  We extend the semiclassical theory of scarring of quantum eigenfunctions
psi_{n}(q) by classical periodic orbits to include situations where these
orbits undergo generic bifurcations. It is shown that |psi_{n}(q)|^{2},
averaged locally with respect to position q and the energy spectrum E_{n}, has
structure around bifurcating periodic orbits with an amplitude and length-scale
whose hbar-dependence is determined by the bifurcation in question.
Specifically, the amplitude scales as hbar^{alpha} and the length-scale as
hbar^{w}, and values of the scar exponents, alpha and w, are computed for a
variety of generic bifurcations. In each case, the scars are semiclassically
wider than those associated with isolated and unstable periodic orbits;
moreover, their amplitude is at least as large, and in most cases larger. In
this sense, bifurcations may be said to give rise to superscars. The
competition between the contributions from different bifurcations to determine
the moments of the averaged eigenfunction amplitude is analysed. We argue that
there is a resulting universal hbar-scaling in the semiclassical asymptotics of
these moments for irregular states in systems with a mixed phase-space
dynamics. Finally, a number of these predictions are illustrated by numerical
computations for a family of perturbed cat maps.
