Phase reduction approach to elastohydrodynamic synchronization of beating flagella

We formulate a theory for the phase reduction of a beating flagellum. The theory enables us to describe the dynamics of a beating flagellum in a systematic manner using a single variable called the phase. The theory can also be considered as a phase reduction method for the limit-cycle solutions in infinite-dimensional dynamical systems, namely, the limit-cycle solutions to partial differential equations representing beating flagella. We derive the phase sensitivity function, which quantifies the phase response of a beating flagellum to weak perturbations applied at each point and at each time. Using the phase sensitivity function, we analyze the phase synchronization between a pair of beating flagella through hydrodynamic interactions at a low Reynolds number.