  We consider a Lurie system obtained via a connection of a linear
time-invariant system and a nonlinear feedback function. Such systems often
have more than a single equilibrium and are thus not contractive with respect
to any norm. We derive a new sufficient condition for $k$-contraction of a
Lurie system. For $k=1$, our sufficient condition reduces to the standard
stability condition based on the bounded real lemma and a small gain condition.
For $k=2$, our condition guarantees well-ordered asymptotic behaviour of the
closed-loop system: every bounded solution converges to an equilibrium, which
is not necessarily unique. We apply our results to derive a sufficient
condition for $k$-contractivity of a networked system.
