Dr. Gilmore is director of the Nonlinear Dynamics Group at Drexel University. A common theme throughout his work is the desire to bring modern mathematics into physics. A first incarnation of this goal was his detailed study of Lie groups and their applications for Physics (see the book page for his Lie Groups text publication). Dr. Gilmore next turned to the subject of Catastrophe Theory (again, see the book page for this text). The text that resulted was translated into Russian on the recommendation of V. I. Arnol'd. This work would lay the foundation for Dr. Gilmore's entry into Nonlinear Dynamics where he, his colleagues, and his students would create a suite of topological tools for the analysis of nonlinear data. His long-range goal is to develop an analysis procedure for nonlinear data beyond its current three dimensional limit.
Mr. Romanazzi is in his third year of graduate studies under the advisement of Dr. Gilmore. He has previously studied the characterization of inequivalant templates of strange attractors in R^3. He is currently studying the effect of embeddings, showing that the stretch and fold mechanism is an invarient of embeddings. Mr. Romanazzi plans to do his thesis work on time serious and embeddings, with an eye towards their applications for empirical data.
Mr. Jones is in his final year of graduate studies. He did his Oral Qualifier on Quantum Decoherence with a committee that included Dr. Lorenzo Narducci. He has recently published a paper with Dr. Robert Gilmore and presented a poster on this paper in Lille France at the ECC11 conference. Further information on his research, including programs, simulations, journal articles, and reports, can be found on his web site.
Mr. Coy is in his final year of graduate studies and is advised by Dr. Gilmore. Ben has done previous research in the nonlinear dynamics field with a project investigating the calculation of local Lyapunov exponents. He is currently studying strange attractors generated by nonlinear oscillators.
Mr. Michaluk is in his second year of graduate studies under Dr. Robert Gilmore. Mr. Michaluk has previously researched harmonic knot embeddings and other techniques for analyzing time series. Ryan enjoys strange attractors and long walks on the beach.
Sam Kennerly is studying entropy production in linear dynamical systems subject to random inputs. His current focus is the effect of stochastic Hamiltonians on von Neumann entropy in density matrix quantum mechanics. Other research and/or random noise can be found at einstein.drexel.edu/~skennerly/.
Dr. Cross graduated with his Ph.D. under the advisement of Dr. Gilmore in 2010. His research interests are in mathematical physics, Field Theory, Mach's Principle, and higher dimensional nonlinear dynamics. He is currently using insights in Nonlinear Dynamics to investigate the magnetic field lines of closed knotted wires.
Mr. Ramos recently received his PhD under Dr. Gilmore. His has worked in quantization of constrained systems such as free electromagnetic fields and Dirac fields. His PhD work was on a Lagrangian Formalism of Gravitoelectromagnetism including new guage transformations.
Dr. Gilmore and Dr. Tsankov continue to collaborate in the search for new topological methods to classify higher dimensional chaotic systems. Dr. Tsankov is also interested in applications of Differential Geometry and Topology to problems in Classical and Celestial Mechanics, Biomechanics, and Control Theory.
Prof. Lefranc carries out both experimental and theoretical work. He has developed an extensive data base from a series of laser experiments. Chaotic data sets have been developed on periodically driven lasers. These data sets contain attractors of three dimensions and higher. He has also developed a suite of powerful tools for extracting unstable periodic orbits from data, assigning a symbolic name to these orbits, and computing their topological entropy in a simple, straightforward way. He and Prof. Gilmore are together attempting to extend topological analysis methods into higher dimensions. Together he has written a book with Prof. Gilmore: R. Gilmore and M. Lefranc, The Topology of Chaos, NY: Wiley, 2002.
Prof. Letellier has an interest in determining the mechanisms that contribute to the generation of distinct types of chaotic attractors. He is also interested in the effects of symmetry on strange attractors. Prof. Letellier has used symmetry to generate a large number of strange attractors with different global topological structures. Together with Prof. Gilmore, he is completing a text: R. Gilmore and C. Letellier, The Symmetry of Chaos, (nearing completion).
Mr. Byrne is a 2004 graduate from Drexel University where he received his B.S. in Physics. As an undergraduate, he collaborated with Dr. Gilmore and Dr. Letellier in an NSF sponsored project investigating the global relationships between cover (symmetric) and image (non-symmetric) chaotic strange attractors. Under Dr. Gilmore's supervision, Mr. Byrne completed his undergraduate thesis "First Return Maps and Bounding Tori as Tools for Topological Analysis of Chaotic Data". The project developed a procedure which facilitates the application of topological methods of analysis to noisy experimental data. His current research focuses on understanding how the basic mechanisms responsible for chaotic behavior in strange attractors change under symmetry transformation and control parameter variation.
Daniel J. Cross, Ryan Michaluk, and R. Gilmore
R. Gilmore, Jean Marc Ginoux, Timothy Jones, C. Letellier, and U. S. Freitas
Daniel J. Cross and R. Gilmore