Oliver Tearne: Collapse of attractors under small random perturbations

Crauel and Flandoli show that the random attractor corresponding to a onedimensional random dynamical system (RDS) is a one point set, provided there exists a compact absorbing set and a unique invariant Markov measure. For example, adding arbitrarily small additive white noise to the one-dimensional ODE ẋ = x−x^{3} causes the deterministic attractor to collapse from an interval to a single point. I will present an algorithm used to investigate this phenomenon in two dimensions, and analytical results showing the random

attractor is a one point set for n-dimensional RDS generated by a class of gradient stochastic differential equations. Central to the algorithm is the result

∂Ω_{B} ⊆ Ω_{∂B} for omega-limit sets of compact sets B. This is used to prove that boundaries of deterministic and random attractors are covered by omegalimit

sets of boundaries of compact absorbing sets. To apply the algorithm I will show that any neighbourhood of an absorbing set for the deterministic system is also absorbing for the corresponding stochastic system with positive probability.