Journal of Nonlinear Science

, Volume 12, Issue 2, pp 85–112

# Normal Vectors on Manifolds of Critical Points for Parametric Robustness of Equilibrium Solutions of ODE Systems

• M. Mönnigmann
• W. Marquardt
Article

## Summary.

{Equilibrium solutions of systems of parameterized ordinary differential equations \dot x = f(x, α) , x ∈ R n , α∈ R m can be characterized by their parametric distance to manifolds of critical solutions at which the behavior of the system changes qualitatively. Critical points of interest are bifurcation points and points at which state variable constraints or output constraints are violated. We use normal vectors on manifolds of critical points to measure the distance between these manifolds and equilibrium solutions as suggested in I. Dobson [J. Nonlinear Sci., 3:307-327, 1993], where systems of equations to calculate normal vectors on codimension-1 bifurcations were presented. We present a scheme to derive systems of equations to calculate normal vectors on manifolds of critical points which (i) generalizes to bifurcations of arbitrary codimension, (ii) can be applied to state variable constraints and output constraints, (iii) implies that the normal vector defining system of equations is of size c 1 n+ c 2 m+ c 3 , c i ∈ R , i.e., no bilinear terms nm or higher-order terms occur, (iv) reduces the number of equations for normal vectors on Hopf bifurcation manifolds compared to previous work, and (v) simplifies the proof of regularity of the normal vector system. As an application of this scheme, we present systems of equations for normal vectors to manifolds of output/state variable constraints, to manifolds of saddle-node, Hopf, cusp, and isola bifurcations, and we give illustrative examples of their use in engineering applications.}

Key words. bifurcation, Hopf, saddle-node, cusp, isola, robustness, optimization, design

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© Springer-Verlag New York Inc. 2002

## Authors and Affiliations

• M. Mönnigmann
• 1
• W. Marquardt
• 1
1. 1.Lehrstuhl für Prozesstechnik, RWTH Aachen, Aac hen, GermanyDE