  Motivated to understand the asymptotic behavior of periodically driven
thermodynamic systems, we study the prototypical example of Brownian particle,
overdamped and underdamped, in harmonic potentials subjected to periodic
driving. The harmonic strength and the coefficients of drift and diffusion are
all taken to be $T$-periodic. We obtain the asymptotic distributions almost
exactly treating driving nonperturbatively. In the underdamped case, we exploit
the underlying $SL_2$ symmetry to obtain the asymptotic state, and study the
dynamics and fluctuations of energies and entropy. We further obtain the
two-time correlation functions, and investigate the responses to drift and
diffusion perturbations in the presence of driving.
