  Microscopic strategy update rules play an important role in the evolutionary
dynamics of cooperation among interacting agents on complex networks. Many
previous related works only consider one \emph{fixed} rule, while in the real
world, individuals may switch, sometimes periodically, between rules. It is of
particular theoretical interest to investigate under what conditions the
periodic switching of strategy update rules facilitates the emergence of
cooperation. To answer this question, we study the evolutionary prisoner's
dilemma game on regular networks where agents can periodically switch their
strategy update rules. We accordingly develop a theoretical framework of this
periodically switched system, where the replicator equation corresponding to
each specific microscopic update rule is used for describing the subsystem, and
all the subsystems are activated in sequence. By utilizing switched system
theory, we identify the theoretical condition for the emergence of cooperative
behavior. Under this condition, we have proved that the periodically switched
system with different switching rules can converge to the full cooperation
state. Finally, we consider an example where two strategy update rules, that
is, the imitation and pairwise-comparison updating, are periodically switched,
and find that our numerical calculations validate our theoretical results.
