  We investigate how the dynamical production of quantum entanglement for
weakly coupled, composite quantum systems is influenced by the chaotic dynamics
of the corresponding classical system, using coupled kicked tops. The linear
entropy for the subsystem (a kicked top) is employed as a measure of
entanglement. A perturbative formula for the entanglement production rate is
derived. The formula contains a correlation function that can be evaluated only
from the information of uncoupled tops. Using this expression and the
assumption that the correlation function decays exponentially which is
plausible for chaotic tops, it is shown that {\it the increment of the strength
of chaos does not enhance the production rate of entanglement} when the
coupling is weak enough and the subsystems (kicked tops) are strongly chaotic.
The result is confirmed by numerical experiments. The perturbative approach is
also applied to a weakly chaotic region, where tori and chaotic sea coexist in
the corresponding classical phase space, to reexamine a recent numerical study
that suggests an intimate relationship between the linear stability of the
corresponding classical trajectory and the entanglement production rate.
