  We introduce three forms of generalized contraction (GC). Roughly speaking,
these are motivated by allowing contraction to take place after small
transients in time and/or amplitude. Indeed, contraction is usually used to
prove asymptotic properties, like convergence to an attractor or entrainment to
a periodic excitation, and allowing initial transients does not affect this
asymptotic behavior.
  We provide sufficient conditions for GC, and demonstrate their usefulness
using examples of systems that are not contractive, with respect to any norm,
yet are GC.
