Complex dynamics in simple models of shear banding

Complex fluids commonly undergo flow instabilities and flow-induced transitions that result in spatially heterogeneous ``shear banded'' states. Often, these banded states display oscillatory or chaotic dynamics, measured in the bulk rheological signals and in the motion of the interface between the bands. Until recently, however, theory predicted a steady state comprising stationary bands separated by a flat interface. We discuss recent theoretical progress in capturing complex dynamics of the banded state: first in a model in which the interface (or interfaces) remains flat but moves in a chaotic way; second in a model that explicitly allows for undulations along the interface.