Defining equations for singular solutions and numerical applications. (English) Zbl 0556.65044

Numerical methods for bifurcation problems, Proc. Conf., Dortmund/Ger. 1983, ISNM 70, 42-56 (1984).
[For the entire collection see Zbl 0535.00021.]
The present paper establishes a relation between the concept of a universal unfolding in singularity theory and direct numerical methods for singular points in bifurcation diagrams. A solution \(z_ 0\in {\mathbb{R}}^ M\) of a nonlinear system \[ T(z)=0,\quad T: {\mathbb{R}}^ M\to {\mathbb{R}}^ N\quad (M\geq N) \] is called singular if \(rank(T'(z_ 0))<N\). In order to compute such a singular solution in a stable way one needs additional control variables c, i.e. a system of the type \(T(z,c)=0,\) \(c\in {\mathbb{R}}^ p\) where \(T(z,c_ 0)=T(z)\) for some \(c_ 0\). The singular point \((z_ 0,c_ 0)\) can then be calculated as a regular solution of a so called defining equation \(D_ T(z,c)=0,\) \(D_ T: {\mathbb{R}}^{M+p}\to {\mathbb{R}}^{M+p}.\)
The author presents a list of defining equations for some of the most common singularities - such as folds, cusps, swallow-tails, simple bifurcation points, isola formation points and multiple bifurcation points. It is shown that in these cases the number p of stabilizing parameters equals the so called codimension of the singularity.
Singular solutions are obtained numerically by a combination of Newton’s method for a low dimensional defining equation together with a Lyapunov- Schmidt like reduction process. The basis for this method is a key result stating that the choice of projectors in the Lyapunov-Schmidt reduction does not affect the type of the singularity and hence the defining equation. The method is applied to a discrete cell model with diffusion and an inhibited Michaelis-Menton reaction. Here a multiple bifurcation point is computed at which four steady state branches intersect.


65H10 Numerical computation of solutions to systems of equations
58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
80A32 Chemically reacting flows


Zbl 0535.00021