Despite its importance for applications, relatively little progress has been made towards the development of a bifurcation theory for random dynamical systems. In this talk, I will demonstrate that adding noise to a deterministic mapping with a pitchfork bifurcation does not destroy the bifurcation, but leads to two different types of bifurcations. The first bifurcation is characterised by the a breakdown
of uniform attraction, while the second bifurcation can be described topologically. Both bifurcations do not correspond to a change of sign of the Lyapunov exponents, but I will explain that these bifurcations can be characterised by qualitative changes in the dichotomy spectrum and collisions of attractor-repeller pairs.
This is joint work with M. Callaway, T.S. Doan, J.S.W. Lamb (Imperial College London) and C.S. Rodrigues (MPI Leipzig)