  Phase synchronization between the vortex shedding behind a two-dimensional
circular cylinder and its vibrations is investigated using the phase-reduction
analysis. Leveraging this approach enables the development of a
one-dimensional, linear model with respect to the limit-cycle attractor of the
laminar wake, which accurately describes the phase dynamics of the
high-dimensional, nonlinear fluid flow and its response to rotational,
transverse and longitudinal vibrations of the cylinder. This phase-based model
is derived by assessing the phase-response and sensitivity of the wake dynamics
to impulse perturbations of the cylinder, which can be performed in simulations
and experiments. The resulting model in turn yields the theoretical conditions
required for phase-locking between the cylinder vibrations and the wake. We
furthermore show that this synchronization mechanism can be employed to
stabilize the wake and subsequently reduce drag. We also uncover the
circumstances under which the concurrent occurrence of different vibrational
motions can be used to promote or impede synchronization. These findings
provide valuable insights for the study of vortex-induced body oscillations,
the enhancement of aerodynamic performance of flyers, or the mitigation of
structural vibrations by synchronizing or desynchronizing the oscillatory
motions of body to the periodic wake.
