  Contraction theory is a powerful tool for proving asymptotic properties of
nonlinear dynamical systems including convergence to an attractor and
entrainment to a periodic excitation. We consider three generalizations of
contraction with respect to a norm that allow contraction to take place after
small transients in time and/or amplitude. These generalized contractive
systems (GCSs) are useful for several reasons. First, we show that there exist
simple and checkable conditions guaranteeing that a system is a GCS, and
demonstrate their usefulness using several models from systems biology. Second,
allowing small transients does not destroy the important asymptotic properties
of contractive systems like convergence to a unique equilibrium point, if it
exists, and entrainment to a periodic excitation. Third, in some cases as we
change the parameters in a contractive system it becomes a GCS just before it
looses contractivity with respect to a norm. In this respect, generalized
contractivity is the analogue of marginal stability in Lyapunov stability
theory.
