Constructing global bifurcation diagrams by the parametric representation method (Q1807801)

The problem of determination the number of solutions for the equation \(f_{0}(x)+u_{1}f_{1}(x)+u_{2}f_{2}(x)=0, x\in \mathbb{R},\) as a function of the parameter values \((u_{1},u_{2})\), i.e. construction of the bifurcational curve, is considered. The parametric representation method [\textit{P. Gray} and \textit{S. K. Scott}, Chemical oscillations and instabilities: Nonlinear chemical kinetics, Clarendon Press, Oxford (1994)] is used. This method has two advantages: 1) the singularity set can be easily constructed as a curve parametrized by \(x\), called \(D\)-curve, 2) the solutions belonging to a given parameter pair can be determined by a simple geometric algorithm based on the so-called tangential property.