  Exploiting the rich phenomenology of periodically-driven many-body systems is
notoriously hindered by persistent heating in both the classical and quantum
realm. Here, we investigate to what extent coupling to a large thermal
reservoir makes stabilisation of a non-trivial steady state possible. To this
end, we model both the system and the reservoir as classical spin chains where
driving is applied through a rotating magnetic field, and simulate the
Hamiltonian dynamics of this setup. We find that the intuitive limits of
infinite and vanishing frequency, where the system dynamics is governed by the
average and the instantaneous Hamiltonian, respectively, can be smoothly
extended into entire regimes separated only by a small crossover region. At
high frequencies, the driven system stroboscopically attains a Floquet-type
Gibbs state at the reservoir temperature. At low frequencies, a synchronised
Gibbs state emerges, whose temperature may depart significantly from that of
the reservoir. Although our analysis in some parts relies on the specific
properties our setup, we argue that much of its phenomenology should be generic
for a large class of systems.
