Critical Points and Local Behavior

Abstract

Bifurcation, which means the emergence of multiple solutions for the same value of parameter f, is induced by the criticality of the Jacobian matrix of the system, as demonstrated using examples in the previous chapter (cf., §1.2.2). The “bifurcation equation” is a standard means to describe bifurcation behavior. In a neighborhood of a simple critical point, for example, a set of equilibrium equations is reduced to a single bifurcation equation, by condensing the influence of a number of independent variables into a single scalar variable by the implicit function theorem.