  Topological defects in active liquid crystals can be confined by introducing
gradients of activity. Here, we examine the dynamical behavior of two defects
confined by a sharp gradient of activity that separates an active circular
region and a surrounding passive nematic material. Continuum simulations are
used to explain how the interplay among energy injection into the system,
hydrodynamic interactions, and frictional forces governs the dynamics of
topologically required self-propelling $+1/2$ defects. Our findings are
rationalized in terms of a phase diagram for the dynamical response of defects
in terms of activity and frictional damping strength. Different regions of the
underlying phase diagram correspond to distinct dynamical modes, namely
immobile defects (ID), steady rotation of defects (SR), bouncing defects (TB),
bouncing-cruising defects (BC), dancing defects (DA), and multiple defects with
irregular dynamics (MD). These dynamic states raise the prospect of generating
synchronized defect arrays for microfluidic applications.
