A codimension-four singularity with potential

Hinke Osinga
Department of Mathematics, Auckland University

Periodic alternations of spiking and quiescence, known as bursting oscillations, are often modelled by systems that exhibit both slow and fast time scales. In such systems, one or more slow variables carry the fast variables through a sequence of bifurcations that mediate transitions between oscillations and steady states. A classification of different bursting types can be obtained by characterising the bifurcations found in the neighbourhood of a singularity; a measure of the complexity of the bursting oscillation is then given by the smallest codimension of the singularities near which it occurs. We investigate bursting oscillations that occur near the central codimension-four singularity of a conjectural unfolding of the full family of cubic Liénard equations. Our motivation is that a particular type of bursting, called fold/subHopf or pseudo-plateau bursting, had not yet been properly classified. We find that the codimension-four doubly-degenerate Bodganov-Takens singularity is an excellent candidate for the organising center that unifies almost all known bursting types. The subsequent numerical investigation of the respective bifurcation diagrams led, in turn, to new insight on how this codimension-four unfolding manifests itself as a sequence of bifurcation diagrams on the surface of a sphere.

Joint work with Arthur Sherman (NIH), Krasimira Tsaneva-Atanasova (Exeter), and Bernd Krauskopf (Auckland).