  Consider a chaotic dynamical system generating Brownian motion-like
diffusion. Consider a second, non-chaotic system in which all particles
localize. Let a particle experience a random combination of both systems by
sampling between them in time. What type of diffusion is exhibited by this {\em
random dynamical system}? We show that the resulting dynamics can generate
anomalous diffusion, where in contrast to Brownian normal diffusion the mean
square displacement of an ensemble of particles increases nonlinearly in time.
Randomly mixing simple deterministic walks on the line we find anomalous
dynamics characterised by ageing, weak ergodicity breaking, breaking of
self-averaging and infinite invariant densities. This result holds for general
types of noise and for perturbing nonlinear dynamics in bifurcation scenarios.
