Genericity analysis of split bifurcations

This paper analyzes the genericity of bifurcations of one-parameter families of smooth (C1) vector fields that are embedded in an underlying multi-dimensional parameter space. Bifurcations with crossing equilibrium loci are called 'split bifurcations.' They include, for example, the pitchfork bifurcation and the transcritical bifurcation. In a regular parameter space where the system's Jacobian matrix with respect to endogenous variables and parameters has full rank at every equilibrium for all parameter values, there is a generic (open and dense) set of one-parameter C1 families of vector fields without split bifurcations. It is not difficult to obtain a regular parameter space when there are enough parameters. A regional migration model (a la Fujita, Krugman and Venables 1999) featuring the pitchfork bifurcation is presented as an example.