Classification of parametrically constrained bifurcations

If the constraint boundary relates to a bifurcation parameter, a bifurcation is said to be parametrically constrained. Relying upon some substitution, a parametrically constrained bifurcation is transformed to an unconstrained bifurcation about new variables. A general form of transition sets of the parametrically constrained bifurcation is derived. The result indicates that only the constrained bifurcation set is influenced by parametric constraints, while other transition sets are the same as those of the corresponding nonparametrically constrained bifurcation. Taking parametrically constrained pitchfork bifurcation problems as examples, effects of parametric constraints on bifurcation classification are discussed.