Normal vectors on manifolds of critical points for parametric robustness of equilibrium solutions of ODE systems.
type: Item
instance of: scholarly article
title: Normal vectors on manifolds of critical points for parametric robustness of equilibrium solutions of ODE systems.
zbMATH Open document ID: 1043.34039
published in: Journal of Nonlinear Science
publication date: 2002-01-01T00:00:00Z
review text: Considering the parameter space of a physical system described by a nonlinear autonomous ODE, this paper introduces the notion of parametric distance to ``critical'' manifolds. Such manifolds include both the bifurcation sets and the point sets at which the state (or ``phase'' in the dynamics terminology) variables constraints, or output constraints, are violated. The interest of this original approach is to define a physical system by choosing a point of the parameter space insuring the maintenance of a given qualitative behavior with low sensitivity to unavoidable parameter variations due to environmental modifications and the system aging. The point so chosen must be sufficiently far from the ``critical'' manifold, leading to what is called ``robustness'' in engineering, i.e., a kind of structural stability. For this purpose, the authors use normal vectors on manifolds of critical points to measure the distance between these manifolds and a given steady state of the corresponding dynamical system. So, they follow a suggestion of I. Dobson who has given systems of equations to calculate normal vectors for codimension-1 bifurcations. The author's contribution consists more in an extension to ``critical'' manifolds (taking into account physical constraints), than in a generalization to bifurcations of arbitrary codimension. Compared to a previous work, this approach also has the advantage to reduce the number of equations for normal vectors on the Poincaré-Andronov bifurcation manifolds (here, wrongly called ``Hopf'' bifurcation manifold), and to simplify the proof for the regularity of the normal vector system. Illustrative examples in engineering applications are given, formulating an optimization problem, leading to system synthesis.
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34D99
Mathematics Subject Classification ID: 37G10
Mathematics Subject Classification ID: 34D30
zbMATH DE Number: 1931215
zbMATH Keywords: nonlinear ordinary differential equations
zbMATH Keywords: nonlinear dynamics
zbMATH Keywords: bifurcations
zbMATH Keywords: robustness
zbMATH Keywords: optimization
author: Martin Mönnigmann
author: Wolfgang Marquardt
reviewed by: Christian Mira
describes a project that uses: ADIC
describes a project that uses: ADIFOR
MaRDI profile type: Publication
full work available at URL: s00332-001-0400-1
OpenAlex ID: W1987811288
Wikidata QID: Q59029587
DOI: 10.1007/S00332-001-0400-1
DBLP publication ID: journals/jns/MonnigmannM02
Recommended article: Robust optimisation with normal vectors on critical manifolds of disturbance-induced stability loss
Recommended article: Normal Vectors on Critical Manifolds for Robust Design of Transient Processes in the Presence of Fast Disturbances
Recommended article: On the stability of integral manifolds of a system of ordinary differential equations in the critical case
Recommended article: Robust normal forms for saddles of analytic vector fields
Recommended article: Q3400454
Recommended article: Robust Stability for Parametric Linear ODEs
Recommended article: Normal forms near critical points for differential equations and maps
Recommended article: PARAMETER DEPENDENCE OF STABLE MANIFOLDS FOR IMPULSIVE EQUATIONS
Recommended article: A note on the structural stability of the equilibrium manifold
Recommended article: Stable Lyapunov manifolds for autonomous systems of nonlinear ordinary differential equations
description: scientific article; zbMATH DE number 1931215