Local models and normal forms of slow-fast systems near Hopf and Bogdanov-Takens bifurcations (R-6821)

Slow-fast systems are dynamical systems in which dynamics at different time scales can be distinguished and are studied in the asymptotic limit where the ratio of the time scales (the singular parameter) tends to 0. It may be helpful to think of population dynamics of certain species that evolve over years versus the dynamics of geological processes, which occurs over millions of years; in the asymptotic limit, one would fix the geological state while describing the population dynamics in time. From a mathematical point of view, slow-fast systems are families of vector fields that have a non-isolated set of singular points when the singular parameter is zero. The dynamics is quite often governed by successions of rapid and slow movements. Well-known phenomena such as Hopf bifurcations have a slow-fast analogon, and slow-fast Hopf bifurcations (or singular Hopf) arise quite frequently in models that arise from applications in life sciences. These models are often simplified toy models. Our aim is to study to which extent the simplified models capture all relevant dynamics, in other words whether or not they can be seen as an actual normal form. In short, we study local normal forms of singularities in slow-fast systems and their unfoldings.

Keywords:SINGULARITIES

Disciplines:Analysis