A kind of structural frequency locking in generalized spatial automata

The classical concept of synchronization is usually related to the locking of the basic frequencies and instantaneous phases of regular oscillations, and this question is addressed by studying specific kinds of coupled systems. This work presents a different point of view. We do not study the convergence of coupled systems to a synchronized behaviour, but try to answer the following question: in a population of coupled differential systems, when each cell (subsystem) exhibits a periodic behaviour, is the whole trajectory periodic? We define generalized spatial automata, with reference to continuous spatial automata, by means of coupling maps and associated measures on the set of cells: the main idea is the fact that a cell interprets its own environment via the states of the whole population and according to its own state. A natural partition of periods is such that cells belong to the same class if their trajectories share a common period. We demonstrate that in a general case where cells belong to an a priori unstructured set, and their trajectories evolve in possibly distinct Banach spaces, the set of classes of periods is generally countable. In particular, when the set of cells is endowed with a Borelian structure, all the cells necessarily share a common period.