Abstract

It is well known that certain properties of the pitchfork bifurcation are destroyed when noise is added (see Crauel and Flandoli, Additive noise destroys a pitchfork bifurcation, Journal of Dynamics and Differential Equations, 1998). In this talk, I would like to present three different points of view that demonstrate that the bifurcation is still present. The first two approaches concern a finite-time and local analysis of the (unbounded noise) random dynamical system, revealing structural changes at the bifurcation point that can be described by the dichotomy spectrum and conditional Lyapunov exponents. In the last part of the talk, we study the situation for bounded noise, and show that the pitchfork bifurcation can be described by means of a discontinuous bifurcation of minimal invariant sets. Joint work with Mark Callaway, Thai Son Doan, Maximilian Engel, Jeroen Lamb, and Christian Rodrigues.