Limit-cycle oscillation of an elastic filament and caterpillar motion

Soft biological materials can exhibit various active motion such as limit-cycle oscillation. The limit-cycle oscillation of active matter might be used for a mechanism of locomotion. We propose a simple model of an active elastic chain as an extension of the van der Pol equation. The uniform state is unstable, and exhibits a limit cycle of breathing motion. If the breathing motion is mirror symmetric, the elastic chain does not move as a whole. However, the breathing motion becomes a caterpillar motion and a unidirectional motion is induced, if an additional heterogeneity is involved or the chain is set on a spatially periodic sawtooth potential. We also analyze the model equation with coupled mode equations, and try to understand the bifurcation to the collective oscillation and the directional motion.