Chaotic self-similar wave maps coupled to gravity

It was shown recently that the $\mathrm{SU}\left(2\right)\mathrm{}$ sigma model coupled to gravity has periodic spherically symmetric self-similar solutions for small values of the coupling constant. In this article we extend the analysis of Bizoń, Szybka, and Wasserman to larger values of the coupling constant $(\alpha >~0.426)$ and present numerical evidence for the chaotic solution and the fractal threshold behavior. We explain this phenomenon in terms of horseshoelike dynamics and heteroclinic intersections.