Equivalence between asynchronous and delayed dynamics in coupled maps

Since the 80's, coupled map lattices have been investigated as paradigmatic models of many collective phenomena. While in the early versions, the states were updated synchronously (all maps simultaneously), there is, in recent years, a concern to consider more realistic updating schemes, where elements do not change all at once. % Then, either asynchronous or time-delayed dynamics have been implemented instead, putting into evidence noticeable differences in the emergent patterns. Usually, asynchronous and delayed dynamics, are introduced as distinct cases. Here, we investigate the correspondence between them. For that goal, aside from (random sequential) asynchronous updating, we implemented different delayed dynamics, with discrete or pseudo-continuous, fixed or random, delays. We show, both numerically and analytically, that the asynchronous case can be exactly matched by an adequate delayed dynamics. This equivalence provides a unified framework and opens an enlightening approach for understanding issues related to control of chaos such as the stabilization of locally unstable states.