Autori
Rocca, MatteoMiglierina, EnricoMolho, ElenaTitolo
A Morse-type index for critical points of vector functionsPeriodico
Università degli Studi dell'Insubria. Dipartimento di Economia. Quaderni di ricercaAnno:
2007 - Fascicolo:
2 - Pagina iniziale:
1 - Pagina finale:
23In this work we study the critical points of vector functions form Rn to Rm with n >_ m,
following the definition introduced by S. Smale in the context of vector optimization. The
local monotonicity properties of a vector function around a critical point which are invariant
with respect to local coordinate changes are considered. We propose a classification
of critical points through the introduction of an index for a critical point consisting of a
triple of nonnegative integers. The proposed index is based on the ”sign” of an appropriate
vector-valued second-order differential, that is proved to be invariant with respect to local
coordinate changes. In order to avoid anomalous behaviours of the Jacobian matrix, the
analysis is partially restricted to the proper critical points, a subset of critical points which
enjoy stability properties with respect to perturbations of the order structure. Under nondegeneracy
conditions, the index is proved to be locally constant. Moreover, the stability
properties of the index with respect to perturbations both of the ordering cone and of the
function are considered. Finally, the consistency of the proposed classification with the one
given by Whitney for stable maps from the plane into the plane is proved.
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