The dichotomy spectrum for random dynamical systems and pitchfork bifurcations with additive noise

This is the final version of the article. Available from the publisher via the DOI in this record.We develop the dichotomy spectrum for random dynamical systems and demonstrate its use in the characterization of pitchfork bifurcations for random dynamical systems with additive noise. Crauel and Flandoli (J. Dynam. Differential Equations 10 (1998) 259–274) had shown earlier that adding noise to a system with a deterministic pitchfork bifurcation yields a unique attracting random equilibrium with negative Lyapunov exponent throughout, thus “destroying” this bifurcation. Indeed, we show that in this example the dynamics before and after the underlying deterministic bifurcation point are topologically equivalent. However, in apparent paradox to (J. Dynam. Differential Equations 10 (1998) 259–274), we show that there is after all a qualitative change in the random dynamics at the underlying deterministic bifurcation point, characterized by the transition from a hyperbolic to a non-hyperbolic dichotomy spectrum. This breakdown manifests itself also in the loss of uniform attractivity, a loss of experimental observability of the Lyapunov exponent, and a loss of equivalence under uniformly continuous topological conjugacies.The second author was supported by a Marie Curie IEF Fellowship, the third author acknowledges the support by Nizhny Novgorod University through the grant RNF 14-41-00044, and the fourth author was supported by an EPSRC Career Acceleration Fellowship EP/I004165/1. This research has been supported by EU Marie-Curie IRSES Brazilian–European Partnership in Dynamical Systems (FP7-PEOPLE-2012-IRSES 318999 BREUDS) and EU Marie-Skłodowska-Curie ITN Critical Transitions in Complex Systems (H2020-MSCA-2014-ITN 643073 CRITICS)