M. Golubitsky, K. Josic and T.J. Kaper
An unfolding theory approach to bursting in fast-slow systems
In: Global Analysis of Dynamical Systems: Festschrift dedicated to Floris Takens on the occasion of his 60th birthday.
(H.W. Broer, B. Krauskopf and G. Vegter, eds.)
Institute of Physics Publ., 2001, 277-308.
Many processes in nature are characterized by periodic bursts
of activity separated by intervals of quiescence. In this chapter we
describe a method for classifying the types of bursting that occur in
models in which variables evolve on two different timescales, i.e.,
fast-slow systems. The classification is based on the observation that
the bifurcations of the fast system that lead to bursting can be collapsed
to a single local bifurcation, generally of higher codimension. The
bursting is recovered as the slow variables periodically trace a closed
path in the universal unfolding of this singularity.
The codimension of a periodic bursting type is then defined to be
the codimension of the singularity in whose unfolding it first
appears. Using this definition, we systematically analyze all of the
known universal unfoldings of codimension-one and -two bifurcations
to classify the codimension-one and -two bursters. Takens was the first
to analyze the unfolding spaces of a number of these. In addition,
we identify several codimension-three bursters that arise in the
unfolding space of a codimension-three degenerate Takens-Bogdanov
point. Among the periodic bursters encountered in mathematical models
for nerve cell electrical activity, so-called elliptical, or
type III, bursters are shown to have codimension two.
Other bursters studied in the literature are shown to first appear in
the unfolding of the degenerate Takens-Bogdanov
point and thus have codimension three.
In contrast with previous classification
schemes, our approach is local, provides an intrinsic notion of
complexity for a bursting system, and
lends itself to numerical implementation.