Folds in solutions of two parameter systems and their calculation. I. (English) Zbl 0576.65052

Jepson, A.; Spence, A.

SIAM J. Numer. Anal. 22, 347-368 (1985).

This paper studies paths of turning points in solutions of nonlinear systems of equations (*) \(f(x,\lambda,\mu)=0\) depending on two free parameters, where f is a \(C^ 3\) mapping from \(X_ 1\times R^ 2\) into \(X_ 2\), for some Banach spaces \(X_ 1\) and \(X_ 2\). First a detailed analysis of the theory of fold curves is given. This theory is based on the development of algorithms for computing the fold curves and for determining the behaviour of the solution surface \(S=f^{-1}(0)\) near a fold curve. It is shown that certain singular points of (*) such as cusp points, bifurcation points, and points of isola formation, are all simple turning points in the fold curve. Fold curves are defined to be solutions of a certain ”extended” system of equations which provides existence results and information about the geometric behaviour of S near a fold curve. The authors verify that simple quadratic turning points of (*) correspond to regular points of the ”extended” systems of equations and demonstrate how to compute a fold curve consisting of simple quadratic turning points of (*) using standard continuation techniques (specially, Euler-Newton continuation). The theory of two parameter problems is applied both to perturbed bifurcation problems and to the formation of isolas. Finally, to support the theoretical results the simple aircraft equilibrium equations are numerically examined.

Reviewer: O.Vaarmann

MSC:

65J15	Numerical solutions to equations with nonlinear operators (do not use 65Hxx)
58E07	Variational problems in abstract bifurcation theory in infinite-dimensional spaces
47J25	Iterative procedures involving nonlinear operators

Keywords:

paths of turning points; Banach spaces; fold curves; singular points; cusp points; bifurcation points; points of isola formation; Euler-Newton continuation; perturbed bifurcation problems; aircraft equilibrium equations