The Numerical Solution of Nonlinear Equations Having Several Parameters I: Scalar Equations
Allan D. Jepson, Alastair Spenc · SIAM Journal on Numerical Analysis · 1985
We consider nonlinear problems of the form $f(x,\lambda ,{\bf \alpha }) = 0$, where $x \in \mathbb{R}$ is a state variable, $\lambda \in \mathbb{R}$ is a bifurcation parameter, $\alpha \in \mathbb{R}^p $ is a vector of auxiliary parameters, and f is a sufficiently smooth scalar function. A numerical approach is developed to calculate the regions in auxiliary parameter space for which the problem has qualitatively similar bifurcation diagrams. The approach is based on recent work in singularity theory, which is used to construct equations and inequalities characterizing various types of singular points for the above equation. These defining equations have several desirable properties which are fully utilized in the numerical calculation of paths of singular points using standard continuation techniques. Numerical results, given for a model of a stirred tank chemical reactor, illustrate the power of the approach.