Genericity analysis of split bifurcations

This paper analyzes the genericity of bifurcations of one-parameter families of smooth (C1) vector …elds that are embedded in an underlying multidimensional parameter space. Bifurcations with crossing equilibrium loci are called “split bifurcations.”They include, for example, the pitchfork bifurcation and the transcritical bifurcation in one-dimensional systems. 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 …elds without split bifurcations. It is not di¢ cult to obtain a regular parameter space when there are enough parameters. A regional migration model (à la Fujita, Krugman and Venables 1999) featuring the pitchfork bifurcation is presented as an example. Keywords and Phrases: Bifurcation; Genericity analysis; Regular parameterization; Migration dynamics JEL Classi…cation Numbers: C61; R23; F12 The author thanks Gaetano Antinol…, William A. Barnett, Marcus Berliant, and John Nachbar for their helpful comments and discussions but retains responsibility for remaining errors.