  A Lurie system is the interconnection of a linear time-invariant system and a
nonlinear feedback function. 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. However, Lurie systems often have more than a single
equilibrium and are thus not contractive with respect to any norm. For $k=2$,
our condition guarantees a well-ordered asymptotic behaviour of the closed-loop
system: every bounded solution converges to an equilibrium, which is not
necessarily unique. We demonstrate our results by deriving a sufficient
condition for $k$-contraction of a general networked system, and then applying
it to guarantee $k$-contraction in a Hopfield neural network, a nonlinear
opinion dynamics model, and a 2-bus power system.
