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**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.

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.

### MSC:

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 |