Bifurcation analysis of a nonlinear field model of regeneration
The analysis of bifurcating solutions in the Totafurno and Trainor [23] model of supernumerary limb production in salamanders is re-examined using the symmetry analysis developed by Totafurno [22]. In particular, we show analytically that the appearance of field solutions possessing 2 and 4 singularities (the 2- and 4-centered solutions, respectively) also correspond to true bifurcations with reduced symmetries, just as had been previously found for a solution to the field equations not possessing such singularities (the twist solution). While the results have significance primarily for the biological problem, this work serves as an instructive example of the application of symmetry groups to the bifurcation analysis of nonlinear field equations arising from a variational principle. The relationship between the solutions of the nonlinear equations and the corresponding linear equations is discussed.