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Instability analysis of saddle points by a local minimax method


Author: Jianxin Zhou
Journal: Math. Comp. 74 (2005), 1391-1411
MSC (2000): Primary 58E05, 58E30; Secondary 35A40, 35A15
DOI: https://doi.org/10.1090/S0025-5718-04-01694-1
Published electronically: July 20, 2004
MathSciNet review: 2137008
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Abstract: The objective of this work is to develop some tools for local instability analysis of multiple critical points, which can be computationally carried out. The Morse index can be used to measure local instability of a nondegenerate saddle point. However, it is very expensive to compute numerically and is ineffective for degenerate critical points. A local (weak) linking index can also be defined to measure local instability of a (degenerate) saddle point. But it is still too difficult to compute. In this paper, a local instability index, called a local minimax index, is defined by using a local minimax method. This new instability index is known beforehand and can help in finding a saddle point numerically. Relations between the local minimax index and other local instability indices are established. Those relations also provide ways to numerically compute the Morse, local linking indices. In particular, the local minimax index can be used to define a local instability index of a saddle point relative to a reference (trivial) critical point even in a Banach space while others failed to do so.


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Additional Information

Jianxin Zhou
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: jzhou@math.tamu.edu

DOI: https://doi.org/10.1090/S0025-5718-04-01694-1
Keywords: Saddle point, instability index, Morse index, (weak) local linking, local minimax method
Received by editor(s): May 4, 2003
Received by editor(s) in revised form: December 11, 2003
Published electronically: July 20, 2004
Article copyright: © Copyright 2004 American Mathematical Society

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