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An unfolding theory approach to bursting in fast-slow systems. (English) Zbl 1215.34043

Broer, Henk W. (ed.) et al., Global analysis of dynamical systems. Festschrift dedicated to Floris Takens for his 60th birthday. Bristol: Institute of Physics Publishing (ISBN 0-7503-0803-6/hbk). 277-308 (2001).
Summary: 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.
For the entire collection see [Zbl 0971.00062].

MSC:

34C23 Bifurcation theory for ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
34E15 Singular perturbations for ordinary differential equations
37G05 Normal forms for dynamical systems
58K60 Deformation of singularities
92C20 Neural biology
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