  We derive a sufficient condition for $k$-contraction in a generalized Lurie
system (GLS), that is, the feedback connection of a nonlinear dynamical system
and a memoryless nonlinear function. For $k=1$, this reduces to a sufficient
condition for standard contraction. For $k=2$, this condition implies that
every bounded solution of the GLS converges to an equilibrium, which is not
necessarily unique. We demonstrate the theoretical results by analyzing
$k$-contraction in a biochemical control circuit with nonlinear dissipation
terms.
