Synchronized states of hydrodynamically coupled filaments and their stability

Cilia and flagella are organelles that play central roles in unicellular locomotion, embryonic development, and fluid transport around tissues. In these examples, multiple cilia are often found in close proximity and exhibit coordinated motion. Inspired by the flagellar motion of biflagellate cells, we examine the synchrony exhibited by a filament pair surrounded by a viscous fluid and tethered to a rigid planar surface. A geometrically-switching base moment drives filament motion, and we characterize how the stability of synchonized states depends of the base torque magnitude. In particular, we study the emergence of bistability that occurs when the anti-phase, breast-stroke branch becomes unstable. Using a bisection algorithm, we find the unstable edge-state that exists between the two basins of attraction when the system exhibits bistability. We establish a bifurcation diagram, study the nature of the bifurcation points, and find that the observed dynamical system can be captured by a modified version of Adler’s equation. The bifurcation diagram and presence of bistability reveal a simple mechanism by which the anti-phase breast stroke can be modulated, or switched entirely to in-phase undulations through the variation of a single bifurcation parameter