Computing a closest bifurcation instability in multidimensional parameter space
type: Item
instance of: scholarly article
title: Computing a closest bifurcation instability in multidimensional parameter space
zbMATH Open document ID: 0789.58055
DOI: 10.1007/BF02429868
published in: Journal of Nonlinear Science
publication date: 1994-06-13T00:00:00Z
review text: The author's summary: ``Engineering and physical systems are often modeled as nonlinear differential equations with a vector \(\lambda\) of parameters and operated at a stable equilibrium. However, as the parameters \(\lambda\) vary from some nominal value \(\lambda_ 0\), the stability of the equilibrium can be lost in a saddle-node or Hopf bifurcation. The spatial relation in parameter space of \(\lambda_ 0\) to the critical set of parameters at which the stable equilibrium bifurcates determines the robustness of the system stability to parameter variations and is important in applications. We propose computing a parameter vector \(\lambda_ *\) at which the stable equilibrium bifurcates which is locally closest in parameter space to the nominal parameters \(\lambda_ 0\). Iterative and direct methods for computing these locally closest bifurcations are described. The methods are extensions of standard, one- parameter methods of computing bifurcations and are based on formulas for the normal vector to hypersurfaces of the bifurcation set. Conditions on the hypersurface curvature are given to ensure the local convergence of the iterative method and the regularity of solutions of the direct method. Formulas are derived for the curvature of the saddle node bifurcation set. The methods are extended to transcritical and pitchfork bifurcations and parametrized maps, and the sensitivity to \(\lambda_ 0\) of the distance to a closest bifurcation is derived. The application of the methods is illustrated by computing the proximity to the closest voltage collapse instability of a simple electric power system''.
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 37N99
Mathematics Subject Classification ID: 65L99
zbMATH DE Number: 490591
zbMATH Keywords: bifurcation
zbMATH Keywords: saddle node
zbMATH Keywords: Hopf
zbMATH Keywords: stability
zbMATH Keywords: robustness
zbMATH Keywords: optimization
zbMATH Keywords: numerical methods
zbMATH Keywords: transcritical
zbMATH Keywords: pitchfork
zbMATH Keywords: extended systems
author: Ian Dobson
reviewed by: C. Chicone
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02429868
OpenAlex ID: W2020554592
description: scientific article; zbMATH DE number 490591