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Singular points and their computation. (English) Zbl 0579.65048

Numerical methods for bifurcation problems, Proc. Conf., Dortmund/Ger. 1983, ISNM 70, 195-209 (1984).
[For the entire collection see Zbl 0535.00021.]
Let \(f: {\mathbb{R}}^ n\times {\mathbb{R}}\times {\mathbb{R}}^ p\to {\mathbb{R}}^ n\) be a smooth function and consider the nonlinear multi-parameter equation (*) \(f(x,\lambda,\alpha)=0\) with the bifurcation parameter \(\lambda\in {\mathbb{R}}\) and the control parameters \(\alpha \in {\mathbb{R}}^ p\). The aim is to obtain a complete decomposition of the control parameter space into regions for which the bifurcation diagrams of (*) are qualitatively similar, and to provide initial points for each component of the bifurcation diagram at any given \(\alpha\). For the scalar case \(n=1\) the respective results of the authors [SIAM J. Numer. Anal., submitted] are summarized in this paper. The discussion of their generalization to the vectorial case includes a numerically convenient form of the Lyapunov- Schmidt decomposition.
Reviewer: H.Jeggle

MSC:

65H10 Numerical computation of solutions to systems of equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces

Citations:

Zbl 0535.00021