Gradient Flow Line Near Birth-Death Critical Points

2017/06/23 by Antony, Charel
#34C23 (Secondary) #37D15 (Primary) #37G10 #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.1706.07746

Abstract

Near a birth-death critical point in a one-parameter family of gradient flows, there are precisely two Morse critical points of index difference one on the birth side. This paper gives a self-contained proof of the folklore theorem that these two critical points are joined by a unique gradient trajectory up to time-shift. The proof is based on the Whitney normal form, a Conley index construction, and an adiabatic limit analysis for an associated fast-slow differential equation.

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