  We study the spatio-temporal dynamics of a model of polar active fluid in two
dimensions. The system exhibits a transition from an isotropic to a polarized
state as a function of density. The uniform polarized state is, however,
unstable above a critical value of activity. Upon increasing activity, the
active fluids displays increasingly complex patterns, including traveling
bands, traveling vortices and chaotic behavior. The advection arising from the
particles self-propulsion and unique to polar fluids yields qualitatively new
behavior as compared to that obtain in active nematic, with traveling-wave
structures. We show that the nonlinear hydrodynamic equations can be mapped
onto a simplified diffusion-reaction-convection model, highlighting the
connection between the complex dynamics of active system and that of excitable
systems.
