Quadfurcation is a bifurcation when four fixed points are created from none at a single location in phase space upon variation of one parameter. This bifurcation is not at all well studied in dynamical systems literature. In this paper Bäcker and Meiss put forth that the quadfurcation is an organizing center for the dynamics of a four-dimensional map, the quadratic diffeomorphism introduced by Moser in 1994.
Bates, Fusco, and Smyrnelis present a systematic study of entire symmetric solutions of the vector Allen-Cahn equation using a variational approach based on a mapping property of the parabolic vector Allen-Cahn equation and on a point-wise estimate for vector minimizers. The relevance of the the entire solutions to tiling of space observed in physical experiments is also discussed.