  We consider diffusion of an inertial Brownian particle moving in a symmetric
spatially periodic potential and driven by an unbiased time periodic force of a
moderate frequency. The study is focused on the dependence of its diffusion
coefficient $D$ on the damping $\gamma$ ranging from the weak to strong one. We
reveal the mechanisms of counterintutive effects in which diffusion can
increase despite the grow of friction and exhibits both minima and maxima.
Their origin is explained by analysis of the deterministic dynamics perturbed
by thermal fluctuations. In the weak damping regime the maximum of diffusion is
induced by a non-monotonic change of population of the running orbits when the
friction increases. In the strong dissipation regime the effect is caused by
bifurcation of periodic orbits oscillating symmetrically around the potential
maximum and their displacement towards the potential minimum when the damping
coefficient increases. It induces the change of the effective barrier of the
periodic potential and its vanishing at the maximum of $D$.
