Conley-Zehnder index and bifurcation of fixed points of Hamiltonian maps

Yanxia Deng, Zhihong Xia

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

We study the bifurcations of fixed points of Hamiltonian maps and symplectic diffeomorphisms. We are particularly interested in the bifurcations where the Conley-Zehnder index of a fixed point changes. The main result is that when the Conley-Zehnder index of a fixed point increases (or decreases) by one or two, we observe that there are several bifurcation scenarios. Under some non-degeneracy conditions on the one-parameter family of maps, two, four or eight fixed points bifurcate from the original one. We give a relatively detailed analysis of the bifurcation in the two-dimensional case. We also show that higher-dimensional cases can be reduced to the two-dimensional case.

Original languageEnglish (US)
Pages (from-to)2086-2107
Number of pages22
JournalErgodic Theory and Dynamical Systems
Volume38
Issue number6
DOIs
StatePublished - Sep 1 2018

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Hamiltonians
Bifurcation (mathematics)
Bifurcation
Fixed point
Nondegeneracy
Diffeomorphisms
High-dimensional
Decrease
Scenarios

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

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Conley-Zehnder index and bifurcation of fixed points of Hamiltonian maps. / Deng, Yanxia; Xia, Zhihong.

In: Ergodic Theory and Dynamical Systems, Vol. 38, No. 6, 01.09.2018, p. 2086-2107.

Research output: Contribution to journalArticle

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AB - We study the bifurcations of fixed points of Hamiltonian maps and symplectic diffeomorphisms. We are particularly interested in the bifurcations where the Conley-Zehnder index of a fixed point changes. The main result is that when the Conley-Zehnder index of a fixed point increases (or decreases) by one or two, we observe that there are several bifurcation scenarios. Under some non-degeneracy conditions on the one-parameter family of maps, two, four or eight fixed points bifurcate from the original one. We give a relatively detailed analysis of the bifurcation in the two-dimensional case. We also show that higher-dimensional cases can be reduced to the two-dimensional case.

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