## 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 language | English |
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Pages (from-to) | 4199-4213 |

Number of pages | 15 |

Journal | Electronic Research Archive |

Volume | 29 |

Issue number | 6 |

Early online date | 31 Oct 2021 |

DOIs | |

Publication status | Published - 31 Dec 2021 |

## Keywords

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

## ASJC Scopus subject areas

- Mathematics(all)