Emergence of beating in a magnetic flagellum consisting of active bots

Abstract

We investigate the emergence of flagellar beating in chains of magnetic self--propelled particles (MSPPs) built from centimeter--scale vibrating robots (Hexbugs) with embedded neodymium dipoles. When one end of the chain is anchored and self--propulsion is activated, longitudinal stress accumulates along the chain until it overcomes the magnetic bending stiffness, triggering a buckling instability that drives sustained flagellar beating. Using a combination of experiments and numerical simulations, we identify three distinct dynamical regimes straight chain, stable flagellar beating, and fission governed by the competition between active force, chain length, and magnetic bending stiffness. The onset of beating requires a seed misalignment set by the balance between magnetic torques and rotational noise, and we show that the transition corresponds to a supercritical Hopf bifurcation. A kinematic model reproduces the observed orientation dynamics with excellent agreement. The magnetic bending stiffness, which arises directly from dipole--dipole interactions, is fully tunable via dipole strength and chain length, offering independent experimental control over both activity and rigidity. Our results establish a macroscopic platform for studying force-induced buckling and self--oscillations in active filaments, with direct connections to flagellar motion in biological and synthetic microswimmers.