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
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:
- 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
- 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].
- 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].
- 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 .
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.
- 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).
- Daniel J. Cross and R. Gilmore, Differential Embedding of the Lorenz Attractor, Phys. Rev. E, 81, 066220 (2010).
- J. Katriel and R. Gilmore, Entropy of bounding tori , Entropy, 12(4), 953-960 (2010).
- Daniel J. Cross and R. Gilmore, Representation theory for strange attractors, Phys. Rev. E, 80(1), 056207 (2009).
- T. D. Tsankov, A. Nishtala, and R. Gilmore, Embeddings of a strange attractor into R^3, Phys. Rev. E, 69, 056215 (2004).
- T. D. Tsankov and R. Gilmore, Topological aspects of the structure of chaotic attractors in R^3, Phys. Rev. E, 69, 056206 (2004).
- T. D. Tsankov and R. Gilmore, Strange attractors are classified by bounding tori, Phys. Rev. Lett., 91(13), 134104 (2003).
- C. Letellier and R. Gilmore, Dressed symbolic dynamics, Phys. Rev. E, 67, 036205 (2003).
- R. Gilmore, Topological analysis of chaotic dynamical systems, Revs. Mod. Phys., 70, 1455-1529 (1998).
- G. B. Mindlin, R. Lopez-Ruiz, H. G. Solari, and R. Gilmore, Horseshoe implications, Phys. Rev. E, 48, 4297-4304 (1993).
- 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).
- 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).
- 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).
- N. B. Tufillaro, H. G. Solari, and R. Gilmore, Relative rotation rates: Fingerprints for strange attractors, Phys. Rev. A, 41, 5717-5720 (1990).
- H. G. Solari and R. Gilmore, Relative rotation rates for driven dynamical systems, Phys. Rev. A, 37, 3096-3109 (1988).