Path-Connectedness In Global Bifurcation Theory

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Abstract

A celebrated result in bifurcation theory is that, when the operators involved are compact, global connected sets of non-trivial solutions bifurcate from trivial solutions at non-zero eigenvalues of odd algebraic multiplicity of the linearized problem. This paper presents a simple example in which the hypotheses of the global bifurcation theorem are satisfied, yet all the path-connected components of the connected sets that bifurcate are singletons. Another example shows that even when the operators are everywhere infinitely differentiable and classical bifurcation occurs locally at a simple eigenvalue, the global continua may not be path-connected away from the bifurcation point. A third example shows that the non-trivial solutions which bifurcate at non-zero eigenvalues, irrespective of multiplicity when the problem has gradient structure, may not be connected and may not contain any paths except singletons.

Original languageEnglish
Pages (from-to)4199-4213
Number of pages15
JournalElectronic Research Archive
Volume29
Issue number6
Early online date31 Oct 2021
DOIs
Publication statusPublished - 31 Dec 2021

Keywords

  • Global bifurcation
  • hereditarily indecomposable continua
  • path-connectedness
  • point-set topology
  • topological degree
  • variational methods

ASJC Scopus subject areas

  • Mathematics(all)

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