Abstract
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.
Highlights
Many natural systems are characterized by interactions between dynamical processes that run at very different timescales
If the fast dynamics is one dimensional, it is typically confined to a manifold, though at bifurcation it may lose its manifold structure at isolated points
Many examples of bifurcations of relaxation oscillations have been considered [2], including some associated with bifurcations of the critical set [3, 15, 16] it seems that no exhaustive list of scenarios has been proposed
Summary
Many natural systems are characterized by interactions between dynamical processes that run at very different timescales. In the simplest case of one fast and one slow variable, bifurcations of the critical set can be directly tackled using a global version of the singularity theory with distinguished parameter in [17]. We introduce a global singular equivalence for the singular trajectories and use this to classify persistence (proposition 4) and codimension one bifurcations (proposition 5) of these simple relaxation oscillations These codimension one bifurcations naturally split into those caused by bifurcations of the critical set, and those caused by interaction of singularities of the slow flow with the critical set: in section 5.3 we present some numerical examples of various types. We include several Appendices that give more details of the tools used for the classification and the examples
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