Conservative, dissipative and super-diffusive behavior of a particle propelled in a regular flow

A recent model of Ariel et al. (2017) for explaining the observation of Lévy walks in swarming bacteria suggests that self-propelled, elongated particles in a periodic array of regular vortices perform a super-diffusion that is consistent with Lévy walks. The equations of motion, which are reversible in time but not volume preserving, demonstrate a new route to Lévy walking in chaotic systems. Here, the dynamics of the model is studied both analytically and numerically. It is shown that the apparent super-diffusion is due to “sticking” of trajectories to elliptic islands, regions of quasi-periodic orbits reminiscent of those seen in conservative systems. However, for certain parameter values, these islands coexist with asymptotically stable periodic trajectories, causing dissipative behavior on very long time scales.