles. These can be tracked as perturbation is increased and eventually they reach criticality. What remains of them (are they "cantori") is not known. We have studied several of these breakups and revealed the universal behavior of the critical circles of many different winding numbers. (links: Apte-Wurm-Morrison03, Fuchss-Wurm-Apte-Morrison06) We have studied the regularity of the critical invariant circles. (links: Apte-Llave-Petrov05)
This universal behavior leads to the renormalization framework. A lot of work has focused on renormalization for critical circles in twist maps and in circle maps. We have constructed renormalization group operators that are inspired by the self-similar structure of the periodic orbits near the critical circle. These operators are intimately associated to the transformations that relate successive continued fraction convergents of the frequency of the critical circle. (links: Apte-Wurm-Morrison03, and 05)
In my dissertation, I show how these operators predict the critical behavior of sequences of periodic orbits other than the continued fraction convergents. In fact, there is a countable infinity of such sequences, generated by these operators. I show that for the noble circles in nontwist maps, these sequences indeed show the predicted behavior. (likns: dissertation)
I believe that this framework can be fruitfully applied to other criticalities, such as those in twist maps and circle maps. I also hope that this will lead to making more concrete the "residue map coordinates" that were used in an intuitive sense by John Greene, whose work has been a great inspiration to me.
Now, this part of the research does not follow naturally from the above two! Data assimilation refers to the set of tools and techniques for combining the data about a system with the output of its model to produce a prediction. It is not a unified mathematical or physical theory [yet! I am working on it!! :)]
Here is a naive description: take a chaotic system, evolve it in time starting with an initial condition (say, our best guess based on past observations), and compare the observations of the system with the predictions. It is clear that in all likelihood, the predictions will not exactly match the observations (even within the error bounds of the data). Now, the challenge is: what do we do with either the initial conditions or the model equations so that we get "better" predictions! The real problem can be a lot worse in the sense that the dynamical model might not be a good representation of all the dynamics of the system such as the atmosphere (model error); we might not have a good idea about whether the mathematical model even has an attractor; and what such a attractor of the model has to do with the real system!
In the statistical language, this problems is naturally formulated in terms of sampling from the posterior distribution of the state, given the data and the model. We are developing Monte Carlo techniques based on Langevin equation to sample this posterior. The results from a linear shallow water model with two Fourier modes shows that this distribution has interesting structure that incorporates the data as well as the dynamics of the model. One of the big challenges is the adaption of this method for bigger complex models.(links: Apte-Hairer-Stuart-Voss06, Apte-Jones-Stuart07)
"Model errors" pose a very significant conceptual challenge in the data assimilation problems. We are studying this problem within the Lagrangian data assimilation setting. The model is taken to be that of Lagrangian particles, but the data comes from inertial particles with small mass. The analytical understanding comes from the geometric singular perturbation theory while the numerical experiments will be designed to understand how this "model error" affects the assimilation of the inertial particle data. The eventual goal is to formulate a coherent mathematical framework for the data assimilation in the presence of model errors. (Yes, I was only half joking at the beginning of this section!)
For my Ph.D. qualifier, I worked with Li Jiang under the supervision of Professor Fischler. We were calculating the two loop contribution to the entropy of the N=4 supersymmetric Yang-Mills theory, which is also related to the entropy of black holes. The same week as our qualifier talks, the following paper appeared on arxiv with exactly the same results!! :) hep-th/9811224
As a fellow of the wonderfully enjoyable GFD summer school at Woods Hole, in addition to learning to play softball, I was also working with Oliver Buhler from Courant to learn about the methods that Andrews and McIntyre('78) and later Buhler ('00) [and other too?] developed for studying the effect of waves on the mean flow, and was applying them to non-Newtonian flows. We were also having fun watching many "counterintuitive phenomena" (if one has an intuition for Newtonian fluids) such as rod climbing and bubbles with a cusp.
While at Austin, I also got interested in the Ph.D. dissertation topic of another Phil Morrison student, Todd Krause. I am working with him and Phil on the Darwin system - the electromagnetic potentials expanded up to second order in v/c - and the action for these approximate equations. There are very interesting subtleties, including non-gauge-invariant equations and charge non-conservation!