  We investigate the synchronization properties of the two-dimensional periodic
flow over a circular cylinder using the principles of phase-reduction theory.
The influence of harmonic external forcings on the wake dynamics, and the
possible synchronization of the vortex shedding behind the cylinder to these
forcings, is determined by evaluating the phase response of the system to weak
impulse perturbations. These horizontal and vertical perturbations are added at
different phase values over a period, in order to develop a linear
one-dimensional model with respect to the limit cycle that describes the
high-dimensional and nonlinear dynamics of the fluid flow via only a single
scalar phase variable. This model is then utilized to acquire the theoretical
conditions for the synchronization between the cylinder wake and the harmonic
forcings added in the global near-wake region. Valuable insights are gained by
comparing the findings of the present research against those rendered by the
dynamic mode decomposition and adjoint analysis of the wake dynamics in earlier
works. The present analysis reveals regions in the flow which enable phase
synchronization or desynchronization to periodic excitations for applications
such as active flow control and fluid-structure interactions.
