Diffusive dynamics in systems with translational symmetry: A one-dimensional-map model

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Abstract
A one-dimensional, one-parameter-map model for dissipative systems with translational symmetry is studied. The map possesses confined periodic and chaotic solutions which form an infinite array on the real line, periodic or chaotic running solutions which propagate coherently to the left or right, and a variety of diffusive motions where iterates wander over the entire interval like a random walk. The onset of diffusion in various regions of parameter space is studied in detail and simple dynamical models for the behavior of the diffusion coefficient near bifurcation points are constructed.