Bifurcation of critical sets and relaxation oscillations in singular fast-slow systems

Fast-slow dynamical systems have subsystems that evolve on vastly different timescales, and bifurcations in such systems can arise due to changes in any or all subsystems. We classify bifurcations of the critical set (the equilibria of the fast subsystem) and associated fast dynamics, parametrized b...

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Bibliographic Details
Published in:Nonlinearity 2020-06, Vol.33 (6), p.2853-2904
Main Authors: Nyman, Karl H M, Ashwin, Peter, Ditlevsen, Peter D
Format: Article
Language:English
Subjects:
bifurcation
fast-slow dynamics
relaxation oscillation
singularity
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Summary:Fast-slow dynamical systems have subsystems that evolve on vastly different timescales, and bifurcations in such systems can arise due to changes in any or all subsystems. We classify bifurcations of the critical set (the equilibria of the fast subsystem) and associated fast dynamics, parametrized by the slow variables. Using a distinguished parameter approach we are able to classify bifurcations for one fast and one slow variable. Some of these bifurcations are associated with the critical set losing manifold structure. We also conjecture a list of generic bifurcations of the critical set for one fast and two slow variables. We further consider how the bifurcations of the critical set can be associated with generic bifurcations of attracting relaxation oscillations under an appropriate singular notion of equivalence.
ISSN:0951-7715
1361-6544
DOI:10.1088/1361-6544/ab7292