  The effect of the chaotic dynamical states of the agents on the coevolution
of cooperation and synchronization in a structured population of the agents
remains unexplored. With a view to gaining insights into this problem, we
construct a coupled map lattice of the paradigmatic chaotic logistic map by
adopting the Watts--Strogatz network algorithm. The map models the agent's
chaotic state dynamics. In the model, an agent benefits by synchronizing with
its neighbours and in the process of doing so, it pays a cost. The agents
update their strategies (cooperation or defection) by using either a stochastic
or a deterministic rule in an attempt to fetch themselves higher payoffs than
what they already have. Among some other interesting results, we find that
beyond a critical coupling strength, that increases with the rewiring
probability parameter of the Watts--Strogatz model, the coupled map lattice is
spatiotemporally synchronized regardless of the rewiring probability. Moreover,
we observe that the population does not desynchronize completely -- and hence
finite level of cooperation is sustained -- even when the average degree of the
coupled map lattice is very high. These results are at odds with how a
population of the non-chaotic Kuramoto oscillators as agents would behave. Our
model also brings forth the possibility of the emergence of cooperation through
synchronization onto a dynamical state that is a periodic orbit attractor.
