Normal vectors on manifolds of critical points for parametric robustness of equilibrium solutions of ODE systems.
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.
nonlinear ordinary differential equations
scientific article; zbMATH DE number 1931215
scientific article