Motivated by the observation of highly unstable flowing states in suspensions
of microtubules and kinesin, we analyze a model of mutually-propelled filaments
suspended in a solvent. The system undergoes a mean-field isotropic-nematic
transition for large enough filament concentrations when the nematic order
parameter is allowed to vary in space and time. We analyze the model in two
contexts: a quasi-one-dimensional channel with no-slip walls and a
two-dimensional box with periodic boundaries. Using stability analysis and
numerical calculations we show that the interplay between non-uniform nematic
order, activity, and flow results in a variety of complex scenarios that
include spontaneous banded laminar flow, relaxation oscillations, and chaos.Comment: 15 pages, 15 figure