Nonlocal maps

A new kind of area-preserving map of the phase plane is introduced to represent the dynamics of interacting particles on a line. Unlike the familiar point map, the map is nonlocal in the sense that the evolution of a point in the phase plane depends not only on its position but also on the positions of other points, weighted by an evolving phase-plane density. In the case where this density is uniform and confined within a closed boundary B in the phase plane, the evolution of B is followed for a great variety of interaction potentials. Numerical experiments and analytical arguments show that a simple B develops great complexity. The resulting morphologies, incorporating fission and fusion of particle densities, are illustrated by high resolution graphics.