Blue sky catastrophies (part 1)

Palestrante: Dmitri Turaev (Imperial College London)

Título: Blue sky catastrophies (part 1)

Resumo: One of the first questions of classical bifurcation theory was to  determine stability boundaries for periodic orbits of systems of  differential equations. In other words, how a stable periodic  orbit of a system of differential equations can disappear as parameters  of the system vary?

Up to date, 7 primary (i.e. codimension-1) stability boundaries are  known: saddle-node periodic, orbit, period-doubling, torus creation,  reverse Andronov-Hopf bifurcation, homoclinic loop to a saddle,  homoclinic loop to a saddle-node, and the so-called blue-sky catastrophe. Conjecturally, this list is complete, i.e. no other  stability boundaries of codimension-1 exist. The blue-sky catastrophe is the only primary stability boundary where the periodic orbit disappears by transforming into an infinite set of
orbits. It was discovered as a purely mathematical construction, however it occurred later that such sets emerge naturally in slow-fast systems and, in particular, the blue-sky catastrophe is responsible for transitions from tonic spiking (sustainable fast oscillations) to bursting (a regime where fast oscillations are regularly interrupted by the quiescence phase) in Hodgkin-Huxley type models of neurons. The great variability of the duration of the bursting phase near the moment of the blue-sky catastrophe can be responsible for switching between behavior patterns in small networks of neurons.

In the lectures, the analysis of the blue-sky catastrophe in slow-fast systems will be carried on based on the invariant manifold theory augmented with basic tools from classical theory of bifurcations on the plane, averaging theory, and Lyapunov exponents.