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 | References | Similar Articles | Additional Information

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