  The nonequilibrium dynamics of a cycling three state Potts model is studied
on a square lattice using Monte Carlo simulations and continuum theory. This
model is relevant to chemical reactions on a catalytic surface and to molecular
transport across a membrane. Several characteristic modes are formed depending
on the flipping energies between successive states and the contact energies
between neighboring sites. Under cyclic symmetry conditions, cycling
homogeneous phases and spiral waves form at low and high flipping energies,
respectively. In the intermediate flipping energy regime, these two modes
coexist temporally in small systems and/or at low contact energies. Under
asymmetric conditions, we observed small biphasic domains exhibiting
amoeba-like locomotion and temporal coexistence of spiral waves and a dominant
non-cyclic one-state phase. An increase in the flipping energy between two
successive states, say state 0 and state 1, while keeping the other flipping
energies constant, induces the formation of the third phase (state 2), owing to
the suppression of the nucleation of state 0 domains. Under asymmetric
conditions regarding the contact energies, two different modes can appear
depending on the initial state, due to a hysteresis phenomenon.
