  We study diffusion in a one-dimensional periodic array of scatterers modeled
by a simple map. The chaotic scattering process for this map can be changed by
a control parameter and exhibits the dynamics of a crisis in chaotic
scattering. We show that the strong backscattering associated with the crisis
mechanism induces a crossover which leads to different asymptotic laws for the
parameter-dependent diffusion coefficient. These laws are obtained from exact
diffusion coefficient results and are supported by simple random walk models.
We argue that the main physical feature of the crossover should be present in
many other dynamical systems with non-equilibrium transport.
