  Atmospheric self-organization and activator-inhibitor dynamics in biology
provide examples of checkerboard-like spatio-temporal organization. We study a
simple model for local activation-inhibition processes. Our model, first
introduced in the context of atmospheric moisture dynamics, is a
continuous-energy and non-Abelian version of the fixed-energy sandpile model.
Each lattice site is populated by a non-negative real number, its energy. Upon
each timestep all sites with energy exceeding a unit threshold re-distribute
their energy at equal parts to their nearest neighbors. The limit cycle
dynamics gives rise to a complex phase diagram in dependence on the mean energy
$\mu$: For low $\mu$, all dynamics ceases after few re-distribution events. For
large $\mu$, the dynamics is well-described as a diffusion process, where the
order parameter, spatial variance $\sigma$, is removed. States at intermediate
$\mu$ are dominated by checkerboard-like period-two phases which are however
interspersed by much more complex phases of far longer periods. Phases are
separated by discontinuous jumps in $\sigma$ or $\partial_{\mu}\sigma$ - akin
to first and higher-order phase transitions. Overall, the energy landscape is
dominated by few energy levels which occur as sharp spikes in the single-site
density of states and are robust to noise.
