Constructing global bifurcation diagrams by the parametric representation method

摘要

The parameter dependence of the solution x of equation f0(x)+u1f1(x)+u2f2(x)=0 is considered. Our aim is to divide the parameter plane (u1,u2) according to the number of the solutions, that is to construct a bifurcation curve. This curve is given by the singularity set, but in practice it is difficult to depict it, because it is often derived in implicit form. Here we apply the parametric representation method which has the following 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 tangential property; (3) the global bifurcation diagram, that divides the parameter plane according to the number of solutions can be geometrically constructed with the aid of the D-curve.