Prof. Gilmore and his colleagues have been studying chaos in physical systems for many years. He is particularly interested in establishing an understanding of the structure of chaos.

Prof. Gilmore's group has created a hierarchy of discrete classifications for chaotic dynamical systems - and their strange attractors - in three dimensional spaces. There are four levels in this hierarchy. The first two levels have been described in Refs. 1 and 2.

From the smallest to the largest, these levels are:

1. Basis sets of orbits. These are sets of orbits in a strange attractor whose presence forces all other (unstable) periodic orbits that exist in the attractor. Up to any finite period, a finite basis set can be constructed algorithmically [1,2,3,4].
2. Branched manifolds. These are two-dimensional structures that are manifolds almost everywhere. They are not manifolds because they possess two types of singularities: splitting points and branch lines. The splitting points describe the stretching process that is ultimately responsible for the sensitivity to initial conditions exhibited by chaotic systems. The branch lines describe the return process that is ultimately responsible for maintaining bounded motion in the attractor. Branched manifolds organize all the unstable periodic orbits in a strange attractor in a very specific way, which is teased out by computing the gauss linking numbers of pairs of orbits. A more refined topological invariant, the relative rotation rates, offer even more information. We have created an algorithm for determining the branched manifold underlying physical experiments and applied it to determine the mechanism underlying some chemical data. Other laboratories throughout the world have adopted this procedure for analyzing chaotic data [1,2,5-9].
3. Bounding tori. In the same way that branched manifolds organize periodic orbits, bounding tori organize branched manifolds. Every branched manifold can be embedded in a ``minimal'' three dimensional manifold, called an inertial manifold. Its boundary is a torus. The flow generating the strange attractor, restricted to the torus surface, provides a canonical form that can be/has been used to classify every strange attractor studied in three dimensions [10-13].
4. Embeddings of bounding tori. The bounding tori described above are as seen from the ``inside'' (the classification is intrinsic). In analyses of data, we see the bounding torus as it is embedded in 3-space. Any bounding torus can be embedded in the surrounding space in a discrete number of ways. This enumeration is currently being worked out [14].

Prof. Gilmore and his colleagues have created the analog of Fourier analysis for nonlinear dynamical systems in three dimensions. Strange attractors, or their caricature, branched manifolds/bounding tori, are build up Lego-style from two basic building blocks, one containing splitting singularities, the other joining singularities, in a way that is systematic yet with sufficient degrees of freedom to allow an even richer variety of behavior in physical systems than has yet been seen.

Selected Publications:

1. R. Gilmore, Jean Marc Ginoux, Timothy Jones, C. Letellier, and U. S. Freitas, Connecting curves for dynamical systems, J. Phys. A: Math. Theor., 43(25), 255101 (2010).
2. Daniel J. Cross and R. Gilmore, Differential Embedding of the Lorenz Attractor, Phys. Rev. E, 81, 066220 (2010).
3. J. Katriel and R. Gilmore, Entropy of bounding tori , Entropy, 12(4), 953-960 (2010).
4. Daniel J. Cross and R. Gilmore, Representation theory for strange attractors, Phys. Rev. E, 80(1), 056207 (2009).
5. T. D. Tsankov, A. Nishtala, and R. Gilmore, Embeddings of a strange attractor into R^3, Phys. Rev. E, 69, 056215 (2004).
6. T. D. Tsankov and R. Gilmore, Topological aspects of the structure of chaotic attractors in R^3, Phys. Rev. E, 69, 056206 (2004).
7. T. D. Tsankov and R. Gilmore, Strange attractors are classified by bounding tori, Phys. Rev. Lett., 91(13), 134104 (2003).
8. C. Letellier and R. Gilmore, Dressed symbolic dynamics, Phys. Rev. E, 67, 036205 (2003).
9. R. Gilmore, Topological analysis of chaotic dynamical systems, Revs. Mod. Phys., 70, 1455-1529 (1998).
10. G. B. Mindlin, R. Lopez-Ruiz, H. G. Solari, and R. Gilmore, Horseshoe implications, Phys. Rev. E, 48, 4297-4304 (1993).
11. F. A. Papoff, A. Fioretti, E. Arimondo, G. B. Mindlin, H. G. Solari, and R. Gilmore, Structure of chaos in the laser with saturated absorber, Phys. Rev. Lett., 68, 1128-1131 (1992).
12. G. B. Mindlin, H. G. Solari, M. A. Natiello, R. Gilmore, and X.-J. Hou, Topological analysis of chaotic time series data from the Belousov-Zhabotinskii reaction, J. Nonlinear Science, 1, 147-173 (1991).
13. G. B. Mindlin, X.-J. Hou, H. G. Solari, R. Gilmore, and N. B. Tufillaro, Classification of strange attractors by integers, Phys. Rev. Lett., 64, 2350-2353 (1990).
14. N. B. Tufillaro, H. G. Solari, and R. Gilmore, Relative rotation rates: Fingerprints for strange attractors, Phys. Rev. A, 41, 5717-5720 (1990).
15. H. G. Solari and R. Gilmore, Relative rotation rates for driven dynamical systems, Phys. Rev. A, 37, 3096-3109 (1988).