New transition between discrete and continuous self-similarity in critical gravitational collapse

Abstract

We analyze a bifurcation phenomenon associated with critical gravitational collapse in a family of self-gravitating SU(2) $\ensuremath{\sigma}$ models. As the dimensionless coupling constant decreases, the critical solution changes from discretely self-similar (DSS) to continuously self-similar (CSS). Numerical results provide evidence for a bifurcation which is analogous to a heteroclinic loop bifurcation in dynamical systems, where two fixed points (CSS) collide with a limit cycle (DSS) in phase space as the coupling constant tends to a critical value.