  Oscillations in nonequilibrium noisy systems are important physical
phenomena. These oscillations can happen in autonomous biochemical oscillators
such as circadian clocks. They can also manifest as subharmonic oscillations in
periodically driven systems such as time-crystals. Oscillations in autonomous
systems and, to a lesser degree, subharmonic oscillations in periodically
driven systems have been both thoroughly investigated, including their relation
with thermodynamic cost and noise. We perform a systematic study of
oscillations in a third class of nonequilibrium systems: feedback driven
systems. In particular, we use the apparatus of stochastic thermodynamics to
investigate the role of noise and thermodynamic cost in feedback driven
oscillations. For a simple two-state model that displays oscillations, we
analyze the relation between precision and dissipation, revealing that
oscillations can remain coherent for an indefinite time in a finite system with
thermal fluctuations in a limit of diverging thermodynamic cost. We consider
oscillations in a more complex system with several degrees of freedom, an Ising
model driven by feedback between the magnetization and the external field. This
feedback driven system can display subharmonic oscillations similar to the ones
observed in time-crystals. We illustrate the second law for feedback driven
systems that display oscillations. For the Ising model, the oscillating
dissipated heat can be negative. However, when we consider the total entropy
that also includes an informational term related to measurements, the
oscillating total entropy change is always positive. We also study the
finite-size scaling of the dissipated heat, providing evidence for the
existence of a first-order phase transition for certain parameter regimes.
