  We develop a Magnus formalism for periodically driven systems which provides
an expansion both in the driving term and the inverse driving frequency,
applicable to isolated and dissipative systems. We derive explicit formulas for
a driving term with a cosine dependence on time, up to fourth order. We apply
these to the steady state of a classical parametric oscillator coupled to a
thermal bath, which we solve numerically for comparison. Beyond dynamical
stabilization at second order, we find that the higher orders further
renormalize the oscillator frequency, and additionally create a weakly
renormalized effective temperature. The renormalized oscillator frequency is
quantitatively accurate almost up to the parametric instability, as we confirm
numerically. Additionally, a cut-off dependent term is generated, which
indicates the break-down of the hierarchy of time scales of the system, as a
precursor to the instability. Finally, we apply this formalism to a
parametrically driven chain, as an example for the control of the dispersion of
a many-body system.
