  A dynamical system is called contractive if any two solutions approach one
another at an exponential rate. More precisely, the dynamics contracts lines at
an exponential rate. This property implies highly ordered asymptotic behavior
including entrainment to time-varying periodic vector fields and, in
particular, global asymptotic stability for time-invariant vector fields.
Contraction theory has found numerous applications in systems and control
theory because there exist easy to verify sufficient conditions, based on
matrix measures, guaranteeing contraction.
  Here, we provide a geometric generalization of contraction theory called
k-order contraction. A dynamical system is called k-order contractive if the
dynamics contracts k-parallelotopes at an exponential rate. For k=1 this
reduces to standard contraction.
  We describe easy to verify sufficient conditions for k-order contractivity
based on matrix measures and the kth additive compound of the Jacobian of the
vector field. We also describe applications of the seminal work of Muldowney
and Li, that can be interpreted in the framework of 2-order contraction, to
systems and control theory.
