  For many physical systems the transition from a stationary solution to
sustained small amplitude oscillations corresponds to a Hopf bifurcation. For
systems involving impacts, thresholds, switches, or other abrupt events,
however, this transition can be achieved in fundamentally different ways. This
paper reviews 20 such `Hopf-like' bifurcations for two-dimensional ODE systems
with state-dependent switching rules. The bifurcations include boundary
equilibrium bifurcations, the collision or change of stability of equilibria or
folds on switching manifolds, and limit cycle creation via hysteresis or time
delay. In each case a stationary solution changes stability and possibly form,
and emits one limit cycle. Each bifurcation is analysed quantitatively in a
general setting: we identify quantities that govern the onset, criticality, and
genericity of the bifurcation, and determine scaling laws for the period and
amplitude of the resulting limit cycle. Complete derivations based on
asymptotic expansions of Poincare maps are provided. Many of these are new,
done previously only for piecewise-linear systems. The bifurcations are
collated and compared so that dynamical observations can be matched to
geometric mechanisms responsible for the creation of a limit cycle. The results
are illustrated with impact oscillators, relay control, automated balancing
control, predator-prey systems, ocean circulation, and the McKean and
Wilson-Cowan neuron models.
