Stability of temporally periodic states of classical many-body systems

We study the stability of spatially coherent, time-periodic states in noisy, classical, discrete-time, many-body systems with short-range interactions. Generic stability of periodic k cycles with k>2 can be achieved only by rules carefully constructed to exploit lattice anisotropy and so suppress droplet growth. For ordinary rules which do not utilize spatial anisotropy in this way, periodic k cycles with periods k>2 are metastable rather than stable under generic conditions, losing spatial coherence through nucleation and growth of droplets. The unusual dynamical properties of the periodic states stabilized by anisotropy are described.