Defining equations for bifurcations and singularities

Singularity theory and bifurcation theory lead us to consider varieties in jet spaces of mappings. Explicit defining equations for these varieties are complex and sometimes difficult to compute numerically. This paper considers two examples: saddle-node bifurcation of periodic orbits and the Thom–Boardman stratification of singularity theory. Saddle-node bifurcation of periodic orbits is determined by their monodromy matrices. The bifurcation occurs when the difference between the monodromy matrix and the identity has a two dimensional nilpotent subspace. We discuss numerical methods for computing this nilpotency. The usual definitions of the Thom–Boardman stratification of a map involve computing the rank of the map restricted to submanifolds. Without explicit formulas for these submanifolds, determination of the rank is a difficult numerical problem. We reformulate the defining equations for the submanifolds of the stratification here, producing a minimal set of regular defining equations for each stratum.