In the paper

M. Frankel, G. Kovacic, V. Roytburd and I. Timofeyev, Finite-dimensional dynamical system modeling thermal instabilities, Physica D 137 (2000), 295-315. ( Complete text in pdf format)

we uncovered a whole zoo of very interesting dynamic scenarios for a relatively simple 3-by-3 system of ODEs. I believe that these scenarios can be explained through a very novel bifurcation mechanism. To justify this conjecture a series of numerical computations should be done on the system in search of special solutions. There exists a package in the public domain, AUTO, which is especially appropriate for the task.

A substantial part of the project is to achieve proficiency in using AUTO and probably Matlab. The research will include study of important aspects of modern qualitaive theory of ODEs (or dynamical systems), in particular the role of homoclinic orbits and bifurcation will be elucidated. I also should note that the paper mentioned above originated as an undergraduate research project several years ago.

Another project involves qualitative study of ordinary differential equations (4-by-4 in this case) that come from a stability investigation for detonation waves. It will involve symbolic (Maple or Mathematica) as well as numerical computations.

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RPI Math