A New Transition between Discrete and Continuous Self-Similarity in Critical Gravitational Collapse

TL;DR

This paper investigates a bifurcation in critical gravitational collapse where the self-similarity of solutions transitions from discrete to continuous as a parameter varies, revealing a novel dynamical systems phenomenon.

Contribution

It identifies and analyzes a new transition mechanism between discrete and continuous self-similarity in gravitational collapse models, supported by numerical evidence.

Findings

Evidence for a bifurcation analogous to a heteroclinic loop in dynamical systems.

Transition from DSS to CSS as coupling constant decreases.

Numerical confirmation of the bifurcation phenomenon.

Abstract

We analyze a bifurcation phenomenon associated with critical gravitational collapse in a family of self-gravitating SU(2) $\sigma$-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.