$k$-Contraction in a Generalized Lurie System

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.