Criticality and bifurcation in the gravitational collapse of a self-coupled scalar field

We examine the gravitational collapse of a nonlinear $\sigma$ model in spherical symmetry. There exists a family of continuously self-similar solutions parametrized by the coupling constant of the theory. These solutions are calculated together with the critical exponents for black hole formation of these collapse models. We also find that the sequence of solutions exhibits a Hopf-type bifurcation as the continuously self-similar solutions become unstable to perturbations away from self-similarity.