A geometric approach to the synchronization of a pair of two-state switching rowers interacting hydrodynamically

Two-state switching rowers, or simply rowers, are model self-sustained oscillators in a fluid at a low Reynolds number, introduced in Cosentino Lagomarsino et al (2003 Phys. Rev. E 68 021908) and realized experimentally in Kotar et al (2010 Proc. Natl Acad. Sci. USA 107 7669–73). Here we present a new approach for investigating the hydrodynamic synchronization of a pair of rowers with the same and different frequencies. Our analysis is geometrical, in the spirit of the qualitative theory of dynamical systems. By taking advantage of the separation of timescales in the model, the dynamics can be decomposed into a sequence of fast changes followed by slow relaxations. In this framework we discuss how synchronization is determined by the dominant mode of the relaxation dynamics. For rowers with the same frequencies, our analysis recovers naturally the anti-phase synchronized motion, its stability and the basin of attraction (Cosentino Lagomarsino et al 2003 Phys. Rev. E 68 021908; Kotar et al 2010 Proc. Natl Acad. Sci. USA 107 7669–73); in the case of rowers with different frequencies we are able to provide upper bounds for the phase-locked solution and to determine a critical value of the frequency mismatch after which the coordination is lost. Our estimates are in good agreement with numerical simulations. In this way we provide a simple and robust explanation for the transition from the phase-locked state to the loss of synchrony. (paper)